Accurate and Efficient Evaluation of the Second … of Tables 4.1 Tabulated data for the classical...

Accurate and Efficient Evaluation of the SecondVirial Coefficient Using Practical

Intermolecular Potentials for Gases

by

Maciej K. Hryniewicki

A thesis submitted in conformity with the requirements

for the degree of Masters of Applied Science

Graduate Department of Aerospace Engineering

University of Toronto

Copyright c© 2011 by Maciej K. Hryniewicki

Abstract

Accurate and Efficient Evaluation of the Second Virial

Coefficient Using Practical Intermolecular Potentials for Gases


Masters of Applied Science

Graduate Department of Aerospace Engineering

University of Toronto

2011

The virial equation of state p = ρRT[

1 + B(T ) ρ + C(T ) ρ2 + · · ·]

for high pressure and den-

sity gases is used for computing chemical equilibrium properties and mixture compositions of

strong shock and detonation waves. The second and third temperature-dependent virial coeffi-

cients B(T ) and C(T ) are included in tabular form in computer codes, and they are evaluated

by polynomial interpolation. A very accurate numerical integration method is presented for

computing B(T ) and its derivatives for tables, and a sophisticated method is introduced for

interpolating B(T ) more accurately and efficiently than previously possible. Tabulated B(T )

values are non-uniformly distributed using an adaptive grid, to minimize the size and storage of

the tables and to control the maximum relative error of interpolated values. The methods intro-

duced for evaluating B(T ) apply equally well to the intermolecular potentials of Lennard-Jones

in 1924, Buckingham and Corner in 1947, and Rice and Hirschfelder in 1954.

ii

Acknowledgements

I would like to thank my supervisor, Professor James J. Gottlieb, for all of his guidance and

support throughout the duration of this thesis. His enthusiasm, care and insight are much

appreciated.

I would like to acknowledge the research stipend provided by the University of Toronto

Institute for Aerospace Studies during my studies.

Finally, I would like to thank my parents, Waldemar and Izabela, and my sister, Magdalena,

for their ongoing love and encouragement. They have instilled in me the importance of educa-

tion, and I am forever grateful to them for supporting my desire to further my education in the

field of aerospace science and engineering.


University of Toronto Institute for Aerospace Studies

February 2011

iii

Contents

Abstract ii

Acknowledgements iii

Contents v

List of Tables vi

List of Figures viii

1 Introduction 1

2 The Virial Equation of State 4

2.1 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Second Virial Coefficient B(T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Intermolecular Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Solution Methodology for B(T) and its Derivatives 15

3.1 Non-Dimensionalization of B(T) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Derivatives of B∗(T∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Derivatives of the Classical Second Virial Coefficient Bc(T∗) . . . . . . . . 16

iv

3.2.2 Derivatives of the Quantum Correction Bq(T∗) . . . . . . . . . . . . . . . 17

3.3 Transformation for Improper to Proper Integrals . . . . . . . . . . . . . . . . . . 18

3.4 Method of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Method of Interpolation for Tabulated Virial Coefficients . . . . . . . . . . . . . 21

3.6 Plots of the Integrands of B∗(T∗) and its Derivatives . . . . . . . . . . . . . . . . 24

4 Results for B∗(T∗) and its Derivatives 33

4.1 Numerical Solutions for B∗(T∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Classical Second Virial Coefficient Bc(T∗) . . . . . . . . . . . . . . . . . . 34

4.1.2 Quantum Correction Bq(T∗) . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.3 Selected Tabulated Solutions for B∗(T∗) . . . . . . . . . . . . . . . . . . . 42

4.2 Quantum Correction to B∗(T∗) for H2 and N2 . . . . . . . . . . . . . . . . . . . . 49

5 Concluding Remarks 51

References 56

A Assessment of New Interpolation Method 57

v

List of Tables

4.1 Tabulated data for the classical second virial coefficient and its first four deriva-

tives, using the Lennard-Jones 6-12 model, interpolated using a nonic interpolant

(m = 9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


tives, using the Buckingham and Corner modified Buckingham model (α = 13.5

and β = 0.0), interpolated using a nonic interpolant (m = 9). . . . . . . . . . . . 44


tives, using the Rice and Hirschfelder modified Buckingham (exponential-six)

model (α = 13.5 and β = 0.0), interpolated using a nonic interpolant (m = 9). . . 45

4.4 Tabulated data for the quantum correction to the second virial coefficient and

its first four derivatives, using the Lennard-Jones 6-12 model, interpolated using

a nonic interpolant (m = 9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Tabulated data for the quantum correction to the second virial coefficient and its

first four derivatives, using the Buckingham and Corner modified Buckingham

model (α = 13.5 and β = 0.0), interpolated using a nonic interpolant (m = 9). . . 47

4.6 Tabulated data for the quantum correction to the second virial coefficient and

its first four derivatives, using the Rice and Hirschfelder modified Buckingham

(exponential-six) model (α = 13.5 and β = 0.0), interpolated using a nonic

interpolant (m = 9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A.1 Numbers of nodes and tabulated data required for interpolation of tabulated

data by various polynomial interpolation methods of different degrees (m). . . . 59

vi

List of Figures

2.1 Various intermolecular potential energy models. . . . . . . . . . . . . . . . . . . . 8

3.1 Pascal’s triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Integrands of Bc(T∗) and its first three derivatives with T∗ = 1.0. . . . . . . . . . 27

3.3 Integrands of Bc(T∗) and its first three derivatives with T∗ = 10.0. . . . . . . . . 28

3.4 Integrands of Bc(T∗) and its first three derivatives with T∗ = 100.0. . . . . . . . . 29

3.5 Integrands of Bq(T∗) and its first three derivatives with T∗ = 1.0. . . . . . . . . . 30

3.6 Integrands of Bq(T∗) and its first three derivatives with T∗ = 10.0. . . . . . . . . 31

3.7 Integrands of Bq(T∗) and its first three derivatives with T∗ = 100.0. . . . . . . . . 32

4.1 The classical second virial coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Adaptive grid variation for new interpolation method for m = 3 , 5 , . . . , 17, using

the Lennard-Jones 6-12 model for the classical second virial coefficient. . . . . . . 35


the Buckingham and Corner modified Buckingham model (α = 13.5 and β = 0.0)

for the classical second virial coefficient. . . . . . . . . . . . . . . . . . . . . . . . 36


the Rice and Hirschfelder modified Buckingham (exponential-six) model (α =

13.5 and β = 0.0) for the classical second virial coefficient. . . . . . . . . . . . . . 37

4.5 The quantum correction to the second virial coefficient. . . . . . . . . . . . . . . 39

vii


the Lennard-Jones 6-12 model for the quantum correction to the second virial

coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


the Buckingham and Corner modified Buckingham model (α = 13.5 and β = 0.0)

for the quantum correction to the second virial coefficient. . . . . . . . . . . . . . 40


the Rice and Hirschfelder modified Buckingham (exponential-six) model (α =

13.5 and β = 0.0) for the quantum correction to the second virial coefficient. . . . 41

4.9 The significance of the quantum correction to the second virial coefficient. . . . . 49

A.1 Plots of the test function and its first four derivatives. . . . . . . . . . . . . . . . 58

A.2 Test function and illustration of adaptive grid variation for new interpolation

method for m = 3 , 5 , . . . , 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

viii

Chapter 1

Introduction

Major computational codes have been developed worldwide over the past century to predict

chemical equilibrium mixture compositions and their thermodynamic properties of the com-

bustion products of energetic materials for civilian and military applications. These computa-

tional codes are used in the design of combustion engines, heat exchangers, projectile launchers

and shock tubes, in the classification of energetic materials such as explosives, propellants,

fuses, primers and igniters, as well as in the equilibrium computations for rocket performance,

and shock and detonation wave calculations. The most notable computational codes include

the CEA (Chemical Equilibrium with Applications) code developed at the NASA Lewis (now

Glenn) Research Center in the United States [10], the Blake code developed at the United States

Army Ballistic Research Laboratory [9], and the Bagheera code developed at the Bouchet Re-

search center in France. At the University of Toronto Institute for Aerospace Studies (UTIAS),

the CERV (Chemical Equilibrium using Reaction Variables) code was developed in the 1990s

for fairly general chemical equilibrium applications [38].

Most codes can predict solutions for problems using a specified temperature and pressure

(T -P problem) or using a specified energy and volume (U -V problem), and some codes can also

predict solutions for problems using a specified enthalpy and pressure (H-P problem) or using a

specified entropy and volume (S-V problem). The solution methods used by these codes involve

namely the minimization of Gibbs free energy and use either a compositional formulation based

on mole numbers or a stoichiometric formulation based on reaction variables. The formulation

based on reaction variables has proven to be the more computationally efficient approach be-

cause it does not suffer from convergence failures commonly encountered by codes using the

formulation based on mole numbers. Comprehensive summaries of these major computational

codes and their advantages and shortcomings, as well as detailed descriptions regarding their

1

Chapter 1. Introduction 2

equilibrium solution algorithms, are available in the book by Smith and Missen [29] as well as

in the report by Wong, Gottlieb and Lussier [38].

Despite having well-developed solution methods for solving chemical equilibrium prob-

lems, better physical models for these computational codes are required. For example, an

imperfect equation of state that can more accurately describe the thermodynamic relation-

ships between pressure, volume and temperature (p-ν-T relationships) for gaseous species at

high densities and pressures is desirable. The three-term truncated virial equation of state

p = ρRT[

1 + B(T ) ρ + C(T ) ρ2]

accounts for the deviations from the equation of state for an

ideal gas through the second and third temperature-dependent virial coefficients, B(T ) and

C(T ) respectively. Smith and Missen’s stoichiometric algorithm [29] is formulated using the

equation of state for a thermally perfect gas, for which B(T ) and C(T ) are zero. In Wong, Got-

tlieb and Lussier’s work [38], the thermally imperfect equation of state is a modified form of the

virial equation of state in which both B(T ) and C(T ) are based on the conventional Lennard-

Jones 6-12 potential. However, more appropriate intermolecular potentials are available today

to give improvements to B(T ) and C(T ).

The virial coefficients B(T ) and C(T ) are typically stored sparsely in tabular form in com-

putational codes and evaluated using linear, quadratic or cubic polynomial interpolants. When

this methodology is used for the second virial coefficient, sparsely stored in tabular form, inter-

polations are accurate to as few as only three significant digits. Hence, it is desirable to improve

the accuracy of the numerical integration of the second virial coefficient at various tempera-

tures for the assembly of the tables, as well as the accuracy and efficiency of the evaluation

process of the second virial coefficient by using higher-degree polynomial interpolants once the

sparse tables are constructed. These improvements must not only be accurate and precise in

a physical sense, but they must also be capable of being easily incorporated into previous and

new computational codes in an efficient manner. The development of more computationally-

efficient methods for calculating the thermodynamic properties of an imperfect gas not only

ensures that better physical models are used, but it also offers the promise of faster and more

robust computations.

An accurate and efficient method is developed in this thesis for the evaluation of the second

temperature-dependent virial coefficient B(T ) using more practical intermolecular potential

energy models for gases. This study begins in chapter 2 with a relevant review of imperfect

equations of state, including the virial equation of state and its virial coefficients. Simple and

practical intermolecular potential energy models are also reviewed and the quantum correction

to B(T ) is included. The study continues with the methodology of numerically evaluating

Chapter 1. Introduction 3

B(T ) and its derivatives, both accurately and efficiently, in chapter 3. This begins with the

method of integration that is used to accurately obtain the solutions to B(T ), and ends with

the development of a special method of accurately interpolating sparse tabulated data of B(T )

versus temperature. The solutions to B(T ) are stored non-uniformly in tables using adaptive

grids for use in computational codes such as the UTIAS CERV code, and this sophisticated

interpolation method for a function and its lower derivatives is used to control the accuracy

of the interpolated values. Numerical results for B(T ) and its derivatives, calculated using

practical intermolecular potential energy models, are reported in chapter 4, as is the significance

of the quantum correction to B(T ). In closing, several concluding remarks regarding this

research are included in chapter 5.

Chapter 2

The Virial Equation of State

The virial equation of state and its virial coefficients are introduced at the beginning of this

chapter. Of particular interest in this thesis is the second virial coefficient B(T ), as mentioned

in chapter 1. Hence, B(T ) is defined fully as the sum of two terms: the first is the classical

term and the second is the correction for quantum effects. Both terms are integrals and the

integrands contain the intermolecular potential or the force between two molecules undergoing

a collision. This definition of B(T ) leads further to a review of various models of intermolecular

potentials, ranging from the simplest to the most complex models known today. The compli-

cated intermolecular potentials are needed and used in subsequent work to determine physically

realistic solutions of B(T ).

2.1 Equations of State

The equation of state for a thermally perfect gas can be expressed by

pν = RT or p = ρRT , (2.1)

which can be derived directly from the kinetic theory of gases [36] for the case when atoms and

molecules are assumed to undergo simple structureless molecular collisions. The symbols p, ν,

T and ρ = 1/ν denote the pressure, molar volume, temperature and molar density, respectively,

and the symbol R denotes the universal gas constant. This equation of state for an ideal gas is

useful for describing the p-ν-T properties of gases at low densities and moderate temperatures.

At very high pressures and temperatures (such as in the case of detonations and explosions)

the compressibility factor Z = pν/RT can be significantly larger than unity, and at very low

4

Chapter 2. The Virial Equation of State 5

pressures and temperatures the compressibility factor Z can be much smaller than unity [6].

To account for deviations from the ideal gas model given by equation 2.1, imperfect gas models

must be used to accurately represent the p-ν-T behaviour for the gas phase. For high density and

pressure gases that typically occur in strong shock waves and detonations related to explosions,

empirical equations of state are often used.

One example for an empirical equation of state is the equation of state given by Redlich

and Kwong [23] with modifications by Soave [30], which is given by

p =RT

v − b− a(T )

v(v − b), (2.2)

in which the symbol a(T ) is a temperature-dependent parameter for intermolecular attraction,

and the symbol b is a constant that accounts for intermolecular repulsion. These parameters

are determined normally on the basis of a best fit of the equation of state to experimental and

theoretical data. The previous equation of state is an improved version of the original equation

p =RT

v − b− a

v2(2.3)

of van der Waals [35], in which the symbols a and b were both constant. Both equations 2.2 and

2.3 reduce to the equation of state for an ideal gas given by equation 2.1 when the parameters

a → 0 and b → 0.

Another example of an empirical equation of state is

p = ρRT + (B0RT − A0 −C0

T 2+

D0

T 3− E0

T 4)ρ2 + (bRT − a − d

T)ρ3

+ α(a +d

T)ρ6 +

c

T 2(1 + γρ2)ρ3 exp(−γρ2)

(2.4)

from the work of Benedict, Webb and Rubin [2] and Starling [31]. The symbols A0, B0, C0,

D0, E0, a, b, c, d, α and γ are empirical constants determined by means of a best fit of the

equation of state to experimental data. Although these and other empirically based imperfect

gas models exist, they are typically only accurate for a particular application over a small to

large range of temperatures and pressures for which there is experimental and theoretical data.

Hence, it is still preferable to use an equation of state that has a rigorous theoretical basis and

often a correspondingly extended or extrapolated range of applicability.

The virial equation of state of interest in this thesis can be expressed as

p = ρRT[

1 + B(T ) ρ + C(T ) ρ2 + D(T ) ρ3 + · · ·]

. (2.5)

The virial equation of state can accurately model imperfect gas behaviour in gaseous species over

a wide range of temperatures and pressures, when compared to other equations of state. It is


most often used for high pressure and temperature gases typical of strong shock and detonation

waves. The previous equation of state is derived on the basis of kinetic theory and statistical

mechanics [20, 13, 26]. Unlike the equation of state for an ideal gas, the virial equation of state

includes the effects of molecular volume and intermolecular collisions between two molecules,

three molecules, and so on and so forth.

The correction term in the virial equation of state, given by equation 2.5, is the polynomial

equation included in the square brackets. The first virial coefficient in equation 2.5 is constant

and equal to unity. The second temperature-dependent virial coefficient is B(T ) and takes

into account binary molecular collisions in which the structural details of the two interacting

molecules are important. The third temperature-dependent virial coefficient is C(T ) and takes

into account tertiary molecular collisions in which the structural details of the three interacting

molecules are important, and so on and so forth. When the temperature-dependent virial

coefficients are equal to zero or asymptote to zero, the equation of state for an ideal gas is

obtained. The elegance of the virial equation of state is that it should become more accurate

by simply including terms in the power series expansion based on the molar density. However,

higher-order virial coefficients are very difficult to determine and are typically neglected. In

many cases only the second virial coefficient is incorporated, and in some cases the third virial

coefficient is also included.

2.2 Second Virial Coefficient B(T)

The second temperature-dependent virial coefficient in the virial equation of state given by

equation 2.5 takes into account binary molecular collisions. This virial coefficient is defined by

B(T ) = −2πn

∫

∞

0f(r, T ) r2 dr +

nh2

24πm(kT )3

∫

∞

0F 2(r)

[

f(r, T ) + 1]

r2 dr . (2.6)

The first term is the classical second virial coefficient and the second term is the correction for

quantum or relativistic effects. The symbols n, r, h, m and k denote the number of molecules

(typically taken as either Avogadro’s number or Loschmidt’s number), the intermolecular sep-

aration, Planck’s constant, the molecular mass and the Boltzmann constant, respectively.

The function f(r, T ) in equation 2.6 is a relationship for the forces between molecules. It is

called the Mayer function and given by

f(r, T ) = exp

(−φ(r)

kT

)

− 1 , (2.7)

which is simply the Boltzmann factor minus one. The function φ(r) in equation 2.7 is the

potential energy of the interactions between two molecules, and it is discussed further in section


2.3. The intermolecular force F (r) is related to the intermolecular potential energy φ(r) in

equation 2.7 through

F (r) = −dφ(r)

dror φ(r) =

∫

∞

rF (r) dr , (2.8)

which are equivalent expressions.

Extensive research has been done to include quantum corrections to the virial equation of

state in an effort to model the behaviour of gaseous species at low temperatures, mainly for

the second virial coefficient [33, 12, 4, 16, 17, 5]. Buckingham and Corner [4] have summarized

the work of their predecessors [28, 34, 37, 18, 3] on the quantum correction to the second virial

coefficient, showing that of the three major quantum corrections only the one which accounts

for the probability of intermolecular potential energy configurations not being proportional to

the exponential term in equation 2.7 needs to be taken into account. Kim and Henderson [17]

later provided a general expression for the quantum correction to the third and fourth virial

coefficients which has been derived from the quantum correction to the Helmholtz free energy,

that can also be used to derive the quantum correction to the second virial coefficient. The

quantum correction to the second virial coefficient used in this thesis has been adopted from

the work of Buckingham and Corner [4] and verified by using the work of Kim and Henderson

[17]. Note that the quantum correction to the second virial coefficient has never previously been

included in any major computational code for the calculation of the thermodynamic properties

of an imperfect gas; only the classical second virial coefficient given by the first term in equation

2.6 has been used previously.

2.3 Intermolecular Potentials

The second virial coefficient B(T ) depends on the intermolecular potential energy model φ(r),

as illustrated in equation 2.6. A theoretically exact potential energy is not currently known

or available. Extensive work has been done on developing realistic potential energy models

which accurately represent the repulsive forces between two molecules at small distances from

each other, whereas the attractive forces between two molecules at large separation distances

are much better known [14, 15, 27]. The most important simple to complicated intermolecular

potential energy models which have been important for modeling angle-independent spherically

symmetrical molecules are illustrated in figure 2.1, in which the non-dimensional intermolecular

potential φ/ǫ is plotted versus a non-dimensional separation distance, either r/σ or r/rmin. The

symbol rmin denotes the location where the intermolecular potential is a minimum and given

by ǫ. The symbol σ is a force constant called the collision diameter, and it denotes the location


0 1 2−1

0

1

2

4

φ(r)

r /σ

rigid impenetrablesphere potential

∞

0 1 2−1

0

1

2

4

φ(r)

r /σ

point centersof repulsion

potential

∞

10 2−1

0

1

2

4φ(r)——ε

r /σ

Sutherland’spotential

ε

∞

10 2−1

0

1

2

4φ(r)——ε

r /σ

square-wellpotential

ε

∞

10 2−1

0

1

2

4φ(r)——ε

r /σ

Lennard-Jones’m-n potential

(m = 6)

ε

∞

n =18

8

8

1812

0 1 2−1

0

1

2

4φ(r)——ε

r /rmin

Buckingham’spotential(β = 0)

ε

–∞

α =

7

8

8.6

13.5

0 1 2−1

0

1

2

4φ(r)——ε

r /rmin

Buckingham andCorner’s modified

Buckingham potential(β = 0)

ε

∞

α =

7

7

8

13.5

0 2−1

0

1

2

4φ(r)——ε

r /rmin

Rice and Hirsch-felder’s modified

Buckingham potential(exponential-six

with β = 0)ε

α =

7

7

8

8.6

8.6

13.5

13.5

Figure 2.1: Various intermolecular potential energy models.

of balanced repulsive and attractive interactions between molecules. The symbol ǫ is a force

constant called the characteristic energy, and it denotes the depth of the potential well when

the intermolecular potential is a minimum. These force constants are determined through the

manipulation of either experimental second virial coefficient data or experimental transport

property measurements (dynamic viscosity and thermal conductivity), over a range of different

temperatures, and they are unique for a given gaseous species of interest in the sense of a curve

fit.

The decision of choosing which intermolecular potential energy model to use in the calcu-


lations for the second virial coefficient rests on the degree of realism the particular model can

provide as well as the numerical difficulties associated with the use of an intermolecular potential

energy function φ(r). The top four intermolecular potential energy models illustrated in figure

2.1 are theoretically simplistic but are used to give a crude representation of the interactions

between molecules. The potentials representing rigid impenetrable spheres and point centers of

repulsion are historically important, but they are not practical to use. Sutherland’s potential

and the square-well potential are more practical and are sometimes useful from a theoretical

standpoint but they, too, cannot provide the desired level of realism for use in engineering and

science applications. The bottom four intermolecular potential energy models illustrated in

figure 2.1 are far more realistic models; they are much more difficult to integrate numerically

but give the best representations of the interactions between molecules.

The Lennard-Jones m-n potential with m = 6 and n = 12 has been used often and is

important because of its simplicity. Buckingham’s potential with 9 < α < 15 and 0 < β < 0.25

best describes the forces between molecules, but it exhibits aphysical behaviour as the attractive

forces between molecules become infinite at small intermolecular separations. Buckingham’s

potential has been modified firstly by Buckingham and Corner in 1947 [4] and later by Rice

and Hirschfelder in 1954 [27], in an effort to repair the original model’s aphysicalities. These

last two modified Buckingham intermolecular potentials are currently regarded today as the

best models for angle-independent spherically symmetrical molecules. Note that a good general

overview of the various intermolecular potential energy models presented herein is given in the

book by Hirschfelder, Curtiss and Bird [13], which contains substantially more information.

The intermolecular potential energy and the corresponding intermolecular force for the rigid

impenetrable spheres model are given by

φ(r) =

{

∞ if r < σ ,

0 if r > σ ,(2.9)

F (r) =

0 if r < σ ,

undefined if r = σ ,

0 if r > σ ,

(2.10)

respectively. This is a simple model that yields a crude representation of the strong repulsive

forces between molecules at small distances from each other. Owing to its simplicity this model

is typically used only for exploratory calculations because the results for B(T ) are analytical

solutions. There is no temperature dependence of the second virial coefficient when the rigid

impenetrable spheres model is used.

The intermolecular potential energy and the corresponding intermolecular force for the point


centers of repulsion model are given by

φ(r) =(σ

r

)δ, (2.11)

F (r) =δ

σ

(σ

r

)δ+1, (2.12)

respectively, in which the symbol δ denotes the index of repulsion. For most molecules the

index of repulsion is between 9 and 15, although a value of 4 corresponds to the special case

of Maxwellian molecules [13]. The point centers of repulsion model is slightly more realistic

than the impenetrable spheres model, but once again represents only the strong repulsive forces

between molecules at small distances from each other. It is typically used only for exploratory

calculations, because only a simple differentiable function is needed. Nevertheless, the results

obtained using this model are aphysical since molecules do not interact with only repulsive

forces.

The intermolecular potential energy and the corresponding intermolecular force for the

Sutherland model are given by

φ(r) =

{

∞ if r < σ ,

−ǫ(

σr

)γif r > σ ,

(2.13)

F (r) =

0 if r < σ ,


−ǫ γσ

(

σr

)γ+1if r > σ ,

(2.14)

respectively, in which the symbol γ denotes the index of attraction. The Sutherland model is

fairly realistic in comparison to the aforementioned models, and it is still reasonably easy to

handle analytically. It represents molecular interactions according to an inverse power law, and,

unlike the previous two models, it takes into account both attractive and repulsive interactions

between molecules.


square-well model are given by

φ(r) =

∞ if r < σ ,

−ǫ if σ < r < λσ ,

0 if r > λσ ,

(2.15)

F (r) =

0 if r < σ ,


0 if σ < r < λσ ,

undefined if r = λσ ,

0 if r > λσ ,

(2.16)


respectively, in which the symbol λ represents a parameter greater than unity, typically in the

range from 1.25 to 1.75 as reported by McFall, Wilson and Lee [21]. The square-well model is

a slightly better version than the Sutherland model, by representing a finite and more realistic

intermolecular attraction region.


general case of the Lennard-Jones m-n model are given by

φ(r) =m

n − m

( n

m

)( nn−m)

ǫ[(σ

r

)n−

(σ

r

)m]

=m

n − mǫ[(rmin

r

)n− n

m

(rmin

r

)m]

, (2.17)

F (r) =m

n − m

( n

m

)( nn−m) ǫ

σ

[

n(σ

r

)n+1− m

(σ

r

)m+1]

=mn

n − m

ǫ

rmin

[

(rmin

r

)n+1−

(rmin

r

)m+1]

, (2.18)

respectively, in which m denotes the index of attraction and n denotes the index of repulsion.

The second version of φ(r) and F (r) stems from the change of using σ to using rmin. The two

are related by setting φ(r) equal to zero and r = σ, and this yields rmin =(

nm

)1

n−m σ.

The Lennard-Jones model is most commonly used with the index of attraction equal to 6

and the index of repulsion equal to 12. For the specific case of the Lennard-Jones 6-12 model,

the intermolecular potential energy and the corresponding intermolecular force are given by

φ(r) = 4 ǫ

[

(σ

r

)12−

(σ

r

)6]

= ǫ

[

(rmin

r

)12− 2

(rmin

r

)6]

, (2.19)

F (r) = 4ǫ

σ

[

12(σ

r

)13− 6

(σ

r

)7]

= 12ǫ

rmin

[

(rmin

r

)13−

(rmin

r

)7]

, (2.20)

respectively. Note that the location where the intermolecular potential φ(r) is a minimum is

given by rmin, and the location where the intermolecular force F (r) is exactly zero is given σ,

and the two are related by rmin = 21/6σ. Note also that the Lennard-Jones 6-12 model, for

the case of the second virial coefficient, is important because the integration can be performed

analytically to give the exact series solution

B(T ) =2

3πnσ3

∞∑

j=1

bj

(

kT

ǫ

)

−(2j−1)4

=2

3πn

r3min√

2

∞∑

j=1

bj

(

kT

ǫ

)

−(2j−1)4

, (2.21)


with

bj =−2(j− 1

2)

4(j − 1)!Γ

(

2j − 3

4

)

, (2.22)

in which Γ(x) denotes the gamma function of an arbitrary real number x.


Buckingham model are given by

φ(r) = ǫ(6 + 8β) exp

[

α(

1 − rrmin

)]

− α[

(

rmin

r

)6+ β

(

rmin

r

)8]

α − 6 + (α − 8)β, (2.23)

F (r) =ǫα

rmin

(6 + 8β) exp[

α(

1 − rrmin

)]

−[

6(

rmin

r

)7+ 8β

(

rmin

r

)9]

α − 6 + (α − 8) β, (2.24)

respectively. The symbols α and β denote the steepness of exponential repulsion and the signif-

icance of repulsive and attractive terms, respectively. They typically take values of 9 < α < 15

and 0 < β < 0.25, as discussed in the book by Hirschfelder, Curtiss and Bird [13]. The Buck-

ingham model includes the induced-dipole-induced-dipole interaction and the induced-dipole-

induced-quadrapole interaction, and the repulsive interaction between molecules is modeled

using the exponential relationship. The location σ where the intermolecular potential φ(r) is

zero must be solved iteratively by setting φ(r) equal to zero and r = σ, and a good initial guess

is given by σ = rmin (6/α)1/(α−6) for 8 < α < ∞.

Although the Buckingham model provides more promise than the Lennard-Jones 6-12 model

when it comes to modeling the intermolecular interactions, it contains a physical anomaly. The

Buckingham model exhibits aphysical behaviour for very small intermolecular separations, as

illustrated in figure 2.1, where the attractive interactions become infinite. This aphysical nega-

tive infinity in Buckingham’s model has been circumvented in two different ways by Buckingham

and Corner [4] and Rice and Hirschfelder [27], to make a modified Buckingham potential model

that is usable.

The intermolecular potential energy and the corresponding intermolecular force for Buck-

ingham and Corner’s modified Buckingham model are given by

φ(r) = ǫ(6 + 8β) exp

[

α(

1 − rrmin

)]

− α[

(

rmin

r

)6+ β

(

rmin

r

)8]

f

α − 6 + (α − 8) β, (2.25)

F (r) =ǫα

rmin

(6 + 8β) exp[

α(

1 − rrmin

)]

−[

6(

rmin

r

)7+ 8β

(

rmin

r

)9]

f − g

α − 6 + (α − 8) β, (2.26)


respectively, in which the functions f and g are given by

f =

exp[

4(

1 − rmin

r

)3]

if r < rmin ,

1 if r ≥ rmin ,(2.27)

g =

12(

1 − rmin

r

)2[

(

rmin

r

)6+ β

(

rmin

r

)8]

f if r < rmin ,

0 if r ≥ rmin ,(2.28)

respectively. It is evident by direct comparison of equations 2.23 and 2.24 with equations 2.25

with 2.26, that when r ≥ rmin , that is when f = 1 and g = 0, Buckingham and Corner’s

modified Buckingham model simplifies to the original Buckingham model. This ensures that

the modification occurs only at small intermolecular separations r < rmin to circumvent the

aphysicality of infinite attractive forces at very small intermolecular separations present in

the original Buckingham model, as illustrated in figure 2.1. Similar to the original Buckingham

model, Buckingham and Corner’s modified Buckingham model also includes the induced-dipole-

induced-dipole interaction and the induced-dipole-induced-quadrapole interaction, and the re-

pulsive interaction between molecules is once again modeled using an exponential relationship.

The location σ where the intermolecular potential φ(r) is zero in Buckingham and Corner’s

modified Buckingham model must be solved iteratively by setting φ(r) equal to zero and r = σ,

and a good initial guess is, once again, given by σ = rmin (6/α)1/(α−6) for 8 < α < ∞.


modified Buckingham model of Rice and Hirschfelder are given by

φ(r) =

∞ if r < rmax ,

ǫ αα−6

[

6α exp

(

α{

1 − rrmin

})

−(

rmin

r

)6]

if r > rmax ,(2.29)

F (r) =

undefined if r < rmax ,

ǫrmin

6αα−6

[

exp(

α{

1 − rrmin

})

−(

rmin

r

)7]

if r > rmax ,(2.30)

respectively. The modifications made by Rice and Hirschfelder have minimized the induced-

dipole-induced-quadrapole interaction by firstly setting β = 0 in the original Buckingham

model. The negative infinity of the original Buckingham model near zero intermolecular sepa-

ration is eliminated by setting the intermolecular potential energy equal to positive infinity at

intermolecular separation distances of r < rmax, where rmax denotes the location at which the

intermolecular potential exhibits a maximum. This is illustrated in figure 2.1. The location σ

where the intermolecular potential φ(r) is zero in Rice and Hirschfelder’s modified Buckingham

model must be solved iteratively by setting φ(r) equal to zero and r = σ, and a good initial

guess is, once again, given by σ = rmin (6/α)1/(α−6) for 8 < α < ∞. The location rmax where

the intermolecular potential exhibits a maximum in Rice and Hirschfelder’s modified Bucking-

ham model must be solved iteratively by setting φ′(r) or F (r) equal to zero and r = rmax, and


a good initial guess is given by rmax = rmin

[

exp

{

−α7 +

(

51α+44

)7}]

for 7 ≤ α < ∞. Note

that the modified Buckingham model of Rice and Hirschfelder is commonly known or referred

to as the exponential-six model. Also, note that the modifications to the original Buckingham

model which were made by Rice and Hirschfelder are simpler than those made by Buckingham

and Corner. In fact, Rice and Hirschfelder’s changes are simplistic and crude in comparison to

those of Buckingham and Corner.

The intermolecular potential energy models which are of interest in this thesis and which are

used in the calculations of the second virial coefficient are the Lennard-Jones 6-12 model, and

the modified Buckingham models of Buckingham and Corner as well as Rice and Hirschfelder.

Each of these models have been chosen for study and comparison because they provide the

desired degree of realism in the intermolecular interactions required to accurately model binary

collisions and provide realistic values for B(T ). In the case of the Lennard-Jones 6-12 model

there also exists an exact series solution to the classical second virial coefficient that can be

used to facilitate the assessment of the numerical integration accuracy.

The Lennard-Jones 6-12 model has been used frequently to obtain the classical second virial

coefficient in past computational codes for the calculation of chemical equilibrium mixture

compositions associated with strong shock and detonation waves. More recently, the modified

Buckingham models of Rice and Hirschfelder as well as Buckingham and Corner have seen an

increase in popularity. These two models are being used more often today to solve similar

problems, as first seen in the work by Ree [24, 25]. As a result, the goal of this work is to

provide an efficient method to obtain accurate solutions for the second virial coefficient and its

derivatives in the calculation of the properties of gaseous species while using the virial equation

of state, for the Lennard-Jones 6-12 model, as well as the modified Buckingham models of

Buckingham and Corner in 1947 and Rice and Hirschfelder in 1954. Specifically, this includes

solutions to both the classical second virial coefficient as well as its quantum correction, the

latter of which has not been included previously in any major computational code.

Chapter 3

Solution Methodology for B(T) and

its Derivatives

The methodology that was developed to determine accurate integral solutions for the second

virial coefficient B(T ) and its derivatives is presented in this chapter. The non-dimensionalization

of B(T ) and an elegant method of expressing its derivatives are presented first. The transfor-

mation used to map the improper integrals of B(T ) and its derivatives from the unbounded

domain [0,∞] to the bounded domain [0, π2 ] is then given. The highly accurate method of

integration needed in the evaluation of B(T ) and its derivatives is introduced next. Since the

solutions of B(T ) and its derivatives are anticipated to be stored in tabular form (arrays) as a

function of temperature T in computer programs, a sophisticated method is presented for the

accurate interpolation of tabulated B(T ) data and its derivatives on a non-uniform grid. Fi-

nally, selected integrand plots of B(T ) and its derivatives are shown on the transformed domain

[0, π2 ] to illustrate that these are definite integrals (they contain no infinities) with a relatively

smooth integrand.

3.1 Non-Dimensionalization of B(T)

The second virial coefficient B(T ), given earlier by equation 2.6, is written in non-dimensional

form as

B∗(T∗) = Bc(T∗) + Λ∗Bq(T∗) (3.1)

for convenience, where

B∗(T∗) =B(T∗)

b0, b0 =

2

3πnr3

min , (3.2)

15

Chapter 3. Solution Methodology for B(T) and its Derivatives 16

and b0 is known as the co-volume. The classical term Bc(T∗) and the quantum correction term

Bq(T∗) in equation 3.1 are given as

Bc(T∗) = −3

∫

∞

0f(r∗, T∗) r2

∗ dr∗ (3.3)

and

Bq(T∗) =3

T 3∗

∫

∞

0F 2∗ (r∗)

[

f(r∗, T∗) + 1]

r2∗ dr∗ , (3.4)

respectively, and they are also both non-dimensional. Furthermore, f(r, T ) in equation 2.7 is

non-dimensional and given by

f(r∗, T∗) = exp

(−φ∗(r∗)

T∗

)

− 1 , (3.5)

which is in turn contains the non-dimensional intermolecular potential energy, temperature and

separation distance given by

φ∗(r∗) =φ(r∗)

ǫ, T∗ =

kT

ǫ, r∗ =

r

rmin, (3.6)

respectively. The non-dimensional quantum mechanical parameter Λ∗ in equation 3.1 is specific

to a given molecular species and given by

Λ∗ =h2R

12 M k ǫ r2min

, (3.7)

in which h = h2π is the reduced Planck constant, and M = mR

k is the molar mass. The

non-dimensional intermolecular force in equation 3.4 is also written non-dimensionally as

F∗(r∗) =rmin

ǫF (r) = −dφ∗(r∗)

dr∗. (3.8)

This non-dimensionalization of B(T ) gives a more universal representation B∗(T∗), because the

individual species information (such as ǫ, rmin and M) does not affect the integration or the

final results.

3.2 Derivatives of B∗(T∗)

3.2.1 Derivatives of the Classical Second Virial Coefficient Bc(T∗)

First, second and higher derivatives of the classical second virial coefficient Bc(T∗) given by

equation 3.3 are required. The differentiation of Bc(T∗) is performed with respect to the non-

dimensional temperature T∗. The temperature-dependence of Bc(T∗) is initially evident in the

function f(r∗, T∗), and differentiation and subsequent differentiations result in more and more


terms that accumulate into a polynomial-like set of terms in the derivatives of Bc(T∗). The nth

derivative of the classical second virial coefficient Bc(T∗) with respect to the non-dimensional

temperature T∗ can finally be expressed elegantly as

B(n)c (T∗) = −3

∫

∞

0cPn(r∗, T∗)

[

f(r∗, T∗) + 1]

φ∗(r∗) r2∗ dr∗ (3.9)

for n = 1, 2, 3, . . .. The non-dimensional function cPn(r∗, T∗) in equation 3.9 is given by the

series of n terms as

cPn(r∗, T∗) =n

∑

j=1

(−1)n+1−j Cn,j φj−1∗ (r∗)

Tn+j∗

. (3.10)

The coefficients Cn,j in equation 3.10 correspond to entries in the lower triangular matrix C,

given as

C =

1 0

2 1

6 6 1

24 36 12 1...

......

.... . .

Cn,1 Cn,2 Cn,3 Cn,4 . . . Cn,n

, (3.11)

in which the integer entries can be generated sequentially, row after row, using the algorithm

Ci,j =

1 if i = j ,

(i + j − 1)Ci−1,j if j = 1 ,

(i + j − 1)Ci−1,j + Ci−1,j−1 if j > 1 ,

(3.12)

for i = 1, 2, . . . , n and j = 1, 2, . . . , i. Note that the notation used for coefficients Cn,j di-

rectly indicates that the coefficients used for the nth derivative of the second virial coefficient

correspond to entries in the nth row in the matrix C given in equation 3.11.

3.2.2 Derivatives of the Quantum Correction Bq(T∗)

First, second and higher derivatives of the quantum correction Bq(T∗) given by equation 3.4 are

also required. The derivatives of Bq(T∗) with respect to the non-dimensional temperature T∗ are

derived in a manner similar to those for Bc(T∗). The nth derivative of the quantum correction

Bq(T∗) to the second virial coefficient with respect to the non-dimensional temperature T∗ can

be expressed elegantly as

B(n)q (T∗) = 3 Λ∗

∫

∞

0qPn(r∗, T∗) F 2

∗ (r∗)[

f(r∗, T∗) + 1]

r2∗ dr∗ (3.13)


for n = 1, 2, 3, . . .. The non-dimensional function qPn(r∗, T∗) in equation 3.13 is given by the

series of n + 1 terms as

qPn(r∗, T∗) =n+1∑

j=1

(−1)n+1−j Qn+1,j φj−1∗ (r∗)

Tn+j+2∗

. (3.14)

The coefficients Qn+1,j in equation 3.14 correspond to entries in the lower triangular matrix Q,

given as

Q =

1 0

3 1

12 8 1

60 60 15 1...

......

.... . .

Qn+1,1 Qn+1,2 Qn+1,3 Qn+1,4 . . . Qn+1,n+1

, (3.15)

in which the integer entries can be generated sequentially, row after row, using the algorithm

Qi,j =

1 if i = j ,

(i + j)Qi−1,j if j = 1 ,

(i + j)Qi−1,j + Qi−1,j−1 if j > 1 ,

(3.16)

for i = 1, 2, . . . , n + 1 and j = 1, 2, . . . , i. Note that the notation used for coefficients Qn+1,j

directly indicates that the coefficients used for the nth derivative of the quantum correction to

the second virial coefficient correspond to entries in the (n + 1)th row in the matrix Q given in

equation 3.15, which differs slightly from the formulation used for the derivatives of the second

virial coefficient and the matrix C given in equation 3.11.

3.3 Transformation for Improper to Proper Integrals

The second virial coefficient and its derivatives are composed of Bc(T∗) and B(n)c (T∗) for the

classical term and Bq(T∗) and B(n)q (T∗) for the quantum correction. See equations 3.1, 3.3, 3.4,

3.9 and 3.13. These are all improper integrals on the infinite domain [0,∞] in terms of variable

r∗. These integrals are transformed into proper integrals by using the transformation given by

r3∗ = tan(x) , with 3 r2

∗ dr∗ = [1 + tan2(x)] dx , (3.17)

changing the integration variable from r∗ to x on the new finite domain [0, π2 ]. This eliminates

the problem of attempting to perform numerical integrations over an infinite domain.


The previous equations for the second virial coefficient and its derivatives are restated in

non-dimensional form as

B∗(T∗) = Bc(T∗) + Λ∗Bq(T∗) , (3.18)

B(n)∗ (T∗) = B(n)

c (T∗) + Λ∗B(n)q (T∗) , (3.19)

respectively, where

Bc(T∗) = −∫ π

2

0f(x, T∗)

[

1 + tan2(x)]

dx , (3.20)

Bq(T∗) =

∫ π2

0qP0(x, T∗) F 2

∗ (x)[

f(x, T∗) + 1] [

1 + tan2(x)]

dx , (3.21)

B(n)c (T∗) =

∫ π2

0cPn(x, T∗)

[

f(x, T∗) + 1]

φ∗(x)[

1 + tan2(x)]

dx , (3.22)

and

B(n)q (T∗) =

∫ π2

0qPn(x, T∗)F 2

∗ (x)[

f(x, T∗) + 1] [

1 + tan2(x)]

dx , (3.23)

respectively. After the transformation, all of the integrals are proper integrals, meaning they

are restricted to a finite domain. The functions f(x, T∗), φ∗(x), F∗(x), cPn(x, T∗) and qPn(x, T∗)

are all shown with x symbolically replacing r∗. However, inside the functions, r∗ is replaced by

tan13 (x).

Transformations of r∗ to x other than that given in equation 3.17 have been tested. Three

of these are given as

r3∗ =

x

1 − x, r3

∗ = tan(πx

4

)

, and r3∗ =

(2m − 1)x

2m − xm, (3.24)

for m = 1, 2, or 3. The first two transformations are given on the finite domain [0, 1], and the

last is given on [0, 2]. These three transformations given by 3.24 are similar to the transforma-

tion given by 3.17. However, each transformation distributes the integrand differently over its

domain with the variable x, and this affects the accuracy of the integrated results. Integration

tests showed that the transformation given by equation 3.17 gave good results, so it was selected

for this work. It also is the simplest transformation to use.

3.4 Method of Integration

The proper and definite integrals in Bc(T∗), Bq(T∗), B(n)c (T∗) and B

(n)q (T∗) in equations 3.20

to 3.23 can be numerically integrated more accurately by using Gaussian quadrature than

composite integration methods (e.g., trapezoidal, Simpson’s, Boole’s and Newton-Cotes’ rules)


for the same number of function evaluations. Hence, Gaussian quadrature was selected for

performing the integrations in this thesis.

In conventional Gaussian quadrature the integration of a function f(x) on the interval [a, b]

is done alternatively on the interval [−1, 1] by using the linear transformation x = b−a2 z + a+b

2

with dx = b−a2 dz, such that

∫ x=b

x=af(x) dx =

b − a

2

∫ z=1

z=−1f

(

b − a

2z +

a + b

2

)

dz (3.25)

is the result. In the present work f(x) corresponds to the integrands of Bc(T∗), Bq(T∗), B(n)c (T∗)

and B(n)q (T∗). The integral is approximated by the weighted sum of n function evaluations,

which can be expressed as

∫ x=b

x=af(x) dx ≈ b − a

2

n∑

i=1

wi f

(

b − a

2zi +

a + b

2

)

, (3.26)

for the so-called n-point integration rule. The locations zi at which the function evaluations are

done are determined by the roots of the Gauss-Legendre polynomials given by the recurrence

formula

Pn(z) =1

2n n!

dn

dzn

(

z2 − 1)n

, n = 0, 1, 2, 3, . . . , (3.27)

and the weights wi are determined correspondingly by the relationships

wi =2

(

1 − z2i

)

[P ′n(zi)]

2 (3.28)

for i = 1 , 2 , 3 , · · · , n, in which P ′n(zi) is the first derivative of the Gauss-Legendre polyno-

mial. See the mathematical handbook by Abramowitz and Stegun [1] and the book by Press,

Teukolsky, Vetterling and Flannery [22] for more information. The locations zi and weights wi

are determined for Gaussian quadrature so that the integrations are exact for all polynomial

functions of degree (2n − 1) or less. Hence, a 50-point rule will integrate exactly a polynomial

of degree 99 and less (neglecting round off errors), with only 50 function evaluations.

For all numerical integrations including the integrals connected to Bc(T∗) and Bq(T∗) for the

Lennard-Jones 6-12 model, Gaussian quadrature is done over the entire transformed domain

[0, π2 ] with the variable x, so a = 0 and b = π

2 . For the modified Buckingham model of

Buckingham and Corner the quadrature is done firstly over the domain [0, π4 ] and then over the

remaining domain [π4 , π2 ] and the results are added. This composite integration is done because

the underlying equations for the Buckingham and Corner model are somewhat different in the

two domains and are not smooth in terms of the higher derivatives at the location x = π4 . See

equations 2.25 to 2.28. Lastly, for the modified Buckingham model of Rice and Hirschfelder a


composite integration is also done, first over the domain [0, tan−1(

r3max/r3

min

)

] and then over the

remaining domain [tan−1(

r3max/r3

min

)

, π2 ], and once again the results are added. The composite

integration is done because the potential function is infinite and its derivatives are zero within

the first domain [0, tan−1(

r3max/r3

min

)

]. See equations 2.29 and 2.30. This method of using a

composite integration avoids doing an inaccurate Gaussian quadrature across a discontinuity

in B(T ) and its derivatives.

All Gaussian quadratures of B(T ) and its derivatives were done using Wolfram Research’s

software package called Mathematica. This was done so that tabulated results for the second

virial coefficient and its derivatives could be generated potentially as accurate to as many

significant digits as desired by the user, so that the solutions could later be included in arrays

in Fortran and C++ programming language codes for computing strong shock and detonation

wave properties. While other software packages may have limited the accuracy of the solutions

of B(T ) and its derivatives to fifteen significant digits or much less by using double-precision

floating-point arithmetic operations owing to rounding or round-off errors, Wolfram Research’s

Mathematica provided the opportunity to obtain more accurate solutions to a prescribed

number of significant digits through the use of arbitrary-precision calculations.

The integrations for B(T ) and its derivatives have been performed using Gaussian quadra-

ture with an n-point rule of 300. The use of a 300-point rule was sufficiently high that integrated

results obtained with Wolfram Research’s Mathematica were accurate to sixteen significant

digits or more when a working precision of fifty digits was maintained throughout all inter-

nal computations. In comparison, tabulated results for the second virial coefficient have been

reported previously using as few as only three significant digits [6].

3.5 Method of Interpolation for Tabulated Virial Coefficients

The virial coefficient B(T ) and its derivatives are normally stored in tables for later use in

computational codes for calculations involving chemical equilibrium and strong shock and det-

onation waves. Furthermore, these virial coefficients are stored at very small temperature

intervals to reduce the interpolation error of a lower degree polynomial interpolant. However,

fewer virial coefficients need to be stored when a high-degree polynomial interpolant scheme is

used. Such a scheme is given herein.

A special interpolation method has been developed by the Unsteady Gasdynamics group

at UTIAS [11] for polynomial interpolation for a discrete function and its derivatives that are


available from tables. This sophisticated interpolation scheme is described herein, without

extensive derivation. For an interpolant f(x) passing through the adjacent data pairs (xi, fi)

and (xi+1, fi+1) with i = 1, 2, . . . , n for n discrete data pairs, the Taylor series expansion

truncated to a polynomial of degree m is given by

f =1

0!f∣

∣

i∆x0

i η0 +

1

1!

df

dx

∣

∣

∣

∣

i

∆x1i η

1 +1

2!

d2f

dx2

∣

∣

∣

∣

i

∆x2i η

2 + · · · +1

m!

dmf

dxm

∣

∣

∣

∣

i

∆xmi ηm , (3.29)

in which ∆xi = xi+1 − xi is the ith interval width and η = x−xi

xi+1−xi= x−xi

∆xi= 1 − ξ is the

normalized x. The value of f and its derivatives are f∣

∣

i, df

dx

∣

∣

i, d2f

dx2

∣

∣

i, . . . , dmf

dxm

∣

∣

i, evaluated

at location xi, and the number of derivatives used depends directly on the desired degree of

the polynomial interpolant. When the interpolant is written in the form of equation 3.29, the

interpolant automatically passes through fi, f ′i , f ′′

i , . . . , at location xi by its construction for a

degree m interpolant, but it does not necessarily satisfy fi+1, f ′i+1, f ′′

i+1, . . . , at location xi+1.

To force the interpolant to pass through (xi+1, fi+1) and satisfy the higher derivatives at this

location, the interpolant is commonly rewritten in terms of fi+1, along with fi, and includes

information of the higher derivatives at each of these nodes.

For piece-wise linear interpolation between two adjacent data pairs, the interpolant, stem-

ming from a Taylor series expansion of degree m = 1, can be written as

f =1

0!f∣

∣

i(1 − η)1 η0 {1}∆x0

i +(−1)0

0!f∣

∣

i+1(1 − ξ)1 ξ0 {1}∆x0

i . (3.30)

For piece-wise cubic interpolation between two adjacent data pairs, the interpolant, stemming

from a Taylor series expansion of degree m = 3, can be written as

f =1

0!f∣

∣

i(1 − η)2 η0 {1 + 2η}∆x0

i +(−1)0

0!f∣

∣

i+1(1 − ξ)2 ξ0 {1 + 2ξ}∆x0

i

+1

1!

df

dx

∣

∣

∣

∣

i

(1 − η)2 η1 {1}∆x1i +

(−1)1

1!

df

dx

∣

∣

∣

∣

i+1

(1 − ξ)2 ξ1 {1}∆x1i . (3.31)

For piece-wise quintic interpolation between two adjacent data pairs, the interpolant, stemming

from a Taylor series expansion of degree m = 5, can be written as

f =1

0!f∣

∣

i(1 − η)3 η0

{

1 + 3η + 6η2}

∆x0i +

(−1)0

0!f∣

∣

i+1(1 − ξ)3 ξ0

{

1 + 3ξ + 6ξ2}

∆x0i

+1

1!

df

dx

∣

∣

∣

∣

i

(1 − η)3 η1 {1 + 3η}∆x1i +

(−1)1

1!

df

dx

∣

∣

∣

∣

i+1

(1 − ξ)3 ξ1 {1 + 3ξ}∆x1i

+1

2!

d2f

dx2

∣

∣

∣

∣

i

(1 − η)3 η2 {1}∆x2i +

(−1)2

2!

d2f

dx2

∣

∣

∣

∣

i+1

(1 − ξ)3 ξ2 {1}∆x2i , (3.32)

and so on and so forth. More accurate piece-wise interpolants between two adjacent data

pairs can be easily obtained for higher odd degree polynomials by pattern recognition and

straightforward generalization of equations 3.30, 3.31 and 3.32.


1 10 45 120 210 252 210 120 45 10 1

1 9 36 84 126 126 84 36 9 1

1 8 28 56 70 56 28 8 1

1 7 21 35 35 21 7 1

1 6 15 20 15 6 1

1 5 10 10 5 1

1 4 6 4 1

1 3 3 1

1 2 1

1 1

1linear (m = 1)

cubic (m = 3)

quintic (m = 5)

septic (m = 7)

nonic (m = 9)

m = 11

et cetera

Figure 3.1: Pascal’s triangle.

Although the patterns of the factorials and the powers on (1 − η), η, (1 − ξ), ξ and ∆xi

can be recognized easily from one interpolant to the next, the pattern of the integer coefficients

on the terms of the polynomials in the curly brackets is not evident. These coefficients can

be identified with Pascal’s triangle of integers, illustrated in figure 3.1. The rows of Pascal’s

triangle stem from the coefficients from the expansion of

(x + y)i =i

∑

j=0

(

i

j

)

xi−j yj , (3.33)

and are given by the binomial coefficients denoted by(

ij

)

. For computer programs the binomial

coefficients for each row are calculated efficiently by

bi =

(

m−12

)

!

i!(

m−12 − i

)

!, i = 0, 1, . . . ,

m − 1

2, (3.34)

with 0! = 1 and a degree m interpolant. The coefficients on the terms of the polynomials in the

curly brackets of equations 3.30 to 3.32, and for higher odd degree interpolants, correspond to

slanted slices of numbers in Pascal’s triangle, as illustrated in figure 3.1. These slanted sets of

numbers are denoted by gi and calculated by

gi =

(

i + m−32

)

!

(i − 1)!(

m−12

)

!, i = 1, 2, . . . ,

m + 1

2, (3.35)

for a polynomial interpolant of degree m. Note that m is the degree of the interpolant being

used, and it is always odd, that is, m = 1, 3, 5, 7, 9, . . ., for the case of interpolants that are

linear, cubic, quintic, septic, nonic, . . . , and so on and so forth.

The virial coefficient B(T ) is typically stored sparsely in tabular form (arrays) versus tem-

perature for use in computational codes. A piece-wise polynomial interpolant of degree m can


be used to interpolate tabulated data for the second virial coefficient given by equation 3.1 only

when all derivatives up to and including the m−12 th derivative can be calculated.1 To control

the relative interpolation error between two adjacent nodes, an adaptive grid spacing is used.

This maximizes the size of each interval between two adjacent nodes such that the maximum

error in each interval equals a specified error tolerance.

A case study with a test function has been done in appendix A and the results show that

the size and storage space of the tabulated data is minimized when an adaptive grid spacing is

used. From the results presented in appendix A it was possible to derive that for a polynomial

interpolant of degree m that requires n nodes to capture a test function, the total storage space

for this special method of interpolation is given by

s =m + 3

2n . (3.36)

Here, 2n storage spaces are required to store data pairs of (xi, fi) for n nodes, and m−12 n storage

spaces are required to store m−12 derivatives of fi for n nodes, that is, df

dx

∣

∣

i, d2f

dx2

∣

∣

i, . . . , d

m−12 f

dxm−1

2

∣

∣

i,

as can easily be verified by inspection of equations 3.30, 3.31 and 3.32. In comparison to other

interpolation methods (such as those using Lagrange’s polynomials, Newton’s polynomials,

or splines)2 the piece-wise interpolation method typically requires fewer nodes to accurately

capture a given function based on a specified error tolerance, meaning that the storage space

of the tabulated data can be most effectively minimized when this method is used. In addition

to requiring fewer nodes, the piece-wise interpolation method is more accurate in general since

the values of the derivatives at adjacent nodes are known exactly, which is not the case in

comparison to other interpolation methods. For example, a cubic spline interpolation method

approximates the derivative values at the endpoints using either natural or clamped boundary

conditions which, although may ensure that the curvature along the spline in minimized, may

not ensure that the derivative values at each node are necessarily accurate.

3.6 Plots of the Integrands of B∗(T∗) and its Derivatives

Graphical illustrations of the integrands of B∗(T∗) and some of its lower derivatives are presented

in this section. This is done for the most important cases of intermolecular potentials of

Lennard-Jones in 1924, Buckingham and Corner in 1947 and Rice and Hirschfelder in 1954.

1See section 3.2.2The storage space for an interpolation method using Lagrange’s polynomials, Newton’s polynomials, or

splines is s = 2n + m(n − 1) for an interpolant of degree m using n nodes. Here, 2n storage spaces are requiredto store data pairs of (xi, fi) for n nodes, and m(n − 1) storage spaces are required to store m coefficients in(n − 1) node intervals.


The primary reasons for showing these results is to illustrate the behaviour of these integrands,

show that they are relatively smooth (in regions between discontinuities, where applicable), and

further indicate the use of Gaussian quadrature integrations, as explained earlier in section 3.4.

For the classical second virial coefficient, the integrands for Bc(T∗) and its first, second and

third derivatives with respect to temperature are given by

cI0(x, T∗) = −f(x, T∗)[

1 + tan2(x)]

, (3.37)

cI1(x, T∗) =

{−1

T 2∗

}

[

f(x, T∗) + 1]

φ∗(x)[

1 + tan2(x)]

, (3.38)

cI2(x, T∗) =

{−φ∗(x)

T 4∗

+2

T 3∗

}

[

f(x, T∗) + 1]

φ∗(x)[

1 + tan2(x)]

, (3.39)

and

cI3(x, T∗) =

{−φ2∗(x)

T 6∗

+6φ∗(x)

T 5∗

− 6

T 4∗

}

[

f(x, T∗) + 1]

φ∗(x)[

1 + tan2(x)]

, (3.40)

respectively, and have been derived using the methodology presented earlier in section 3.2.1.

For the quantum correction to the second virial coefficient, the integrands for Bq(T∗) and its

first, second and third derivatives with respect to temperature are given by

qI0(x, T∗) =

{

1

T 3∗

}

F 2∗ (x)

[

f(x, T∗) + 1] [

1 + tan2(x)]

, (3.41)

qI1(x, T∗) =

{

φ∗(x)

T 5∗

− 3

T 4∗

}

F 2∗ (x)

[

f(x, T∗) + 1] [

1 + tan2(x)]

, (3.42)

qI2(x, T∗) =

{

φ2∗(x)

T 7∗

− 8φ∗(x)

T 6∗

+12

T 5∗

}

F 2∗ (x)

[

f(x, T∗) + 1] [

1 + tan2(x)]

, (3.43)

and

qI3(x, T∗) =

{

φ3∗(x)

T 9∗

− 15φ2∗(x)

T 8∗

+60φ∗(x)

T 7∗

− 60

T 6∗

}

F 2∗ (x)

[

f(x, T∗)+1] [

1 + tan2(x)]

, (3.44)

respectively, and have been derived using the methodology presented earlier in section 3.2.2.

The graphs for the integrands of Bc(T∗) and its first three derivatives are presented in figures

3.2, 3.3 and 3.4, for non-dimensional temperatures of 1.0, 10.0 and 100.0, respectively. The

graphs for the integrand of Bq(T∗) and its first three derivatives are presented in figures 3.5,

3.6 and 3.7, for non-dimensional temperatures of 1.0, 10.0 and 100.0, respectively.

For the Lennard-Jones 6-12 model, the integration is performed over the entire domain

[0, π2 ] since there are no discontinuities. For the modified Buckingham models of Buckingham

and Corner as well as Rice and Hirschfelder, the use of a composite integration method is

further justified through inspection of these graphs. When the modified Buckingham model


of Buckingham and Corner is used, the discontinuity is present at x = π2 in the integrands of

the second virial coefficient. Although this is not evident graphically in the integrands which

have been presented, the discontinuity nonetheless exists and becomes more apparent in the

integrands of higher derivatives of the second virial coefficient. However, most evident is the

discontinuity present at x = tan−1(r3max/r3

min) in the integrands of the second virial coefficient

when the modified Buckingham model of Rice and Hirschfelder is used. In this specific case, the

integrand for the classical second virial coefficient within the domain [0, tan−1(r3max/r3

min)] is

given by[

1 + tan2(x)]

for Bc(T∗) and equal to zero for its derivatives, since φ(x) = ∞ and thus

f(x, T∗) = −1 when x < tan−1(r3max/r3

min) for this model. For similar reasons, the integrand

for the quantum correction is equal to zero within the domain [0, tan−1(r3max/r3

min)] for Bq(T∗)

and its derivatives because [ f(x, T∗) + 1] = 0.


0 π/8 π/4 3π/8 π/2−4

0

2

x

cI0(x,T

*)

Buckingham and Corner modelRice and Hirschfelder modelLennard−Jones 6−12 model

0 π/8 π/4 3π/8 π/2−2

0

6

x

cI1(x,T

*)

0 π/8 π/4 3π/8 π/2−20

0

x

cI2(x,T

*)

0 π/8 π/4 3π/8 π/2

0

80

x

cI3(x,T

*)

Figure 3.2: Integrands of Bc(T∗) and its first three derivatives with T∗ = 1.0.


0 π/8 π/4 3π/8 π/2

0

1

x

cI0(x,T

*)


0 π/8 π/4 3π/8 π/2

−0.04

0

0.02

x

cI1(x,T

*)

0 π/8 π/4 3π/8 π/2

−0.005

0

0.005

x

cI2(x,T

*)

0 π/8 π/4 3π/8 π/2−0.002

0

0.002

x

cI3(x,T

*)



0 π/8 π/4 3π/8 π/2

0

1

x

cI0(x,T

*)


0 π/8 π/4 3π/8 π/2

−0.004

0

x

cI1(x,T

*)

0 π/8 π/4 3π/8 π/2−0.00002

0

0.00006

x

cI2(x,T

*)

0 π/8 π/4 3π/8 π/2

−0.000001

0

0.000001

x

cI3(x,T

*)



0 π/8 π/4 3π/8 π/20

1500

x

qI0(x,T

*)


0 π/8 π/4 3π/8 π/2

−3000

0

x

qI1(x,T

*)

0 π/8 π/4 3π/8 π/2

0

15000

x

qI2(x,T

*)

0 π/8 π/4 3π/8 π/2

−60000

0

20000

x

qI3(x,T

*)

Figure 3.5: Integrands of Bq(T∗) and its first three derivatives with T∗ = 1.0.


0 π/8 π/4 3π/8 π/20

30

x

qI0(x,T

*)


0 π/8 π/4 3π/8 π/2

−4

0

2

x

qI1(x,T

*)

0 π/8 π/4 3π/8 π/2

−0.5

0

1.5

x

qI2(x,T

*)

0 π/8 π/4 3π/8 π/2−0.5

0

0.5

x

qI3(x,T

*)



0 π/8 π/4 3π/8 π/20

2

x

qI0(x,T

*)


0 π/8 π/4 3π/8 π/2

−0.02

0

0.02

x

qI1(x,T

*)

0 π/8 π/4 3π/8 π/2

−0.0005

0

0.0005

x

qI2(x,T

*)

0 π/8 π/4 3π/8 π/2

−0.00002

0

0.00002

x

qI3(x,T

*)


Chapter 4

Results for B∗(T∗) and its

Derivatives

The numerical evaluation and interpolation methodology for the second virial coefficient and its

derivatives were presented in the previous chapter. Numerical results for B∗(T∗) and its deriva-

tives are given herein using this methodology for gaseous species and practical intermolecular

potential energy models. Second virial coefficient results for both Bc(T∗) and Bq(T∗) are in-

cluded to show their temperature-dependent behaviour and to illustrate the importance of the

quantum correction.

4.1 Numerical Solutions for B∗(T∗)

The application of the sophisticated method of interpolation using polynomial interpolants of

different degree m is presented for both the classical second virial coefficient Bc(T∗) as well as

the quantum correction Bq(T∗) to the second virial coefficient. Included are interpolants that

are cubic (m = 3), quintic (m = 5), septic (m = 7), nonic (m = 9), and so on and so forth,

up to m = 17. The relative error of the interpolated values is controlled to a specified value of

10−6 % using an adaptive grid spacing, meaning that the reconstructed solutions to the second

virial coefficient represented by interpolation are accurate to at least eight significant digits in

every interval between any two adjacent nodes throughout the entire domain.

The results are illustrated graphically for the specific cases of the Lennard-Jones 6-12

model and the modified Buckingham models of Buckingham and Corner as well as Rice and

33

Chapter 4. Results for B∗(T∗) and its Derivatives 34

Hirschfelder. A wide non-dimensional temperature (T∗) domain of [10−1, 104] was selected to

capture both light gases at elevated temperatures (T ≈ 100, 000 K) as well as heavy gases at low

temperatures (T ≈ 100 K). In general, lighter gases have lower values for ǫ/k whereas heavier

gases have larger ones [13], which, through the use of equation 3.6 for the expression represent-

ing the non-dimensional temperature, result in these bounding non-dimensional temperature

values.

4.1.1 Classical Second Virial Coefficient Bc(T∗)

The variation of the classical second virial coefficient Bc(T∗) is given as a function of the non-

dimensional temperature T∗ in figure 4.1, for the Lennard-Jones 6-12 model and the modified

Buckingham models of Buckingham and Corner as well as Rice and Hirschfelder. As illustrated

in this graph, the second virial coefficient is negative at low temperatures (the second virial

coefficient approaches negative infinity as the temperature tends towards zero) and positive at

high ones, passing through the Boyle temperature when the second virial coefficient is equal

to zero. The differences in the Bc(T∗) plots for the three different intermolecular potential

energy models at higher T∗ values are non-trivial. The classical second virial coefficient exhibits

the highest values for the Lennard-Jones 6-12 model, and the lowest values for the Rice and

Hirschfelder modified Buckingham (exponential-six) model. These results are in agreement

100

101

102

103

104

−1

0

0.6

T*

Bc(T

*)

8−


Figure 4.1: The classical second virial coefficient.


with data in the literature [13, 6]. These differences are due to the high molecular speeds at

elevated temperatures, caused by the differences in the regions of the exponential repulsion in

the intermolecular potential energy models. The modified Buckingham model of Buckingham

and Corner is considered to be the most realistic of these three models for the calculation of

Bc(T∗), since it includes the most detailed account of molecular interactions while maintaining

a very high level of sophistication in the construction of the model itself.

The classical second virial coefficient Bc(T∗) is included using adaptive grid spacing for the

generation of tables. Graphs are used to illustrate the node locations, using an adaptive grid

spacing, for the minimum number of n nodes required to interpolate Bc(T∗) using a polynomial

interpolant of degree m. Once the number of n nodes are known for a polynomial interpolant of

degree m, the storage space of the tabulated data is calculated using s = m+32 n. The adaptive

grids for the classical second virial coefficient are illustrated graphically in figures 4.2 to 4.4, for

Lennard-Jones 6-12 model and the modified Buckingham models of Buckingham and Corner

as well as Rice and Hirschfelder, respectively. The adaptive grids and interpolations for Bc(T∗)

are done separately for each intermolecular potential energy model of interest, as a result of the

slight variations in the Bc(T∗) plots when different models are used.

Results for the Lennard-Jones 6-12 model are illustrated in figure 4.2. The number of nodes

10−1

100

101

102

103

104

−1

0

0.6

T*

Bc(T

*)

8−

m = 3 n = 646

m = 5 n = 138

m = 7 n = 65

m = 9 n = 42

m = 11 n = 31

m = 13 n = 25

m = 15 n = 22

m = 17 n = 19

Figure 4.2: Adaptive grid variation for new interpolation method for m = 3 , 5 , . . . , 17, using

the Lennard-Jones 6-12 model for the classical second virial coefficient.


required to generate an adaptive grid when polynomial interpolants of degree m = 3, 5, 7, 9,

11, 13, 15 and 17 are used are 646, 138, 65, 42, 31, 25, 22 and 19, respectively. In terms of

the storage space required in computer programs, the data storage is 1,938, 552, 325, 252, 217,

200, 198 and 190, respectively.

Results for the Buckingham and Corner modified Buckingham model are illustrated in figure

4.3. The number of nodes required to generate an adaptive grid when polynomial interpolants

of degree m = 3, 5, 7, 9, 11, 13, 15 and 17 are used are 667, 142, 66, 42, 31, 26, 22 and 20,

respectively. In terms of the storage space required in computer programs, the data storage

is 2,001, 568, 330, 252, 217, 208, 198 and 200, respectively. In this case, it becomes quite

apparent that it is not advantageous to use a polynomial interpolant of degree m = 17 because

the savings in data storage bottoms out when m = 15. Instead, it appears that the largest

savings in storage space occur when a quintic (m = 5) or septic (m = 7) interpolant is used.

10−1

100

101

102

103

104

−1

0

0.6

T*

Bc(T

*)

8−

m = 3 n = 667

m = 5 n = 142

m = 7 n = 66

m = 9 n = 42

m = 11 n = 31

m = 13 n = 26

m = 15 n = 22

m = 17 n = 20


the Buckingham and Corner modified Buckingham model (α = 13.5 and β = 0.0) for the

classical second virial coefficient.

Results for the Rice and Hirschfelder modified Buckingham (exponential-six) model are

illustrated in figure 4.4. The number of nodes required to generate an adaptive grid when

polynomial interpolants of degree m = 3, 5, 7, 9, 11, 13, 15 and 17 are used are 669, 142,

66, 42, 31, 26, 22 and 20, respectively. In terms of the storage space required in computer


programs, the data storage is 2,007, 568, 330, 252, 217, 208, 198 and 200, respectively. Once

again, it is not advantageous to use an interpolant of degree m = 17 because the savings in data

storage bottoms out when m = 15. Note that although the number of nodes required are equal

to the previous case, specifically for polynomial interpolants of degree m = 5 , 7 , . . . , 17, the

interpolations do not generate the exact same adaptive grids (e.g. non-dimensional temperature

values).

10−1

100

101

102

103

104

−1

0

0.6

T*

Bc(T

*)

8−

m = 3 n = 669

m = 5 n = 142

m = 7 n = 66

m = 9 n = 42

m = 11 n = 31

m = 13 n = 26

m = 15 n = 22

m = 17 n = 20


the Rice and Hirschfelder modified Buckingham (exponential-six) model (α = 13.5 and

β = 0.0) for the classical second virial coefficient.

The results of the adaptive grids for the classical second virial coefficient using polynomial

interpolants of different degree m = 3 , 5 , . . . , 17 show that the number of nodes required

to interpolate the tabulated data decreases as higher-degree interpolants are used. As for the

storage space of the tabulated data, the largest savings are seen when a quintic (m = 5) or septic

(m = 7) interpolant is used, and little to no savings are seen as polynomial interpolants upwards

of degree m = 17 are used. In other words, the savings in storage space eventually bottom out.

In smooth regions of the graph with relatively small curvature, such as when 101 ≤ T∗ ≤ 104

in figures 4.2, 4.3 and 4.4, the grid spacing between two adjacent nodes is large since the new

interpolant can efficiently capture the data over such a wide range of temperatures. As well,

in the specific case of cubic interpolation (m = 3), the large grid spacing near T∗ ≈ 102 can


easily be seen as a result of the function having an inflection point in the center of that region.

Lastly, it should be noted that these results validate that the derivatives of the classical second

virial coefficient have been derived and calculated accurately (as per section 3.2.1). Otherwise,

the results would not have shown a decrease in the number of nodes needed as the degree of

the polynomial interpolant was increasing.

The behaviour of the second virial coefficient can be better understood by taking a closer

look at the potential energy models which govern the behaviour of the interacting molecules.

At low temperatures the average energies of the molecules are on the order of the characteristic

energy, and colliding molecules exhibit behaviour similar to the molecular interactions within

the attractive region of the potential energy model. As a result, the second virial coefficient is

thus negative and there is a decrease in pressure in the gas. At high temperatures the average

energies of the molecules are relatively large in comparison to the characteristic energy, and

colliding molecules exhibit behaviour similar to the molecular interactions within the repulsive

region of the potential energy model. In this case, the second virial coefficient becomes positive

and the pressure of the gas increases. For very high temperatures, the intermolecular collisions

are so forceful that the molecules penetrate each other and behave as if they had a smaller

volume. This explains why the second virial coefficient has a maximum positive value, after

which it tends towards zero [13]. It also highlights the importance of accurately modeling

the repulsive interactions between molecules, and hints that small intricacies amongst different

intermolecular potential energy models do contribute noticeably to variations in Bc(T∗).

4.1.2 Quantum Correction Bq(T∗)

The variation of the quantum correction Bq(T∗) to the second virial coefficient is given as

a function of the non-dimensional temperature T∗ in figure 4.5, for the Lennard-Jones 6-12


Hirschfelder. As mentioned earlier, the quantum correction to the second virial coefficient

accounts for the probability of the intermolecular potential energy configurations not being

proportional to the exponential term in equation 2.7. Hence, the quantum correction is positive

at all temperatures and approaches positive infinity as the temperature tends towards zero.

Similar to Bc(T∗), graphs of the quantum correction Bq(T∗) to the second virial coefficient

are included using adaptive grid spacing. The adaptive grids for the quantum correction to the

second virial coefficient are illustrated graphically in figures 4.6 to 4.8, for Lennard-Jones 6-12



100

101

102

103

104

0

100

T*

Bq(T

*)

8


Figure 4.5: The quantum correction to the second virial coefficient.

Hirschfelder, respectively. The adaptive grids and interpolations of Bq(T∗) are done separately

for each intermolecular potential energy model of interest, as a result of the slight variations in

the Bq(T∗) plots when different models are used.

Results for the Lennard-Jones 6-12 model are illustrated in figure 4.6. The number of nodes

required to generate an adaptive grid when polynomial interpolants of degree m = 3, 5, 7, 9,

11, 13, 15 and 17 are used are 933, 189, 84, 52, 38, 30, 26 and 23, respectively. In terms of

the storage space required in computer programs, the data storage is 2,799, 756, 420, 312, 266,

240, 234 and 230, respectively.

Results for the Buckingham and Corner modified Buckingham model are illustrated in figure

4.7. The number of nodes required to generate an adaptive grid when polynomial interpolants

of degree m = 3, 5, 7, 9, 11, 13, 15 and 17 are used are 967, 194, 86, 53, 39, 31, 26 and 23,

respectively. In terms of the storage space required in computer programs, the data storage is

2,901, 776, 430, 318, 273, 248, 234 and 230, respectively.

Results for the Rice and Hirschfelder modified Buckingham (exponential-six) model are

illustrated in figure 4.8. The number of nodes required to generate an adaptive grid when

polynomial interpolants of degree m = 3, 5, 7, 9, 11, 13, 15 and 17 are used are 980, 196, 87, 54,

39, 31, 26 and 23, respectively. In terms of the storage space required in computer programs,

the storage is 2,940, 784, 435, 324, 273, 248, 234 and 230, respectively.


10−1

100

101

102

103

104

0

100

T*

Bq(T

*)

8m = 3 n = 933

m = 5 n = 189

m = 7 n = 84

m = 9 n = 52

m = 11 n = 38

m = 13 n = 30

m = 15 n = 26

m = 17 n = 23


the Lennard-Jones 6-12 model for the quantum correction to the second virial coefficient.

10−1

100

101

102

103

104

0

100

T*

Bq(T

*)

8

m = 3 n = 967

m = 5 n = 194

m = 7 n = 86

m = 9 n = 53

m = 11 n = 39

m = 13 n = 31

m = 15 n = 26

m = 17 n = 23


the Buckingham and Corner modified Buckingham model (α = 13.5 and β = 0.0) for the

quantum correction to the second virial coefficient.


10−1

100

101

102

103

104

0

100

T*

Bq(T

*)

8m = 3 n = 980

m = 5 n = 196

m = 7 n = 87

m = 9 n = 54

m = 11 n = 39

m = 13 n = 31

m = 15 n = 26

m = 17 n = 23


the Rice and Hirschfelder modified Buckingham (exponential-six) model (α = 13.5 and

β = 0.0) for the quantum correction to the second virial coefficient.

The results of the adaptive grids for the quantum correction to the second virial coefficient

using polynomial interpolants of different degree m = 3 , 5 , . . . , 17 show that both the number

of nodes required to interpolate the tabulated data as well as the storage space decreases as

higher-degree interpolants are used. Similar to the case for the classical second virial coefficient,

the largest savings in the storage space for the tabulated data are seen when a quintic (m = 5)

or septic (m = 7) interpolant is used. In smooth regions of the graph with relatively small

curvature, such as when 101 ≤ T∗ ≤ 104 in figures 4.6, 4.7 and 4.8, the grid spacing between

two adjacent nodes is large since the new interpolant can efficiently capture the data over such

a wide range of temperatures. Lastly, it should be noted that these results validate that the

derivatives of the quantum correction to the second virial coefficient have been derived and

calculated accurately (as per section 3.2.2). Similar to the results for the interpolated classical

second virial coefficient, the results here would otherwise not have shown a decrease in the

number of nodes needed as the degree of the polynomial interpolant was increasing.

It is worthwhile to mention that the results plotted in figures 4.6, 4.7 and 4.8 are species-

independent, meaning that the quantum correction illustrated in these graphs does not take into

account the non-dimensional quantum mechanical parameter Λ∗ given in equation 3.7, which


is specific for a given molecular species. Although the values for the quantum correction to the

second virial coefficient are seemingly several orders of magnitude higher than the values for the

second virial coefficient itself are, the magnitude of the non-dimensional quantum mechanical

parameter Λ∗ is typically on the order of 10−3 for light gases (and smaller for heavier gases) and

balance out the corrective term quite nicely. These results are investigated further in section

4.2.

4.1.3 Selected Tabulated Solutions for B∗(T∗)

A selected set of tabulated second virial coefficient values are presented on the following six

pages for the specific case of nonic (m = 9) interpolation using an adaptive grid. Tables 4.1,

4.2 and 4.3 present the tabulated data for the classical second virial coefficient for the Lennard-

Jones 6-12 model, and the modified Buckingham models of Buckingham and Corner as well as

Rice and Hirschfelder, respectively, whereas tables 4.4, 4.5 and 4.6 present the tabulated data for

the quantum correction, for the respective intermolecular potential energy models mentioned.

This tabulated data for the second virial coefficient and its derivatives has been generated

using Wolfram Research’s Mathematica using a working precision of fifty significant digits, so

that it can be presented in tabular form accurate with a precision of at least fourteen significant

digits. The second virial coefficients have been calculated using non-dimensional temperature

values that are specified exactly to five significant digits. The truncations were done solely for

the purpose of being able to fit all of the numerical values for the second virial coefficient and its

first four derivatives, all of which are required for nonic (m = 9) interpolation, on one page. For

the same reason, the values for B∗(T∗) and its first four derivatives are presented only for the

last twenty-four temperature nodes for the interpolation, for both the second virial coefficient

Bc(T∗) and its quantum correction Bq(T∗).

The tabulated data given in tables 4.1 to 4.6 might be useful as a reference for researchers

when they generate their own tables for their computer programs using the methodology pre-

sented in this thesis. These tables can be constructed by the user and are expected to vary in

both size and storage space, based on the degree m polynomial interpolant which they choose,

as well as the specified error tolerance for the interpolations and the desired precision for the

final solutions.

Chapter

4.

Resu

lts

for

B∗ (T

∗ )and

its

Deriv

ativ

es

43

Table 4.1: Tabulated data for the classical second virial coefficient and its first four derivatives, using the Lennard-Jones 6-12 model,

interpolated using a nonic interpolant (m = 9).

T∗ Bc(T∗) B′

c(T∗) B′′

c (T∗) B(3)c (T∗) B

(4)c (T∗)

1.5859×100−7.5770649671212×10−1 9.9575958982251×10−1

−1.5002119968127×100 3.3295809452847×100−9.8090160634466×100

2.0482×100−4.1678501662758×10−1 5.4510205648732×10−1

−6.1933024679700×10−1 1.0293439768625×100−2.2686866615976×100

2.6377×100−1.7653500340634×10−1 3.0394768944427×10−1

−2.6458346322809×10−1 3.3359327687058×10−1−5.5660197427118×10−1

3.3404×100−1.3358280895660×10−2 1.7689059174696×10−1

−1.2135902914153×10−1 1.1909421153210×10−1−1.5421226693208×10−1

4.2279×100 1.0661394750451×10−1 1.0289766776797×10−1−5.6255690999810×10−2 4.3264229231723×10−2

−4.3730262044929×10−2

5.5643×100 2.0675174859272×10−1 5.4152442586005×10−2−2.3065393923806×10−2 1.3450630338694×10−2

−1.0244099261307×10−2

7.5042×100 2.8034564489710×10−1 2.6211285321585×10−2−8.7163646739440×10−3 3.7959124507214×10−3

−2.1379125736045×10−3

1.0320×101 3.2970006655030×10−1 1.1407475103606×10−2−3.0482915329031×10−3 9.8517103116864×10−4

−4.0535077438327×10−4

1.4439×101 3.5887380266421×10−1 4.1244584334658×10−3−9.7343654212116×10−4 2.3450559145372×10−4

−6.9996724844940×10−5

2.0525×101 3.7198751818992×10−1 8.6277197376693×10−4−2.7319177666771×10−4 5.0407394720257×10−5

−1.0924016023828×10−5

2.9656×101 3.7273099121269×10−1−3.9231485397914×10−4

−5.9705859406272×10−5 9.3452688348393×10−6−1.4969161177679×10−6

4.3575×101 3.6414969167503×10−1−7.2149032540526×10−4

−4.6646354759569×10−6 1.3167080903696×10−6−1.6928507673382×10−7

6.5211×101 3.4870590360012×10−1−6.7536654479036×10−4 4.8682222725935×10−6 5.8513741188906×10−8

−1.2822813540950×10−8

9.9709×101 3.2829606973800×10−1−5.1398846312545×10−4 4.0043469630654×10−6

−4.6031927041401×10−8 2.4190194595730×10−10

1.5736×102 3.0404106448254×10−1−3.4596768057874×10−4 2.0698973027715×10−6

−2.1843935758693×10−8 3.1257149689546×10−10

2.6862×102 2.7478883552673×10−1−2.0269060729994×10−4 7.9080390143547×10−7

−5.6103821108755×10−9 5.7325488087083×10−11

4.4364×102 2.4794767170699×10−1−1.1788480086835×10−4 2.9441175094071×10−7

−1.3419100413380×10−9 8.8806021384069×10−12

7.0826×102 2.2416448805908×10−1−6.9560094126444×10−5 1.1246034419503×10−7

−3.3148162616145×10−10 1.4199350073872×10−12

1.1146×103 2.0258805660272×10−1−4.1136713096169×10−5 4.3198523067264×10−8

−8.2548665910518×10−11 2.2912159102531×10−13

1.7404×103 1.8294085263883×10−1−2.4313889483576×10−5 1.6606887951042×10−8

−2.0602001609157×10−11 3.7092312460095×10−14

2.7080×103 1.6501118667219×10−1−1.4330295101255×10−5 6.3623979320724×10−9

−5.1221992657638×10−12 5.9798529573124×10−15

4.1940×103 1.4877763202060×10−1−8.4494143289073×10−6 2.4428358744867×10−9

−1.2788853147636×10−12 9.7019140294831×10−16

6.4794×103 1.3406332835360×10−1−4.9773310255547×10−6 9.3748296606395×10−10

−3.1937864411275×10−13 1.5757118110284×10−16

1.0000×104 1.2072270784154×10−1−2.9268101771440×10−6 3.5897586497377×10−10

−7.9562463582650×10−14 2.5524962732946×10−17

Chapter

4.

Resu

lts

for

B∗ (T

∗ )and

its

Deriv

ativ

es

44

Table 4.2: Tabulated data for the classical second virial coefficient and its first four derivatives, using the Buckingham and Corner

modified Buckingham model (α = 13.5 and β = 0.0), interpolated using a nonic interpolant (m = 9).

T∗ Bc(T∗) B′

c(T∗) B′′

c (T∗) B(3)c (T∗) B

(4)c (T∗)

1.8950×100−4.8335762118751×10−1 6.3051470700076×10−1

−7.8360829168924×10−1 1.4240190080524×100−3.4312157982458×100

2.4431×100−2.2610154222205×10−1 3.4848128063953×10−1

−3.3065406335271×10−1 4.5404280948364×10−1−8.2505706935959×10−1

3.1144×100−4.9129833722863×10−2 1.9878438305246×10−1

−1.4751174964869×10−1 1.5626519612627×10−1−2.1840162490945×10−1

3.8221×100 6.2009935413131×10−2 1.2368326357057×10−1−7.5300463168082×10−2 6.4448937820987×10−2

−7.2523082954831×10−2

4.9724×100 1.6681228970185×10−1 6.6638296553456×10−2−3.1913321888298×10−2 2.0914456170348×10−2

−1.7911238846617×10−2

6.6473×100 2.4570979889300×10−1 3.2861245997411×10−2−1.2371459527387×10−2 6.0959195059965×10−3

−3.8879041098070×10−3

9.0764×100 2.9952029209265×10−1 1.4548655142562×10−2−4.4243695614676×10−3 1.6260281793251×10−3

−7.6169921341445×10−4

1.2612×101 3.3176141881927×10−1 5.3745409970356×10−3−1.4481202027934×10−3 3.9839252806074×10−4

−1.3605833884486×10−4

1.7804×101 3.4656797716372×10−1 1.1803462506703×10−3−4.1910822014638×10−4 8.8511144558846×10−5

−2.2043721451721×10−5

2.5552×101 3.4773723315555×10−1−4.8255300835031×10−4

−9.5998449629434×10−5 1.7069647494800×10−5−3.1483910269394×10−6

3.7240×101 3.3845812014104×10−1−9.4731392806614×10−4

−9.5193207717154×10−6 2.5766043260861×10−6−3.7825842197230×10−7

5.5226×101 3.2133363519849×10−1−9.1240954001362×10−4 6.8748404369696×10−6 1.6181609459799×10−7

−3.2032276301449×10−8

8.3532×101 2.9839089669295×10−1−7.1307523821374×10−4 6.2699742945010×10−6

−7.6873749012210×10−8−6.1626889234317×10−11

1.2974×102 2.7097764101188×10−1−4.9517795412694×10−4 3.4545050350387×10−6

−4.2251784130216×10−8 6.8716133844333×10−10

2.1304×102 2.3869358536373×10−1−3.0676920285409×10−4 1.4642480158436×10−6

−1.2658500164588×10−8 1.5761046654429×10−10

3.5236×102 2.0617128841876×10−1−1.8025900820851×10−4 5.5884849668461×10−7

−3.1321591557528×10−9 2.5529083860038×10−11

5.5635×102 1.7788282140707×10−1−1.0809022044366×10−4 2.2251640202102×10−7

−8.2121561976008×10−10 4.4012336251292×10−12

8.6199×102 1.5245396266212×10−1−6.4836125540948×10−5 8.9383532964664×10−8

−2.1888223014705×10−10 7.7565076566939×10−13

1.3205×103 1.2960970516626×10−1−3.8741678099271×10−5 3.5959795312441×10−8

−5.8753217624011×10−11 1.3839881203416×10−13

2.0061×103 1.0924223740127×10−1−2.3039868034426×10−5 1.4464412703806×10−8

−1.5842470602313×10−11 2.4927879090325×10−14

3.0236×103 9.1292352877398×10−2−1.3653964316743×10−5 5.8298970185600×10−9

−4.3037898634021×10−12 4.5487526872695×10−15

4.5334×103 7.5550810014222×10−2−8.0439003868310×10−6 2.3453348613188×10−9

−1.1712494067930×10−12 8.3453934816546×10−16

6.7553×103 6.1927700073224×10−2−4.7188706878696×10−6 9.4497938917086×10−10

−3.2091282509493×10−13 1.5493361880671×10−16

1.0000×104 5.0273416779209×10−2−2.7585275023367×10−6 3.8203221019663×10−10

−8.8771797421975×10−14 2.9213187823347×10−17

Chapter

4.

Resu

lts

for

B∗ (T

∗ )and

its

Deriv

ativ

es

45

Table 4.3: Tabulated data for the classical second virial coefficient and its first four derivatives, using the Rice and Hirschfelder

modified Buckingham (exponential-six) model (α = 13.5 and β = 0.0), interpolated using a nonic interpolant (m = 9).

T∗ Bc(T∗) B′

c(T∗) B′′

c (T∗) B(3)c (T∗) B

(4)c (T∗)

1.9351×100−4.6452597068215×10−1 6.0023216955906×10−1

−7.2982423739445×10−1 1.2957868465635×100−3.0497539428540×100

2.4958×100−2.1407795945018×10−1 3.3147648783065×10−1

−3.0804285984513×10−1 4.1345214209639×10−1−7.3405889015386×10−1

3.1795×100−4.2551596961540×10−2 1.8923081865990×10−1

−1.3784045566258×10−1 1.4293860702030×10−1−1.9544668152613×10−1

3.9323×100 6.8925518675781×10−2 1.1547090963529×10−1−6.8619428157714×10−2 5.7074917107145×10−2

−6.2355473676216×10−2

5.1261×100 1.7007808938515×10−1 6.1692904225289×10−2−2.8880952423966×10−2 1.8378433896811×10−2

−1.5261150766316×10−2

6.8594×100 2.4534903531017×10−1 3.0126600218383×10−2−1.1141183526540×10−2 5.3345366157360×10−3

−3.2988138849368×10−3

9.3699×100 2.9599632877420×10−1 1.3120080305012×10−2−3.9627050342915×10−3 1.4183306448677×10−3

−6.4473608617355×10−4

1.3024×101 3.2561736315014×10−1 4.6591842175426×10−3−1.2847645230474×10−3 3.4572912421287×10−4

−1.1476035509269×10−4

1.8409×101 3.3827839777959×10−1 8.3320850448389×10−4−3.6337660593763×10−4 7.5775416941240×10−5

−1.8385135113095×10−5

2.6439×101 3.3762484662023×10−1−6.3486114428492×10−4

−7.8771391256480×10−5 1.4307873963955×10−5−2.5902365398452×10−6

3.8565×101 3.2684459269543×10−1−1.0033404075562×10−3

−4.7850433199074×10−6 2.0433489093945×10−6−3.0263674767762×10−7

5.7188×101 3.0857534316996×10−1−9.2350575771643×10−4 7.8838809804606×10−6 7.9139390759561×10−8

−2.3699151805776×10−8

8.6560×101 2.8478741008247×10−1−7.0513262461182×10−4 6.3418080041021×10−6

−8.4554272885955×10−8 5.7838113816398×10−10

1.3488×102 2.5673579053997×10−1−4.7996639973509×10−4 3.3472509116355×10−6

−4.0907388343931×10−8 6.7695616312880×10−10

2.2510×102 2.2342525668210×10−1−2.8759122295313×10−4 1.3419793728960×10−6

−1.1272624961777×10−8 1.3613249113675×10−10

3.6624×102 1.9248364744137×10−1−1.6947740665734×10−4 5.1757649801648×10−7

−2.8454156258747×10−9 2.2659656944735×10−11

5.7373×102 1.6544901301698×10−1−1.0147544992587×10−4 2.0603388381691×10−7

−7.4923536955368×10−10 3.9452958953327×10−12

8.8426×102 1.4120562975544×10−1−6.0817609139154×10−5 8.2588324292690×10−8

−1.9968230609940×10−10 6.9742435555064×10−13

1.3477×103 1.1952080815301×10−1−3.6435843762312×10−5 3.3247952786119×10−8

−5.3766135726757×10−11 1.2516052672583×10−13

2.0395×103 1.0014667338261×10−1−2.1782126806659×10−5 1.3382961877429×10−8

−1.4503982383821×10−11 2.2688646587949×10−14

3.0616×103 8.3023171031393×10−2−1.3027371776502×10−5 5.4477986926971×10−9

−3.9286636223036×10−12 4.1453649119278×10−15

4.5737×103 6.7926992650981×10−2−7.7157471758532×10−6 2.2543524805633×10−9

−1.0809087613527×10−12 7.5307072631498×10−16

6.7802×103 5.4984346163530×10−2−4.4810516369670×10−6 9.4496307384341×10−10

−3.1186755256187×10−13 1.4447392166415×10−16

1.0000×104 4.4185736504316×10−2−2.5158673631521×10−6 3.8807601035027×10−10

−9.1250649438875×10−14 2.9232686377135×10−17

Chapter

4.

Resu

lts

for

B∗ (T

∗ )and

its

Deriv

ativ

es

46

Table 4.4: Tabulated data for the quantum correction to the second virial coefficient and its first four derivatives, using the

Lennard-Jones 6-12 model, interpolated using a nonic interpolant (m = 9).

T∗ Bq(T∗) B′

q(T∗) B′′

q (T∗) B(3)q (T∗) B

(4)q (T∗)

9.9343×100 3.9653977714294×100−5.1119806289981×10−1 1.2309231709601×10−1

−4.3193062451235×10−2 1.9868552235608×10−2

1.3124×101 2.7902405545052×100−2.6474368854901×10−1 4.7125730419681×10−2

−1.2254951749524×10−2 4.1849234865213×10−3

1.7429×101 1.9681416406918×100−1.3737501821349×10−1 1.8054648931013×10−2

−3.4734630307962×10−3 8.7872853300671×10−4

2.3243×101 1.3914657180622×100−7.1437638157562×10−2 6.9268479965751×10−3

−9.8480408517979×10−4 1.8431498934793×10−4

3.1105×101 9.8562746144169×10−1−3.7214077099732×10−2 2.6605728715321×10−3

−2.7928084790984×10−4 3.8627274428230×10−5

4.1731×101 6.9957941666191×10−1−1.9430707886893×10−2 1.0241545749874×10−3

−7.9348354147838×10−5 8.1062576423391×10−6

5.6123×101 4.9718830024208×10−1−1.0156870597861×10−2 3.9449228489682×10−4

−2.2544188314738×10−5 1.6998559914392×10−6

7.5617×101 3.5380280413375×10−1−5.3161856799305×10−3 1.5211665580684×10−4

−6.4095976653315×10−6 3.5652998826471×10−7

1.0212×102 2.5180803181740×10−1−2.7806717083576×10−3 5.8555646635868×10−5

−1.8170802610572×10−6 7.4470995225846×10−8

1.3802×102 1.7947623330547×10−1−1.4572636945550×10−3 2.2590408079825×10−5

−5.1636981741062×10−7 1.5594559940372×10−8

1.8682×102 1.2795521349166×10−1−7.6354387036273×10−4 8.7078076606926×10−6

−1.4650825622567×10−7 3.2578792007264×10−9

2.5313×102 9.1269184953268×10−2−4.0019774842154×10−4 3.3566482522211×10−6

−4.1553836493087×10−8 6.8007968459529×10−10

3.4325×102 6.5132187858369×10−2−2.0983556689146×10−4 1.2941110320021×10−6

−1.1784367103603×10−8 1.4190271864633×10−10

4.6581×102 4.6493945226870×10−2−1.1003586894873×10−4 4.9884237158489×10−7

−3.3402522701932×10−9 2.9582599823718×10−11

6.3235×102 3.3208427952026×10−2−5.7743320402172×10−5 1.9243701268957×10−7

−9.4751584644912×10−10 6.1716882252028×10−12

8.5844×102 2.3738169980079×10−2−3.0338186945464×10−5 7.4348665332941×10−8

−2.6926149317950×10−10 1.2902155482381×10−12

1.1664×103 1.6963692670397×10−2−1.5926199438774×10−5 2.8682945281872×10−8

−7.6356229072269×10−11 2.6897385646038×10−13

1.5852×103 1.2126537733605×10−2−8.3637832076590×10−6 1.1069853316796×10−8

−2.1660512368959×10−11 5.6090610787661×10−14

2.1543×103 8.6732565201479×10−3−4.3958336886332×10−6 4.2766618805035×10−9

−6.1521053437158×10−12 1.1713315220880×10−14

2.9270×103 6.2075958820233×10−3−2.3129668558900×10−6 1.6547508779519×10−9

−1.7506881897054×10−12 2.4516515604347×10−15

3.9784×103 4.4425633234418×10−3−1.2166644984411×10−6 6.3991457527109×10−10

−4.9777807846604×10−13 5.1257176538951×10−16

5.4104×103 3.1784724325999×10−3−6.3954899406297×10−7 2.4718734138262×10−10

−1.4131356857448×10−13 1.0694823272863×10−16

7.3544×103 2.2758089620373×10−3−3.3664030932101×10−7 9.5667536935980×10−11

−4.0216549152824×10−14 2.2382068050581×10−17

1.0000×104 1.6293011679318×10−3−1.7713997412379×10−7 3.7005050397474×10−11

−1.1436114469344×10−14 4.6791935302025×10−18

Chapter

4.

Resu

lts

for

B∗ (T

∗ )and

its

Deriv

ativ

es

47

Table 4.5: Tabulated data for the quantum correction to the second virial coefficient and its first four derivatives, using the

Buckingham and Corner modified Buckingham model (α = 13.5 and β = 0.0), interpolated using a nonic interpolant (m = 9).

T∗ Bq(T∗) B′

q(T∗) B′′

q (T∗) B(3)q (T∗) B

(4)q (T∗)

1.2456×101 1.9054228373148×100−2.0771844192762×10−1 4.0818146779135×10−2

−1.1529808815410×10−2 4.2417149532369×10−3

1.6417×101 1.3154274304246×100−1.0632308362371×10−1 1.5573937054008×10−2

−3.2850696357375×10−3 9.0336619020696×10−4

2.1729×101 9.1032004678422×10−1−5.4462999047734×10−2 5.9379429547452×10−3

−9.3406334753481×10−4 1.9171524172441×10−4

2.8847×101 6.3199879307263×10−1−2.7956775892642×10−2 2.2663598579360×10−3

−2.6567180564087×10−4 4.0669726108081×10−5

3.8408×101 4.3998818321656×10−1−1.4371107667628×10−2 8.6473362911744×10−4

−7.5445623436123×10−5 8.6045767685531×10−6

5.1239×101 3.0734703552900×10−1−7.4102419717979×10−3 3.3046105277514×10−4

−2.1438396584174×10−5 1.8204538523339×10−6

6.8500×101 2.1523308768306×10−1−3.8312972292121×10−3 1.2635839898958×10−4

−6.0845329996266×10−6 3.8419093160629×10−7

9.1717×101 1.5105835397096×10−1−1.9888809822431×10−3 4.8433840075652×10−5

−1.7279142668423×10−6 8.1021853575417×10−8

1.2304×102 1.0605235191311×10−1−1.0355796315680×10−3 1.8596656495225×10−5

−4.9026454366382×10−7 1.7031227084007×10−8

1.6531×102 7.4381091619134×10−2−5.4089862941443×10−4 7.1663777472641×10−6

−1.3935387644646×10−7 3.5780025743142×10−9

2.2236×102 5.2030058548816×10−2−2.8309653137424×10−4 2.7742196614600×10−6

−3.9781179174263×10−8 7.5361250250708×10−10

2.9940×102 3.6232724011091×10−2−1.4810880319658×10−4 1.0775510129177×10−6

−1.1413206378717×10−8 1.5946963853890×10−10

4.0327×102 2.5097934560486×10−2−7.7326012968505×10−5 4.1948798570766×10−7

−3.2937093211536×10−9 3.4012724389680×10−11

5.4353×102 1.7266948829840×10−2−4.0149540183939×10−5 1.6288456971018×10−7

−9.5128611390545×10−10 7.2809345491114×10−12

7.3207×102 1.1811842127838×10−2−2.0754618068991×10−5 6.3136937024669×10−8

−2.7529326528852×10−10 1.5680610266201×10−12

9.8549×102 8.0313317173796×10−3−1.0668166813996×10−5 2.4363833856655×10−8

−7.9483924116614×10−11 3.3790157667731×10−13

1.3240×103 5.4384382911862×10−3−5.4697341813721×10−6 9.3948205653084×10−9

−2.2987930816553×10−11 7.3163423735795×10−14

1.7763×103 3.6650467630643×10−3−2.7943087713010×10−6 3.6129861975630×10−9

−6.6385186696840×10−12 1.5843047017741×10−14

2.3792×103 2.4588834567192×10−3−1.4236274487431×10−6 1.3873405091239×10−9

−1.9164614696125×10−12 3.4342495697032×10−15

3.1815×103 1.6420227943114×10−3−7.2349371494211×10−7 5.3215186930964×10−10

−5.5335226895338×10−13 7.4547741847363×10−16

4.2461×103 1.0915514371216×10−3−3.6704871219699×10−7 2.0417503635333×10−10

−1.6008671778584×10−13 1.6239908072230×10−16

5.6597×103 7.2128594757926×10−4−1.8558074961043×10−7 7.8198235533680×10−11

−4.6289608810678×10−14 3.5398520045708×10−17

7.5322×103 4.7363066212458×10−4−9.3541605090162×10−8 2.9922135491335×10−11

−1.3395761341842×10−14 7.7341319991164×10−18

1.0000×104 3.0919730944999×10−4−4.7080023241006×10−8 1.1469886132311×10−11

−3.8942736150612×10−15 1.7018506134245×10−18

Chapter

4.

Resu

lts

for

B∗ (T

∗ )and

its

Deriv

ativ

es

48

Table 4.6: Tabulated data for the quantum correction to the second virial coefficient and its first four derivatives, using the Rice and

Hirschfelder modified Buckingham (exponential-six) model (α = 13.5 and β = 0.0), interpolated using a nonic interpolant (m = 9).

T∗ Bq(T∗) B′

q(T∗) B′′

q (T∗) B(3)q (T∗) B

(4)q (T∗)

1.3920×101 1.9910885624200×100−1.9053680429934×10−1 3.2773082996347×10−2

−8.1375588659958×10−3 2.6396511738030×10−3

1.8414×101 1.3756334462553×100−9.8031017273921×10−2 1.2552664102098×10−2

−2.3217615308186×10−3 5.6138685717655×10−4

2.4432×101 9.5120652366190×10−1−5.0534012446031×10−2 4.8192729324982×10−3

−6.6404866363118×10−4 1.1966088259240×10−4

3.2509×101 6.5752394974659×10−1−2.6057006119030×10−2 1.8507894957783×10−3

−1.8993085934149×10−4 2.5494756650600×10−5

4.3348×101 4.5427965864971×10−1−1.3439405913836×10−2 7.1115387320489×10−4

−5.4354370578760×10−5 5.4342043684328×10−6

5.7895×101 3.1358947128009×10−1−6.9315375599062×10−3 2.7334968219470×10−4

−1.5562764759229×10−5 1.1588914962892×10−6

7.7418×101 2.1621531516473×10−1−3.5740942937824×10−3 1.0508685687281×10−4

−4.4578072955009×10−6 2.4728108782173×10−7

1.0364×102 1.4880938730178×10−1−1.8410254775755×10−3 4.0369381724468×10−5

−1.2760591664766×10−6 5.2729275013105×10−8

1.3880×102 1.0225692079134×10−1−9.4808106169589×10−4 1.5516105933128×10−5

−3.6568774481911×10−7 1.1262266621620×10−8

1.8593×102 7.0129096313541×10−2−4.8796301257270×10−4 5.9650258868406×10−6

−1.0488697983834×10−7 2.4088258168420×10−9

2.4902×102 4.7995461259877×10−2−2.5107489534260×10−4 2.2949824542616×10−6

−3.0134456224790×10−8 5.1650015255051×10−10

3.3364×102 3.2734962371830×10−2−1.2891890219860×10−4 8.8152137327905×10−7

−8.6459451733637×10−9 1.1061736448930×10−10

4.4669×102 2.2266207554801×10−2−6.6189428080784×10−5 3.3905673243558×10−7

−2.4872349157914×10−9 2.3782453138670×10−11

5.9783×102 1.5084650290064×10−2−3.3935980030727×10−5 1.3035934203769×10−7

−7.1589933678832×10−10 5.1199166887912×10−12

7.9928×102 1.0176994313926×10−2−1.7395905003449×10−5 5.0184147920426×10−8

−2.0666413903575×10−10 1.1069850727090×10−12

1.0682×103 6.8215328778710×10−3−8.9011053531281×10−6 1.9287893442276×10−8

−5.9649072001372×10−11 2.3946320066874×10−13

1.4263×103 4.5353092163320×10−3−4.5548574428197×10−6 7.4051707765574×10−9

−1.7230560766051×10−11 5.2013107573856×10−14

1.9005×103 2.9827971640231×10−3−2.3416722971220×10−6 2.8547721519632×10−9

−4.9824587226626×10−12 1.1379396511713×10−14

2.5282×103 1.9243614321582×10−3−1.2072870046075×10−6 1.1137134216547×10−9

−1.4421420800972×10−12 2.4714828238152×10−15

3.3554×103 1.2063081086605×10−3−6.1842046001871×10−7 4.4363466408784×10−10

−4.2834209047848×10−13 5.4059710552161×10−16

4.4364×103 7.2972738094339×10−4−3.0976429909368×10−7 1.7842302843782×10−10

−1.3286719339116×10−13 1.2482689333132×10−16

5.8432×103 4.2420795149732×10−4−1.4926416599057×10−7 7.0280362462850×10−11

−4.1810063746098×10−14 3.0445402327031×10−17

7.6689×103 2.3709084738858×10−4−6.8678829921274×10−8 2.6496568767863×10−11

−1.2797672570426×10−14 7.4555448697369×10−18

1.0000×104 1.2890037517233×10−4−3.0518168739997×10−8 9.6142047219172×10−12

−3.7817006152412×10−15 1.7852285013427×10−18


4.2 Quantum Correction to B∗(T∗) for H2 and N2

A comparison of the classical second virial coefficient to the second virial coefficient with the

quantum correction taken into account is illustrated graphically in figure 4.9. This investigation

is performed using the non-dimensional temperature domain of [10−1, 104], using the same

reasoning as described in the previous section. The results illustrated in figure 4.9 are presented

for the gases hydrogen (H2) and nitrogen (N2), using the Lennard-Jones 6-12 model. Similar

results have be obtained when the modified Buckingham models of Buckingham and Corner as

well as Rice and Hirschfelder were used. The values for the force constants (σ and ǫ) are unique

to the Lennard-Jones 6-12 model and taken as defined in the technical report by Svehla [32].

The light gas hydrogen has been chosen to illustrate the significance of the quantum correction

on the second virial coefficient, since the quantum effects are significant for lighter gases at low

temperatures. Nitrogen has been chosen to illustrate the effects of the quantum correction on

a heavier gas, and also due to its abundance in the earth’s atmosphere. It should be noted

that currently none of the major computational codes account for quantum corrections in their

equations of state used to model gaseous species, and use only the classical term in the second

virial coefficient.

100

101

102

103

104

−1

0

0.6

T*

B*(T

*)

8−

B*(T

*) for H

2 B

*(T

*) for N

2 B

c(T

*)

Figure 4.9: The significance of the quantum correction to the second virial coefficient.

It is evident through inspection of figure 4.9 that the quantum correction is not only impor-

tant at low temperatures but that it in fact trails into the results at higher temperatures. The


quantum correction is most influential in the non-dimensional temperature domain of T∗ ≤ 25,

which corresponds roughly to a temperature domain of T ≤ 1, 500 K for the specific case of

hydrogen gas, and T ≤ 1, 750 K for the specific case of nitrogen gas. In comparison to the

quantum correction for hydrogen gas, the quantum correction for nitrogen gas is much smaller

in magnitude and is indistinguishable graphically from the classical curve. This shows that

the quantum correction is sufficiently significant for the second virial coefficient and should be

taken into account at all temperatures for light gases. For relatively heavier gases the quantum

correction is not as significant, and the solution for the second virial coefficient tends towards

the classical curve as the gas becomes heavier. It should be noted that the results presented in

figure 4.9 agree well with similar trends that are found in literature, both graphically [8, 13, 17]

and in tabular form [4], for the quantum correction to the second virial coefficient.

Chapter 5

Concluding Remarks

This study has presented an accurate and efficient method for the evaluation of the second

temperature-dependent virial coefficient B(T ) and its derivatives, for gaseous species and prac-

tical intermolecular potential energy models, used in the virial equation of state for imperfect

gases. This methodology consists of the following primary developments:

• A very accurate numerical integration method has been presented for the computation

of the integrals associated with B(T ) and its derivatives. Gaussian quadrature has been

used in conjunction with the software package developed by Wolfram Research called

Mathematica, using as large of an n-point rule as necessary to obtain final solutions

accurate to as many significant digits as required.

• A new method of polynomial interpolation of arbitrary degree m has been presented for

the evaluation of B(T ) and its derivatives stored in tables as discrete data Ti, Bi(Ti),

B′i(Ti), . . . , B

(m−12 )

i (Ti), for i = 1, 2, 3, . . . , n for n nodes.

• The minimization of stored data for B(T ) and its derivatives in computer programs using

an adaptive grid to control the relative error of interpolated values and make it the same

for each selected segment of the adaptive grid.

It has been shown in this thesis that B(T ) and its derivatives can be computed easily not only

for the conventional classical Lennard-Jones 6-12 model but also for the modified Buckingham

models of Buckingham and Corner as well as Rice and Hirschfelder. This work has also shown

that the quantum correction to the second virial coefficient can be incorporated easily in ad-

ditional calculations and tabulations of B(T ) and its derivatives, and that this correction is a

51

Chapter 5. Concluding Remarks 52

significant adjustment to the classical second virial coefficient for lighter gases such as hydrogen

(H2).

The same methodology can be used to evaluate the third temperature-dependent virial

coefficient C(T ) and its derivatives, along with its quantum corrections, and construct tables

for storage in computer programs, in virtually the same manner as for B(T ) and its derivatives.

One difficulty is that the numerical evaluation of C(T ) and its derivatives involves triple integrals

and therefore requires the use of multi-dimensional numerical integration techniques.

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[2] Benedict, M., Webb, G. B., and Rubin, L. C. An Empirical Equation for Thermody-

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[3] Beth, E., and Uhlenbeck, G. E. The Quantum Theory of the Non-Ideal Gas. II.

Behaviour at Low Temperatures. Physica, 4(10), 915-924, 1937.

[4] Buckingham, R. A., and Corner, J. Tables of Second Virial and Low-Pressure Joule-

Thomson Coefficients for Intermolecular Potentials with Exponential Repulsion. Proc. R.

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[5] Corbin, N., Meath, W. J., and Allnatt, A. R. Second Virial Coefficients, Including

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[6] Dymond, J. H., and Smith, E. B. The Virial Coefficients of Gases; A Critical Compi-

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53

References 54

[10] Gordon, S., and McBride, B. J. Computer Program for Computation of Complex

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[17] Kim, S., and Henderson, D. Quantum Corrections to the Third and Fourth Virial

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Appendix A

Assessment of New Interpolation

Method

The high-degree polynomial interpolation method for tabulated data that was presented in

section 3.5 is illustrated and assessed in this appendix. The theory presented in section 3.5

is not repeated here. Instead, the new method of polynomial interpolation is compared to

conventional interpolation methods using Lagrange’s polynomials [19] and polynomial splines

[7] of the same degree, in order to illustrate and assess the advantages of numerical efficiency.

This includes the ease of using the new interpolant and the resulting reduction in the size of

the tabulated data that needs to be stored in computational codes.

A set of tabulated data is required for the comparisons and assessments of different inter-

polation methods. This tabulated data is generated by the analytical function

y(x) =1 + 8x

1 + (4x)4(A.1)

on the domain [0, 1], so that the function and its derivatives are readily and accurately available

for the assessments. The behaviour of this test function and its first four derivatives is illustrated

in figure A.1. The test function is smooth and has no discontinuities in the function itself or its

higher derivatives. However, the test function has a maximum, a point of inflection and strong

curvature in some places to make accurate interpolation challenging. The higher derivatives are

highly fluctuating, include several local maxima and minima, include several inflection points,

and their amplitudes are not constant or negligibly small. Most importantly, the test function

is continuously differentiable, providing the opportunity to assess the interpolation methods for

an unrestricted range of high-degree interpolants.

57

Appendix A. Assessment of New Interpolation Method 58

Figure A.1: Plots of the test function and its first four derivatives.

The interpolation of tabulated data is studied for both uniformly and non-uniformly dis-

tributed data. For the data yi = y(xi) distributed uniformly with xi (e.g., uniform grid), where

i = 1, 2, . . . , n for n discrete data pairs, the objective is to reduce the constant interval width

∆x = xi+1 − xi such that the maximum relative interpolation error given by

E =

∣

∣

∣

∣

yexact(x) − yinterpolant(x)

yexact(x)

∣

∣

∣

∣

× 100% (A.2)

at any location x in any interval over the entire domain does not exceed 10−9 %. Symbolically,

Emax[0,1] = 10−9 %. For the data yi = y(xi) distributed non-uniformly with xi (e.g., adaptive grid),

the objective is to determine the minimum number n of discrete data pairs, where i = 1, 2, . . . , n,

such that the maximum error at any location x in every interval between two adjacent nodes

equals 10−9 %, that is Emaxi=1 ,2 ,... ,n−1 = 10−9 %. This maximum allowable error ensures that the


reconstructed solutions represented by interpolation are accurate to at least eleven significant

digits. For the case of the adaptive grid, this maximizes each interval and significantly reduces

the storage of the data xi, y(xi), y′(xi), y′′(xi), . . . , for i = 1, 2, . . . , n.

The results of this investigation are summarized in table A.1, and they provide a comparison

and assessment of various polynomial interpolation methods of different degree m. Included

are interpolants that are linear (m = 1), cubic (m = 3), quintic (m = 5), septic (m = 7), and

so on and so forth, up to m = 17. The advantages of using the new method of interpolation

over conventional interpolation methods using Lagrange’s polynomials and polynomial splines

are clearly evident from a study of the data. Consider the specific case of linear interpolation

Table A.1: Numbers of nodes and tabulated data required for interpolation of tabulated data

by various polynomial interpolation methods of different degrees (m).

degree of interpolant interpolation methoduniform grid adaptive grid

nodes (n) data (s) nodes (n) data (s)

m = 1

Lagrange 91,564 274,691 59,577 178,730

spline 91,564 274,691 59,577 178,730

new interpolant 91,564 183,128 59,577 119,154

m = 3

Lagrange 873 4,362 528 2,637

spline 700 3,497 308 1,537

new interpolant 505 1,515 305 915

m = 5Lagrange 222 1,549 128 891

new interpolant 91 364 52 208

m = 7Lagrange 136 1,217 70 623


m = 9Lagrange 107 1,168 52 563


m = 11Lagrange 93 1,198 45 574


m = 13Lagrange 88 1,307 42 617


m = 15Lagrange 87 1,464 38 631


m = 17Lagrange 87 1,636 37 686



(m = 1) first, with Lagrange’s linear polynomial, a linear spline and the new method of linear

interpolation. All three interpolants require the same 91,564 nodes to linearly interpolate the

tabulated data using a uniform grid, and the same 59,577 nodes to linearly interpolate the

tabulated data using an adaptive grid. For simple linear interpolation the three interpolation

methods are obviously equivalent, and this results in exactly the same uniform and non-uniform

grids. The non-uniform grid results in fewer nodes, however, 59,577 instead of 91,564, which is

a saving of 35%.

Now consider the specific case of cubic interpolation (m = 3) with Lagrange’s cubic polyno-

mial, a cubic spline and the new method of cubic interpolation, as reported in table A.1. These

three methods required 873, 700 and 505 nodes, respectively, to interpolate the tabulated data

over the entire domain using a uniform grid. For an adaptive grid, only 528, 308 and 305 nodes

are required, indicating savings of 40%, 56% and 40%, respectively. These trends are similar

for the higher-degree interpolants with m = 5 , 7 , . . . , 17. Programs for higher-degree spline

interpolations with m > 3 were not readily available, so data for these splines are not included.

It is worthwhile to consider the forms in which the three interpolants are used in this

appendix. For illustration purposes only cubic polynomials are presented for comparison. The

conventional interpolant of Lagrange can be expressed alternatively in an equivalent and shorter

form as

y = yi−1 +yi−yi−1

(xi−xi−1) (xi−xi+1) (xi−xi+2)(x−xi−1) (x−xi+1) (x−xi+2)

+yi+1−yi−1

(xi+1−xi−1) (xi+1−xi) (xi+1−xi+2)(x−xi−1) (x−xi) (x−xi+2)

+yi+2−yi−1

(xi+2−xi−1) (xi+2−xi) (xi+2−xi+1)(x−xi−1) (x−xi) (x−xi+1) (A.3)

for more efficient interpolation calculations. The three coefficients in equation A.3, given in

fractional form, need to be determined. These three coefficients are precomputed easily as a

bi, ci and di for each interval and stored along with xi and yi in tabulated form or arrays in a

computer code.

For interpolation using cubic splines, the conventional interpolant is expressed as

y = yi + bi (x − xi) + ci (x − xi)2 + di (x − xi)

3 , (A.4)

in which the coefficients bi, ci and di need to be determined simultaneously and numerically for

each interval between two adjacent data pairs. These coefficients are determined by matching

y(xi) and the derivatives y′(xi) and y′′(xi) at the nodes between adjacent splines, and using

two specifications for the data end conditions. For natural boundary conditions y′′(x) = 0 at


each end node; for clamped boundary conditions y′(x) is specified at each end node; for this

work y′′(x1) and y′′(xn) are determined at each data end from the cubic polynomial fitted to

the limit and last four data pairs, respectively. This latter approach is superior for engineering

and science applications. These three coefficients bi, ci and di are then stored in tabular form

or arrays along with xi and yi in a computer code.

For the new method of interpolation, the cubic interpolant can be expressed as

y =1

0!y∣

∣

i(1 − η)2 η0 {1 + 2η}∆x0

i +(−1)0

0!y∣

∣

i+1(1 − ξ)2 ξ0 {1 + 2ξ}∆x0

i

+1

1!

dy

dx

∣

∣

∣

∣

i

(1 − η)2 η1 {1}∆x1i +

(−1)1

1!

dy

dx

∣

∣

∣

∣

i+1

(1 − ξ)2 ξ1 {1}∆x1i , (A.5)

as derived in section 3.5. Only the data xi, yi and y′(xi) are stored in tabular form or arrays

in a computer code.

When a function is stored in tabular form (arrays) for interpolation in computational codes,

the focus of attention should shift from the total number of nodes used to the number of data

stored in the program. The storage space is usually minimized to reduce the program size. For

interpolation using Lagrange’s polynomials and polynomial splines, the number of stored data

is s = 2n + m(n− 1) for an interpolant of degree m using n nodes. Here, 2n storage spaces are

required to store the data pairs (xi, yi) for n nodes, and m(n − 1) storage spaces are required

to store m coefficients for (n − 1) node intervals. For interpolation using the new method, the

storage is considerably less, given by s = m+32 n. Here, 2n storage spaces are required to store

data pairs (xi, yi) for n nodes, and m−12 n storage spaces are required to store m−1

2 derivatives of

yi for n nodes, that is, dydx

∣

∣

i, d2y

dx2

∣

∣

i, . . . , d

m−12 y

dxm−1

2

∣

∣

i. Revisiting the specific case of cubic interpolation

(m = 3) in table A.1, interpolants based on Lagrange’s polynomials, cubic splines and the new

method of interpolation require 4,362, 3,497 and 1,515 computer storage spaces, respectively,

for a uniform grid. For an adaptive grid, the corresponding computer storage is 2,637, 1,537

and 915, respectively, yielding savings of 40%, 56% and 40%, respectively. The reduction in

storage for interpolations with higher-degree polynomials is obvious from the results in table

A.1.

Fewer nodes are required to interpolate the tabulated data when an adaptive grid is used in

place of a uniform grid. A uniform grid ensures that the maximum error amongst all intervals

in the entire domain is less than or equal to a specified error(

10−9 %)

, which normally occurs at

one location in the entire domain, whereas an adaptive grid maximizes the size of each interval

such that the maximum error in each interval equals the specified error(

10−9 %)

. The trend of

fewer nodes for a non-uniform grid is obvious in the results in table A.1. Furthermore, a direct

comparison of the number of nodes required to interpolate the tabulated data using various


interpolation methods illustrates that significantly fewer nodes are required to capture the test

function over the entire domain when the new method of interpolation is used, in comparison

to either of the other two interpolation methods. These results illustrate directly the reduction

in size and savings in storage of the tabulated data for the new method of interpolation. The

superiority of the new method of interpolation is due to its sophistication (storing derivatives

along with xi and yi). However, the method is applicable only when the derivatives of the

function are available.

The amount of data that needs to be stored in a computer program decreases substantially

for the new interpolation method when the degree m of the polynomial increases. From linear

(m = 1) to cubic (m = 3), cubic (m = 3) to quintic (m = 5), quintic (m = 5) to septic (m = 7),

and septic (m = 7) to nonic (m = 9), the data storage reduces from 119,154 to 915, 915 to 208,

208 to 115, 115 to 78, respectively, resulting in significant storage savings in computer programs.

Each increase in m has a savings corresponding to 12,922% (m = 1 to 3), 340% (m = 3 to 5),

81% (m = 5 to 7), and 47% (m = 7 to 9). The savings in data storage for polynomials of

degree m > 7 becomes less significant and not worth implementing. The gains are achieved

primarily by using the new interpolation method with a polynomial of degree m = 7 or m = 9.

By storing the exact derivatives for the new interpolation method, the interpolation accuracy

is improved, or the non-uniform spacing between data pairs can be increased. Interpolation

methods using Lagrange’s polynomials and cubic and higher-degree splines cannot reproduce

the derivatives exactly at the data nodes, resulting in a degraded accuracy for interpolations.

The test function along with the node locations using the new method of interpolation

on adaptive grids is presented in figure A.2. The purpose is to further illustrate graphically

the advantages of the new interpolation method and to encourage the use of higher-degree

interpolants on adaptive grids. In smooth regions of the test function with relatively small

curvature, such as when 0.5 ≤ x ≤ 1.0 in figure A.2, the grid spacing between two adjacent

nodes is large since the new interpolant can capture the data over a wide range, for any given

polynomial interpolant of degree m. The results in figure A.2 also illustrate that in the specific

case of cubic interpolation (m = 3) there are some relatively larger gaps, or ∆xi’s, in specific

locations in the domain. These corresponding locations have been denoted by the markers a, b

and c in figures A.1 and A.2. These markers have been positioned to show the exact location

where the fourth derivative of the test function is equal to zero(

d4ydx4 = 0

)

. These results

explain that a cubic interpolant, whose fourth derivative is exactly zero, can capture regions

of the function nearly exactly at the locations where its fourth and higher derivatives are very

small or zero. Similar patterns should occur when higher-degree interpolants are used, except

that they are simply not as evident graphically due to the inherently larger grid spacing under


Figure A.2: Test function and illustration of adaptive grid variation for new interpolation

method for m = 3 , 5 , . . . , 17.

such conditions.

Although the use of the new method of interpolation results in significantly lower storage

requirements in computational codes when higher-degree interpolants are used, the equation

for the interpolant can increase in length and complexity. For example, for septic (m = 7)

interpolation between two adjacent data pairs, the interpolant is given by

y =1

0!y∣

∣

i(1−η)

4η0

{

1+4η+10η2+20η3}

∆x0i

+(−1)0

0!y∣

∣

i+1(1−ξ)

4ξ0

{

1+4ξ+10ξ2+20ξ3}

∆x0i

+1

1!

dy

dx

∣

∣

∣

∣

i

(1−η)4η1

{

1+4η+10η2}

∆x1i

+(−1)1

1!

dy

dx

∣

∣

∣

∣

i+1

(1−ξ)4ξ1

{

1+4ξ+10ξ2}

∆x1i

+1

2!

d2y

dx2

∣

∣

∣

∣

i

(1−η)4η2 {1+4η}∆x2

i+

(−1)2

2!

d2y

dx2

∣

∣

∣

∣

i+1

(1−ξ)4ξ2 {1+4ξ}∆x2

i

+1

3!

d3y

dx3

∣

∣

∣

∣

i

(1−η)4η3 {1}∆x3

i+

(−1)3

3!

d3y

dx3

∣

∣

∣

∣

i+1

(1−ξ)4ξ3 {1}∆x3

i, (A.6)

in full symbolic notation from the derivation given in section 3.5. The calculations of this

interpolant, however, can be simplified and minimized in computer codes. The procedure can

be automated for an interpolant of arbitrary degree m by simply using a cleverly-written set of

nested loops.

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