Solution Thermodynamics Lectures

7/28/2019 Solution Thermodynamics Lectures

1/74

BITS PilaniK K Birla Goa Campus

Solution

ThermodynamicsProf. Srinivas Krishnaswamy

Department of Chemical Engineering


2/74

BITS PilaniK K Birla Goa Campus

Chemical EngineeringThermodynamics (CHE C311)2011 - 2012


3/74

BITS Pilani, K K Birla Goa Campus

Thermodynamic Potentials (U, H, A, G)

Fundamental Relations (4)

Partial Derivatives and associated relations

Maxwell relations Departure functions or Residual properties

General expressions for du, dh, ds

Volume Expansivity and Isothermal Compressibility

General expressions for heat capacities

The Clausius Clapeyron equation

Estimating departure functions

Looking Back!!!!!


4/74


Revisiting Gibbs free energy

Gibbs free energy as a generating function

Chemical potential

Fugacity of pure species

Estimating Fugacity of pure species

Estimating Fugacity of component in a mixture

Mixing rules

Concept of partial molar property

Estimating partial molar properties Excess properties

Activity coefficient models

What next??


5/74


Gibbs Free Energy as aGenerating function

An alternative form of a fundamental property relation asdefined in dimensionless terms:

The Gibbs energy when given as a function of T and Ptherefore serves as a generating function for the otherthermodynamic properties, and implicitly represents completeinformation.

P

T

P

RTGT

RT

H

P

RTG

RT

V

dTRT

HdP

RT

V

RT

Gd

2


6/74


Fugacity and Fugacitycoefficient

Residual Gibbs energy:

G, Gig = the actual and ideal gas values of the Gibbs energy

at the same temperature and pressure

Residual volume:

igR GGG

1

ZP

RTV

PRTVVVV

R

igR


7/74BITS Pilani, K K Birla Goa Campus

Fugacity and Fugacitycoefficient (TB page 208 -209)

The fundamental property relation for residual properties

applies to fluids of constant composition

).(T

RT/GT

RT

H

).(P

RT/G

RT

V

).(dTRT

HdP

RT

V

RT

Gd

P

RR

T

RR

RRR

446

436

4262



Fugacity and Fugacitycoefficient (TB page 208 -209)

*Pressure Explicit form




Volume Explicit form

Like departure functions, 4 ways to estimate fugacity

and fugacity coefficient



Free Energy as Generating function

Beattie- Bridgman Equation

where Ao, Bo, C, a, b,c are constants




Fugacity from Generalized Equation of State



Fugacity from Z factor Correlation



ln = ln (f/P) = (Bo + B1)Pr/Tr

where Bo = 0.083 (0.422 / Tr1.6)

B1 = 0.139 (0.172 / Tr4.2)

Problem: Estimate the fugacity and fugacity coefficient for

ethylene using the Virial coefficient correlation

(0.9963 bar)

Fugacity coefficient throughvirial coefficient relation



1. Estimate the fugacity of carbon monoxide at 50 bar and

200 bar if the following data is given:

P (in bar) 25 50 100 200 400

Z 0.9890 0.9792 0.9741 1.0196 1.2482

2. Determine the fugacity and fugacity coefficient of n-

octane at 427.85 K and 0.215 Mpa using the Lee Kesler

data (0.2368 MPa)

Fugacity problems



Till date we have dealt with pure species (focus ongases)

But in most real situations, more than one species are

present resulting in a mixture

Methods discussed earlier need to be modified by

including an additional composition variable to predict

properties of real gas mixtures

Constants which appear in EOS which characterize purespecies behaviour can be averaged to get mixture

constants

Equations which express mixture constants in terms of

pure species constants are called MIXING RULES

Properties of real gas mixtures



A mixing rule expresses a mixture constant am in terms ofcomposition expressed in mole fraction yi and the pure

component constant ai, i.e. am = yiyjaij

aii

= ai,

ajj

= aj,

(based on interaction of like pairs of

molecules) and aij is based on interaction of unlike pairs

of molecules

Equations which provide the interaction constant in terms of

pure component constants are called combining rulesaij = (aii + ajj) / 2 = (ai + aj) / 2 ----- sqrt(aij ajj) = sqrt(aiaj)

am = yiai am = [ yisqrt(ai)]2

Mixing rules



Mixing Rules

For Cubic Equation of State (RK, SRK, PR)

For Beattie-Bridgman EOS

In the absence of

data, Kij can be set

equal to zero



For Benedict-Webb-Rubin For Lee-Kesler Methodrecommended by Knapp et al.

Kij ~1



For Virial EOS



Pseudo-critical constant method

Kays Rule

Prausnitz-Gunn Rule Joffes Relation


23/74


Rework this

problem using

the generalized

virial coefficient

correlation


24/74


Prediction of Departure functions for real gas mixtures

Make use of

Kays rule/

Modified

Prausnitz Gunn/

Joffes relation to

evaluate Tcm, m

In terms of Virial EOS


25/74



26/74


1. The Vander Waals constants for n-butane and n-octaneare

(1) n-butane a = 1.3874 Pa(m3/mol)2; b = 0.1163e- 3 m3/mol

(2) n-octane a = 3.7890 Pa(m3

/mol)2

; b = 0.237e- 3 m3

/molEstimate the Vander Waals constants for an equimolar mixture

of n-butane and n-octane

(am = 2.4405 Pa (m3/mol)2; bm = 0.1796e-3 m

3/mol

2. Estimate the enthalpy and entropy departures of an

equimolar mixture of n-butane and n-octane at 600K and 16

bar using the Vander Waals EOS

( 1.203 kJ/mol and 1.145 J/mol.K)

Practice problems


27/74


Fugacity and Fugacity Coefficient for real gas

mixture

Determine the fugacity and fugacity coefficient of an equimolar mixtureof n- butane and n-octane at 600K and 16 bar using virial coefficient

correlation


28/74


1. Determine the fugacity and fugacity coefficient for an

equimolar mixture of n-butane and n-octane at 600 K

and 16 bar using the van der Waals EOS (0.9018, 14.43

bar)

2. Rework the above problem using the pseudo-critical

constants method. Make use of Kays rule for pseudo-

critical constant and the Lee-Kesler data for the 3

parameter law of corresponding states (0.914, 14.62

bar)

3. Rework Problem 1 using the virial coefficient correlation

(1.104, 17.66 bar)

Estimation of fugacitycoefficient (Problems)


29/74


For a pure substance, the fugacity at constant T is defined

by dg = vdP = RTdlnf

g go = RTln(f/P)

Similarly in mixtures the partial molar fugacity of a

component i is defined as

dgi = dP = RTdlnfi (put a bar and toodle ^ on fi)

= RTln = RTln

The partial fugacity is not a partial molar property

Fugacity of a component in amixture


30/74


Fugacity of a component in a mixture


31/74


Estimate the partial fugacities of the equimolar mixture of n-butane (1) and n-octane (2) at 600 K and 16 bar. Make use of

the RK equation of state

Tc and Pc values for n butane are 425.2 K and 37.97 bar

Tc and Pc values for n butane are 569.4 K and 24.97 bar

1. Estimate the pure component constants based on RK EOS

2. Use the mixing rules to find constants for mixture. If Kij

is

not known, set it equal to zero

3. Estimate Z (In this problem take it equal to 0.8688)

4. The calculate the partial fugacities

Problem

C ff


32/74


By Virial Coefficient

Correlation


33/74


Problem


34/74


Fugacity of liquid and solid

If a liquid or solid is in equilibrium with its own vapor, then,

fv is

evaluated

and equatedto fl at T and

Psat and then

translated to

fl at required

T and P >

Psat

Poynting Pressure

Correction


35/74


The saturation pressure of n-octane at 427.85 K is 0.215

MPa. Estimate the fugacity of liquid n-octane at 427.85 K

and 1.0 MPa.

(0.2475 MPa)

This approximation is not valid for substance like acetic

acid which forms dimers in the vapour phase.

Problem


36/74


RECAP

(Finding real gas volume) / Departure

Functions / Fugacity

Equation of State

Compressibility factor correlations for EOS

Generalized factor correlations Virial coefficients

PURE

SUBSTANCE

(Finding real gas volume) / Departure

Functions / Fugacity

Equation of State

Compressibility factor correlations for EOS

Generalized factor correlations

Virial coefficients

MIXTURES(MIXING

RULES)

GAS / VAPOUR


37/74


Partial Molar Property

The partial molar property is defined as, where M is an extensive

property and represents the

change in the property M of a

mixture per mol of componenti when temperature, pressure

and mole numbers of all other

constituents are held constant

= i (Chemical Potential)

Solution properties, M

Partial properties,

Pure-species properties,

We use smaller

case letters


38/74


The following can be proven or

In general,

Partial Molar Property


39/74


Differentiating G, we get

(Gibbs-Duhem relation)

At constant T and P,

or


40/74



41/74


Method of Intercepts to evaluate partial molar

properties

From graph,

V1 = AB = AC BC

V2 = DF = DE + DF

P bl ti l l


42/74


1. In order to prepare 2 m3 of alcohol water solution,

alcohol of mole fraction X1 = 0.40 is required to be

mixed with water at 25 oC. Determine the volumes of

alcohol and water needed to prepare the mixture

Partial molar volume of alcohol: 38.3 10 6 m3/mol

Partial molar volume of water: 17.2 10 6 m3/mol

Molar volume of alcohol = 39.21 10 6 m3/mol

Molar volume of water = 18 10 6 m3/mol

[1.223 m3, 0.8424 m3]

Problems on partial molarproperty

P bl ti l l


43/74


In a binary liquid system, the enthalpy of species 1 and 2 at

constant temperature and pressure is represented by the

following equation

H = 400 x1 + 600 x2 + x1x2 (40x1 + 20 x2) where H is in J /mol

Determine expressions for partial molar enthalpies for

species 1 and 2 as function of x1 and the numerical values

of the pure species enthalpies H1 and H2

H1 (partial) = 420 60 x12 + 40 x1

3


P bl ti l l


44/74


1. The partial molar volume of water (1) in methanol (2) at

25 oC and 1 atm is approximated by 18.1 + ax22 where a = -

3.2 cm3/mol. Determine an expression for the partial molar

volume of methanol at the same condition given that V2 =

40.7 cm3/mol at 25 oC and 1 atm. Determine the partialmolar volume of methanol at infinite dilution.

2. The molar volume of a binary solution at 25 oC is given a

Calculate partial molar volumes of components 1 and 2 at

X1 = 0.5 and 0,75


X1 0 0.2 0.4 0.6 0.8 1.0

V 106 (m3/mol) 20 21.5 24 27.4 32 40


45/74


Useful intensive thermodynamic property

Partial derivative of H, A and G of the ith component in a

multicomponent system

Rate of increase in Gibbs free energy per mole ofcomponent i added

dGP,T = idni

GP,T = ini

The Chemical potential is thus seen to be the contribution

of that component to the Gibbs free energy of the

solution

Chemical Potential


46/74


Consider 2 phases a and b in equilibrium

GP,T = ini

Role of in equilibrium

At the same temperature

and pressure chemicalpotential or partial molar

free energy of a component

in every phase must be the

same under equilibrium

conditions

Effect of pressure on chemical


47/74


Effect of pressure on chemicalpotential

Effect of temperature on


48/74


Effect of temperature onchemical potential

Do it yourself

Starting with G = H TS prove that


49/74


Let us recap!!!

).(TRT/GT

RTH

).(P

RT/G

RT

V

).(dTRT

HdP

RT

V

RT

Gd

P

RR

T

RR

RRR

446

436

4262

Use lower case for g

P

R

T

R

T

fT

RT

H

P

f

RT

V

ln

ln

Mixture Fugacity


50/74


Let us recap!!!

oi

i

i

i

f

f

RTg

fRTd

lng

lngd

0

i

i

Fugacity of component in a mixture P

ii

o

P

ii

o

T

fT

RT

HH

T

f

TRT

HH

ln

ln

T

ii

P

f

RT

VV ln


51/74


Measure of the effective of a component

Results from interaction of molecules in non-ideal gas or

solution and is dimensionless

Depends on temperature, pressure and composition

The fugacity of component i as a pure substance atthe temperature and pressure is also the standard state

fugacity of component i

Activity and Activity coefficient

The numerator has

a toodle on top


52/74


The activity coefficient is the mixture is defined as theratio of activity ai of a component to its mole fraction xi in

the liquid mixture

Activity and Activity coefficient

The numerator for f

has a toodle on top

and not a bar

Temperature dependence on


53/74


Temperature dependence onactivity coefficient

Pressure dependency


54/74


Gibbs Duhems equation

At constant T and P,


55/74


1. The activity coefficient of component 1 in a binarysolution is represented by

Where a, b and c are independent of concentration. Obtain

an expression for 2 in terms of x1

2. For a binary system if the activity coefficient of

component is ln 1 = ax22, then derive the expression for

component 2

Problems


56/74


1. Estimate the entropy change due to mixing when 2.8 l of oxygen and19.6 L of hydrogen at 1 atm and 25 oC are mixed to prepare a

gaseous mixture

0.754 cal/mol.K

2. At 25 oC, 0.7 moles of helium are mixed with 0.3 moles of argon.

Calculate the free energy and enthalpy change of mixing. Assume

ideal gas behavior (- 1511.79 J and 0)

3. A 20 L vessel is divided into 2 compartments with the help of a

removable partition. The first compartment contains 12 L of

hydrogen and the second compartment contains 10 L of nitrogen.

Now the partition is withdrawn and gases allowed to mix

isothermally at 1 atm and 298 K. Estimate G, H and S of mixing.

Problems


57/74


Excess Property = Actual Property of mixture at given T

and P Property it would have as an ideal solution at

same temperature and mixture

ME = M Mideal

Excess Properties


58/74


Properties of solution

Lewis & Randall Rule

For an ideal gas mixture (solution)

(Amagats Law)

(Daltons Law)


59/74


Entropy change due to mixing

Entropy change due to mixing is given by

Phase equilibrium in Ideal


60/74


Phase equilibrium in Idealsolutions

At equilibrium,

But

(Raoults Law)


61/74


Raoults law: 2 unknowns 1 equation

Constraints

Antoine equation for

calculating Psat (t inCelsius and P in torr


62/74



63/74



64/74



65/74



66/74



67/74


Ph ilib i bl


68/74


Phase equilibrium problems


69/74


Fl h l l ti


70/74


Flash calculations

Rearranging Raoults law expression,

Ki is the equilibrium ratio or K

factor of component i

material balance around

the flash drum


71/74



72/74

Objecti e assessment


73/74


Revisiting Gibbs free energy

Gibbs free energy as a generating function

Chemical potential

Fugacity of pure species

Estimating Fugacity of pure species Estimating Fugacity of component in a mixture

Mixing rules

Concept of partial molar property

Estimating partial molar properties

Excess properties

Activity coefficient models

Objective assessment

Watch your thoughts they become yourwords


74/74

Watch your words they become your

actions

Watch your actions they become your

habits

Watch your habits they become your

character

Watch your character it becomes your

destiny

Solution Thermodynamics Lectures

Documents

Transcript of Solution Thermodynamics Lectures