Thermodynamics

Physical Chemistry for Combustion Kineticists

•State of a System

•Specific Heat of a Molecule

•Enthalpy of Reaction

•Adiabatic Flame Temperature

•Tables

The aim of these lectures is to convey some basic thermodynamic principles, emphasizing those important for combustion.

Much of the basic thermodynamic principles that one encounters in combustion research can be found in most textbooks in physical chemistry, not to mention the introductory chapters in thermodynamics textbooks. These include the state of the system, specific heats and enthalpy which will be discussed here.

Some concepts, such as adiabatic flame temperature, are unique to combustion, but are direct applications of simple thermodynamic laws.

Some of these quantities are calculated, but also (as is mostly the case within combustion) can be found in tables.

Thermodynamics

Combustion is the generation of heat through Chemical Reactions

For example: The complete combustion of Methane

CH4 + 2O2 !" CO2 + H2O

Heat is generated

Why?

Where does the heat come from?

Thermodynamics quantifies these questionsA major thrust of combustion is the generation of heat and how to utilize that heat, and therefore an important topic within combustion is thermodynamics which quantifies all aspects of work and heat release that occurs during a combustion process.

An example is the complete combustion of methane.

We all know that heat is generated because of the reaction of methane in oxygen.

Why does the reaction to carbonmonoxide and water generate heat?

Where does the heat come from?

Thermodynamics quantitatively answers these questions.

First Law

!U = !Q! !WThe change in the internal energy of a thermodynamic system

!U

!W

!Q

is the amount of heat energy added to the system

minus the work done by the system on the surroundings

The first law of thermodynamics basically says that in going from one state to another, all energy has to be accounted for, conservation of energy principle.

Delta U is the change in energy of a system after the entire process. The di!erence between two states.

delta Q keeps track of all the heat release, for example during the combustion of methane in air, and heat loss, for example by absorption of heat by the walls of a container.

delta W is the work done on the outside world by the process, for example moving a piston. It is minus because energy is taken out.

In combustion, a system is, for example, the combustion of propane in air and the associated heat release. If the combustion process is within piston, for example, then some of the energy of the process is lost moving the piston.

First Law

The first law talks only of before and after

!U(This is the difference between initial and final states)

The total change in energy is independent of path!!!

!Qi + !Wi

There are many paths, i, leading to the same result

Reversible

The energy is not lost, just converted to different formsOne of the most useful aspects of the first law of thermodynamics is that it is a formulation that deals with only with the state before the process and the state after the process. It does not care how one gets there.

Just look at the initial and final states.

Each path, as long as the initial and final states are the same, each di!erent path, with a di!erent combination of changes in Q and W, gives the same total energy change.

Another way of saying this, is that the process is reversible.

No energy is lost. The total energy can always be accounted for.

Practical Advantage

A !" Z

A !" B !" Z

A !" B !" C !" Z

A !" C !" B !" Z

A !" .... !" Z

A !" B !" C !" B !" Z

ReversibleTotal energy is not lost or gained

As long as the initial and final states are the same

Choose the computationally simplest one

Does not have to correspond to realityAnother way to say this is that it deals with reversible processes. Anything you do to the system, you can undo.

No matter how you get from A to Z, no matter what path or intermediates you encounter, the total energy is always the same.

What this means computationally, we can use the simpliest computational path. Maybe we do not have all the information to calculate the energy di!erence between A and Z, but if we know how to get from A to B and then B to Z, then, using the first law of thermodyanamics we can compute the energy change from A to Z.

The first law says that it does even matter whether this corresponds to the actual path taken by the system. The answer will be the same.

Special/Typical Cases

Thermodynamics is extremely general

A large portion of combustion kinetics can be described using some special cases.

Q

molecular energy (reactions, with bond making and breaking)Thermal Conduction (Heat conduction by external objects)

WPV work (changes in pressure and volume)

Thermodynamics is extremely powerful general field and describes many more situations that is needed in combustion chemistry. For this reason we can simplify the possible sources of heat and the type of work, tuned especially what is relevant for special cases within combustion.

Heat, Q, is generated and used by the making and breaking of bonds within a chemical reaction. Within a combustion mechanism, each reaction has an associated amount of energy generated or used due to the making and breaking of bonds. Heat can also be transfered by thermal conduction. For zero dimensional calculations, this is from the external sink, for example the wall of the vessel at a constant temperature.

The work ,W, is restricted to mechanical PV work. Expansion/Contraction of gases and increase/decreases in pressure. A typical case is a piston.

Constant Volume

In constant volume, no (Pdv) work

Qv = !U = U1 ! U2

the change in energy is due exclusively to internal energy

A !" BU1 : Internal energy of molecule AU2 : Internal energy of molecule B

We do not need to know what molecular transformationshappened in going from A to B

We only look at the end states and their respective energiesIn constant volume, no PV work can be done, so the only change in energy between state one and state two is the change in internal energy.

Consider the reaction from species A to species B. Each has an inherent internal energy associated with it (in its atoms, bonds, etc.). Under constant volume, the energy change dues to the reaction is, Qv, is the same as the change in in internal energy, Delta U. No new variable (contrast with the definition of Enthalpy which follows). Note that we do not need to know exactly how the molecular transformations, for example the transition states, that occur in going from A to B. We just look at the beginning and end states

How is this useful?

Standard State Tabulations

A !" B

Only need to tabulate transition to standard state

A !" Standard State MoleculesB !" Standard State Molecules

B !" Standard State Molecules !" B

We made an artificial but calculable path from A to B

Often, we have no direct information about how to get from molecule A to molecule B. It would be impossible (or, at least impractical) to tabulation all possible combinations of reactions.

However, if we define a standard intermediate state of molecules, then we only need to tabulate the path in and out of this standard state.

We just calculate the energy of the path to the standard state from reactants.

And the energy of the path from the standard state to the products.

Although this is not the path that nature takes in going from A to B, it can be used to calculate the energy changes involved.

Energy of a Molecule

Energy of the mass of the components(electrons, protons and neutrons)

Energy to bring these components together into an atom(still more interesting for a physicist)

Energy to bring the atoms together into a molecule(this is interesting for the - combustion - chemist)

The energy within a molecule comes in many forms, starting with the most fundamental, which is the energy contained within the mass of the particles or even bringing these components together to form atoms. These are the main concerns of quantum chemists and physicists.

However, to a combustion chemist, the energy contained in the bonds of the molecule, which is a source and sink of heat during a chemical reaction, and the energy of the entire molecule moving through space is of importance. All the other sources of energy remain basically constant and thus can be ignored in the models.

Calorific Equation of State

Relating internal energy to temperature and pressurewe define two macroscopic properties:

u = u(T, V )h = h(T, P )

Thermodynamics is the study of change(We can really only measure the change between states)

How do these energies change?

To relate the internal energy to temperature, pressure and volume changes, we define two macroscopic variables for the calorific equations of state.

u dependent on T and Vandh dependent on T and P

Thermodynamics is fundamentally the study of change. Absolute macroscopic values can not be measured. We can only compare, see the di!erences (before and after). Think of the first law, dealing with the energy of the initial and final states.

A fundamental question in thermodynamics is how does the energy of a system change.

Calorific Equation of State

du = (!u

!T)vdT + (

!u

!V)T dV

dh = (!h

!T)P dT + (

!h

!P)T dP

u = u(T, V )h = h(T, P )

Full differentiation with respect to all dependent variables

To study the changes of energy, let us look how the (energetic) quantities, h, with respect to T and P, and u, with respect to T and V.

Looking at the di!erential with respect to the dependent variables.

h is dependent on T and P and dh is the di!erential with respect to dT and dP.u is dependent of T and V and du is the di!erential with respect to dT and dV.

Heat Capacity

dh = (!h

!T)P dT + (

!h

!P)T dP

cv ! (!u

!T)PcP ! (

!h

!T)P

Macroscopic quantity that gives how the internal energy of a molecule

changes with respect to temperature.

du = (!u

!T)vdT + (

!u

!V)T dV

Constant Pressure

The internal energy quantities, h and u, have not only a dependence on T but also on P and V, respectively. In a system with no PV work, i.e. P is constant, then we see that two useful parameters emerge, Cv and Cp.

These represent how internal energy of a molecule changes with respect to temperature. The molecules capacity to store heat (this quantity is often called heat capacity).

Heat Capacity

cP ! (!h

!T)P cv ! (

!u

!T)P

In the absence of PV workheat capacity

is the energy stored in a molecule(to raise the temperature)

In the absence of PV work, the heat capacity is the energy stored within the molecule, its ability to raise the temperature.

Energy Differences

The energy is 'seen' when there is a changefrom one state to another

Combustion is the study of utilizing the energyof the change from one set of molecules

to another set of a molecules

The absolute mass stays the sameThe nuclei stay the same

The configuration of the atoms changeInterested in the energy difference between these states.

Energy is seen when there is a change from one state to another. During a chemical reaction, the number of atoms does not change and all the energy captured in an atom below the valence shell remains constant. Only the bonding changes between the atoms.

To the combustion chemist the energy di!erence between the products and reactants of a chemical reaction is of primary importance.

Heat Release

In the complete combustion of methane

CH4 + 2O2 !" CO2 + H2O

State 1 : CH4 + 2O2

State 2 : CO2 + H2OThere is more energy in state 1 than state 2

so there is heat release

This is the basis of combustion

The two states of a chemical reaction is that of the reactants and that of the products. For example in the complete combustion of methane, we have the reactants, methane and oxygen and then we have the products. There is more molecular energy in the reactant state 1 than in the product state 2 so there is heat release. This is the basis of combustion.To calculate the heat release, from the first law of thermodynamics, we are only interested in the di!erence in internal energy between the products and the reactants. We need not be concerned of the complex transitional states that occurred to get from the bonding structure of the reactants to the bonding structure of the products.

EXAMPLES

Only Translational Energy

Noble Gas

He 12.5 1.5

Ne 12.5 1.5

Ar 12.5 1.5

Kr 12.5 1.5

Xe 12.5 1.5

CvCv

R

A monotonic molecule is basically a ball. The only energy it can contain is translational energy. A sphere cannot rotate (not matter how it rotates, it looks the same) and cannot vibrate.

Just an aside, why is Cv listed in this table and not Cp (which is more commonly used)? Remember that under constant volume, there is not PV work, so the change in energy between two states is purely due to the internal energy.

Diatomics

Using the same principle on diatomics

In addition to translation

Vibrational states: The atoms moving in and out along an axis

Rotational: The linear object twirling around

CV =3R

2(trans) + R(vib) + R(rot) = 3.5R

The same principle can be applied to diatomics. Diatomics have not only the translational degree of freedom, but also have additional degrees of freedom. There are two vibrational degree of freedom of the atoms moving along a line and there is there are two rotational degree of freedom of the two atoms. As described above, the diatomic molecule has a total of seven degrees of freedom each contributing R/2 to the heat capacity.The heat capacity is expected to be 3.5R.

Molecular energy

Velocity Distribution

One of the sources of energy that a molecule has is translational energy.

The concept of temperature is related to the the velocities of a large number of molecules through statistical mechanics. Looking at the whole system, the whole set of molecules have velocities ranging from very slow to very fast in a statistic distribution.

The higher the temperature, the average velocity increases and the peak of the distribution shifts towards higher velocites, as is to be expected.

Diatomic Examples

CvCv

RH2

CO

N2

Cl2

Diatomic

20.2 2.43

20.2 2.43

19.9 2.39

24.1 2.90

32.0 3.84Br2

Theoretically, if the diatomics had the full internal energy of the degrees of freedom, it would have a Cv of 3.5R. However, we can see here, that the lighter diatomics have considerably less than this value. Why is this?

Energy Distribution

Ep =12mv2

Each molecule with a specific velocity has a specific amount of energy. This graph shows the energy distribution with respect to temperature. As the temperature increases, the probability of the higher energies increases.

Translational Energy

4!(m

2!kT)3/2v2exp(

!2mv2

2kT)

Ep =12Mv2

p =RT

!

vp =!

2RT

!M

From the distribution of velocities

Derive the average velocity from the curve

Compute the average energy

Using Boltzmann statistics, the probability of a given velocity at a given temperature can be calculated (m is the mass of the molecule, and k is a constant).At a given temperature, we can integrate over v to get the average velocity.Substituting this into the velocity energy relationship, the average translational energy at a given temperature of a system of molecules can be computed.

Molecular Model

Even at the level of the chemical bonda molecule is a complex object a complex set of states

•vibrational

•rotational

•electronic

Even at the level of the chemical bond, a molecule is a complex object to model with its complex set of vibrational, rotational and electronic states.

However, these can be parameterized into simple quantities that capture the energy that is stored in these states. For example, the \emph{specific heat} of a molecule parameterizes how much chemical energy that a molecule with respect to temperature. At higher temperatures, the molecule vibrates and rotates faster, for example.

Energy in Molecule

IR Spectrum shows the

vibrational modes

•Two Major Modes

•bending: around 1300 cm-1

•stretching: around 3100cm-1

Just as translational energy is associated with a molecule moving in space, there is energy associated with the vibrational movement of a molecule (relative to itself in space). The each vibrational state has an energy associated with it. The methane molecule has two major modes associated with bending and stretching of the hydrogen relative to the carbon.

Vibrational and Rotational States

In addition to the vibrational states, there are energies associated with the rotation of the entire molecule. You can see that each vibration state has rotational states associated with it. The figure also shows that the vibrational modes has more energy associated with it, i.e. it is harder to vibration the molecule than it is to rotate it.

Electronic States

Associated with even more energy is the excitement of a molecule to an electronically excited state. Within each electronic state there are a set of vibrational and rotational states. Excited states play a role in combustion at high temperatures and in the transition state of a set of molecules between the reactants and products.

Degrees of Freedom

Under Classical Mechanics Heat capacity is dependent on the number of

degrees of freedom

For each degree of freedom:R

2

CP = 3R

2

Monotonics: Only 3 translational degrees of freedom(true for Noble gases)

Under the equipartition principle, each energy state is independent of the other. In other words, The energy of a molecule can be divided into translation, rotational vibrational and electronic energies, each with their corresponding on temperature.

Under classical mechanics, the heat capacity is basically dependent on the number of degrees of freedom a molecule has. Each degree of freedom contributes R/2 to the heat capacity.

For monotonic molecules (a molecule with only one atom) there is only three degrees of freedom, the three translational directions.

EXAMPLES

Only Translational Energy

Noble Gas

He 12.5 1.5

Ne 12.5 1.5

Ar 12.5 1.5

Kr 12.5 1.5

Xe 12.5 1.5

CvCv

R

A monotonic molecule is basically a ball. The only energy it can contain is translational energy. A sphere cannot rotate (not matter how it rotates, it looks the same) and cannot vibrate.

Just an aside, why is Cv listed in this table and not Cp (which is more commonly used)? Remember that under constant volume, there is not PV work, so the change in energy between two states is purely due to the internal energy.

Diatomics

Using the same principle on diatomics

In addition to translation

Vibrational states: The atoms moving in and out along an axis

Rotational: The linear object twirling around

CV =3R

2(trans) + R(vib) + R(rot) = 3.5R

The same principle can be applied to diatomics. Diatomics have not only the translational degree of freedom, but also have additional degrees of freedom. There are two vibrational degree of freedom of the atoms moving along a line and there is there are two rotational degree of freedom of the two atoms. As described above, the diatomic molecule has a total of seven degrees of freedom each contributing R/2 to the heat capacity.The heat capacity is expected to be 3.5R.

Diatomic Examples

CvCv

RH2

CO

N2

Cl2

Diatomic

20.2 2.43

20.2 2.43

19.9 2.39

24.1 2.90

32.0 3.84Br2

Theoretically, if the diatomics had the full internal energy of the degrees of freedom, it would have a Cv of 3.5R. However, we can see here, that the lighter diatomics have considerably less than this value. Why is this?

Temperature Dependence

For lightweight diatomics Only in the lowest vibrational state is filled

CV =3R

2(trans) + 0(vib) + R(rot) = 2.5R

For heavier compounds, quantum spacing lessThus begin to occupy higher vibrational states.

Can see a temperature dependence (Higher temperatures would occupy higher states)

Temperature Dependence

H2

O2

I2

H2O

H2O2

300K 500K 800K 1000K 1500K

6.9 7.0 7.1 7.2 7.7

7.0 7.4 8.1 8.3 8.7

8.8 8.9 9.0 9.9 11.2

8.0 8.4 9.2 9.9 11.2

10.1 12.0 13.8 14.7 16.2

This table shows two trends.

First, as the temperature increases, the internal energy (in this case Cp) increases. This shows that as the temperature increases, more and more vibrational and rotational levels are occupied, giving a higher level of degree of freedoms.

Secondly, as the molecule increases in complexity, it has more degrees of freedom. This means it has more internal energy to store.

Structure Dependence

The enthalpy of a molecule is dependent on structure.

•First Order

•Type of Bonds

•Second Order

•Steric Effects (cyclic, rotations)

For each non-hydrogen atom, there is an associated set of thermodynamic values. The values depend on the atom in question and what atoms it is connected to. One sums up the values for each non-hydrogen atom.

In the table, the heat of formation and the standard entropy are given along with several heat capacity values at di!erent temperatures. Together, one can calculate the necessary thermodynamic quantities at any temperature (or at least within a range).

Additive

The bonding effects are somewhat additive.

Origin of 'Benson Rules'

Associated with each atom bonded in a certain wayis a set of additive thermodynamic properties

Benson rules assume that the e!ect of structure on thermodynamics is additive. Each structure contributes a certain amount of the property.The structure in benson rules is a atom bonded in a certain way. Associated with each of these atoms is a set of thermodynamic properties to be added up.

Benson Rules

There is a rule for each type of atom in a molecule

•Atom Description

•Center atom (carbon, oxygen...)

•Bonding (list of bonded atoms)

•Thermodynamic Information

•Standard Enthalpy

•Standard Entropy

•Heat Capacity at a list of temperatures

The atom descriptions consist of a center atom, such as carbon, oxygen, etc., but not hydrogen and a list of atoms it is bonded to. Associated with each atom description is a standard enthalpy, standard entropy and a list of heat capacity values at a list of temperatures. The set of atom properties found within a molecule are added up to give the final estimate of the thermodynamic properties.

Hydrocarbon Groups

!Hf,298!Sf,298CP,300CP,500CP,1000CP,1500

C(C)(H)(H)(H)Primary Carbon

C(C)(C)(H)(H)Secondary Carbon

-10.20 -4.93

30.41 9.42

6.19 5.50

9.40 8.25

14.77 12.34

17.58 14.25

This is two typical benson rules for two common carbon types, a primary and a secondary carbon. The first atom listed is the center atom. The atoms is parentheses are the atoms this center atom is bonded to. With each rule are the standard enthalpy and entropy and a list of heat capacities.

Heptane

!Hf,298!Sf,298CP,300CP,500CP,1000CP,1500

C(C)(H)(H)(H)

2 Primary Carbons

C(C)(C)(H)(H)

5Secondary

Carbon

Total

-10.20 -4.93 -45.0530.41 9.42 107.926.19 5.50 39.889.40 8.25 60.0514.77 12.34 91.2417.58 14.25 106.41

CH3CH2CH2CH2CH2CH2CH3

Heptane is a linear molecule having seven carbon atoms.

The two end carbons, the primary carbons, are connected to 3 hydrogens and a carbon. This is denoted as C(C)(H)(H)(H) in the tables. The first symbol is the atom in question and the atoms in parentheses are the connected atoms. For heptane, the values in the benson table are multiplied by 2, i.e. for each of the two primary carbon atoms.

Each of the five carbons in the middle heptane, are connected to two carbons and two hydrogens each. This is denoted by C(C)(C)(H)(H). For heptane, the values in the table are multiplied by 5, i.e. for each of the 5 secondary carbon atoms.

The final added total gives the heat of formation, standard entropy and heat capacities at 300, 500, 1000 and 1500 for heptane.

NASA Polynomials

Common Format for kinetic mechanisms(for example, CHEMKIN)

de facto standard for combustion

Two regions (usually centered at 1000 Kelvin)

Represented by 7 coefficients

Another source of thermodynamic information, commonly used in kinetic programs is the NASA polynomial, i.e. a polynomial fit to the thermodynamic information. This is, for example, used with CHEMKIN and is a defacto standard for combustion.

Two temperature regions are considered in the NASA polynomial, the region below 1000K and the region above 1000K (actually, other dividing temperatures can be used, but 1000K is the most common). The polynomial approximation in each region has 7 coe!cients.

NASA Polynomials

S

R= a1lnT + a2T +

a3T 2

2+

a4T 3

3+

a5T 4

4+ a7

H

RT= a1 +

a2T

2+

a3T 2

3+

a4T 3

4+

a5T 4

5+

a6

T

Cp

R= a1 + a2T + a3T

2 + a4T3 + a5T

4

From the seven coe!cients the dimensionless quantities related to the temperature dependent heat capacity, enthalpy and entropy.

Tabulations: NIST

A valuable resource for thermodynamic and kinetic data is the NIST Webbook

http://webbook.nist.gov/National Institute of Standards (USA)

For a large set of compoundslists (among other things) online searchable

thermodynamic and kinetic data

On valuable source of thermodynamic information is the National (USA) Institute of Standards, acronym NIST.

If you look on their website, they have a ‘webbook’ of thermodynamic and kinetic information. In fact the values their values can be considered to be standard.

Tabulations: JANAF

Compilation of Thermodynamic data

JANAF Thermodynamic Tables

Editors:

M.W. Chase, Jr, C. A. Davies, I.R. Downey, Jr., D.J. Frurip, R.A. MacDonald, A.N. Suveryud

American Chemical Society, Washington D.C.1985

An older source in book form is the JANAF tables from 1985. This is actually a book (remember those?) and found in the library (remember that?).