Chapter 5 Solution of Thermodynamics:

Theory and applications

Chemical Engineering Thermodynamics

5.1 Fundamental Property Relation5.2 The Chemical Potential and Phase Equilibria5.3 Partial Properties of Solution 5.4 Ideal Gas Mixture5.5 Fugacity and Fugacity Coefficient: Pure Species and Species in Mixture/Solution5.6 Fugacity Coefficient of Gas Mixture from the Virial Equation of State5.7 Ideal Solution and Excess Properties5.8 Liquid Phase Properties from VLE data5.9 Property Changes of Mixing5.10 Heat Effects of Mixing Process

Chapter Outline

Multi-component gases and liquids commonly undergoes composition changes by separationand mixing processes.

This chapter gives the thermodynamics applications of both gas mixtures and liquid solutions.

5.1 Fundamental Property Relation

The definition of the chemical potential of species i in the mixture of any closed system:

jnTPi

i nnG

,,

the Gibbs energy which is the function of temperature, pressure and number of molesof the chemical species present.

5.2 The Chemical Potential and Phase Equilibria

For a closed system consists of 2 phase inequilibrium, the mass transfer between phases may occur.

At the same P and T, the chemical potential of each species of multiple phases in equilibriumis the same for all species.

iii

.......

5.3 Partial Properties in Solution

Partial molar property of species i in solution:

jnTPi

i nnM

M,,

iM

In a mixture solution:1) A solution properties, 2) The partial properties base on components in a solution,

iiiiiGSHVM ,,,

Pure-species properties, iiiiiGSHVM ,,,

GSHVM ,,,

In a solution of liquids, its properties:

ii iMxM

iiiiiGSHVM ,,,GSHVM ,,,

For a binary solution, its properties:

2211MxMxM

Similarly, for separate x1 and x2;

2121

MMxMM 2211

MMMxM

(See Example 11.3)

In a ideal gas mixture, partial molar propertiesof a species (except volume) is equal to its molar properties of the species as a pure ideal gas when the temperature is the mixture temperature and the pressure equal to its partial pressure in the mixture.

i

ig

i

ig pTMPTM ,,

5.4 Ideal Gas Mixture

Partial pressure of a species i in ideal-gas mixture:

PyVRTy

piig

i

i

Hence, for enthalpy;ig

ii i

ig HyH

For entropy; i ii

ig

ii i

ig yyRSyS ln

For Gibbs energy;

i iiii i

ig PyyRTTyG ln

integration constant

5.5 Fugacity and Fugacity Coefficient: Pure Species and Species in Gas Mixture or Solution of Liquids

PRTTGi

ig

iln

For pure species in ideal-gas state;

For pure species in real-gas state;

iii fRTTG ln

Pfi

iwhere

is called fugacity coefficient of pure species.

R

i

iig

iiG

Pf

RTGG ln

For species i in a mixture of real gases or in asolution of liquids, in equilibrium;

iiifff ˆ....ˆˆ

The fugacity of each species is the same in all phases.

For vapor-liquid equilibrium,

l

i

v

iff ˆˆ

Pyf

i

i

i

ˆˆ

Fugacity coefficient of species i in gas mixture;

For species in gas mixture or solution of liquids,

Pxf

i

i

i

ˆˆ

Fugacity coefficient of species i in solution;

For species i in ideal-gas mixture, 1ˆ ig

i

5.6 Fugacity Coefficient for Gas Mixture from the Virial Equation of State

j ijikjiikkkyyB

RTP 2

21ˆln

i, j, k are run over all species in gas mixture.

kkiiikikBBB 2 jjiiijij

BBB 2

0kk

0jj

0ii

etc.,ikki

cij

ijcij

ij P

BRTB

ˆ

6.1

0 422.0083.0

rij

ij TB

10ˆijijijijBBB

2.4

1 172.0139.0

rij

ij TB

2ji

ij

ijcjcicijkTTT 1

2/1

cij

cijcij

cij V

RTZP

2cjci

cij

ZZZ

33/13/1

2

cjci

cij

VVV

(Examples 11.7, 11.8 & 11.9)

5.7 Ideal Solution and Excess Properties

EM is defined as the difference between theactual value of solution and value from ideal solution;

idE MMM

5.8 Liquid Phase Properties from VLE data

In a vapor which a gas mixture and a liquid solution coexist in vapor/liquid equilibrium,

For species i in vapor mixture,

Similar for species i in solution,

Pyf v

ii

v

iˆ

Pyf v

ii

l

iˆ

In vapor-liquid equilibrium, vapor is assumed ideal gas, hence,

1ˆ ig

i

Pyffi

v

i

l

i ˆˆ

Thus, fugacity of species i (in both the liquidand vapor phases) is equal to the partialpressure of species i in the vapor phase.

Pyf11

Pyf22

In an ideal solution,

ii

id

ifxf ˆ

By introducing a activity coefficient;

id

i

i

ii

i

i f

ffxf

ˆ

ˆˆ

This is a mixing process for a binary system.The 2 pure species both atT and P initially separatedby a partition, and then allow to mix.

5.9 Property Changes of Mixing

As mixing occurs,expansion accompanied by movement of piston so that P is constant.

Heat is added or removed to maintain the constant T.

When mixing is completed,the volume changed as measured by piston displacement.

Thus, the volume change of mixing, and the enthalpy change of mixing are found from the measured quantities and .

Association with , is called the heat effect of mixing per mole of solution.

tV

VH

Q

Q

Property changes of mixing is given by;

ii

iMxMM GSHVM ,,,

H

212211 nn

VVxVxVV

t

For volume in binary system;

For enthalpy in binary system;

212211 nn

QHxHxHH

5.10 Heat Effects of Mixing

Heat of mixing per mole of solution;

ii

iHxHH

Solving for binary systems;

HHxHxH 2211

This equation provides the calculation of theenthalpies of binary mixture for pure species 1 and 2.

Heat of mixing are similar in many respectto heat of reaction. When a mixture is formed,energy change occurs because interaction between the force fields of the molecules.

However, the heat of mixing are generallyMuch smaller than heats of reaction.

Heats of Solution

When solids or gases are dissolved in liquids,the heat effect is called the heat of solution. This heat of solution is based on the dissolution of 1 mol of solute.

If species 1 is the solute, x1 is the moles ofsolute per mole of solution. Since, is the heat effect of mixing per mole of solution, is the heat effect of mixing per mole of solute.

HH~

1

~

x

HH

Mixing processes are presented by physical-change equations, same like chemical-reactionequations.

When 1 mol of LiCl(s) is mixed with 12 mol of H2O, the process;

O)LiCl(12H)O(12H)LiCl( 22 ls

LiCl(12H2O) means a solution of 1 mol of LiCl dissolved in 12 mol of H2O, giving heat effectof the process at 25°C and 1 bar; J 33,614

~ H

(Try Example 12.4, 12.5, 12.6, 12.7, 18.12.9)

Tutorial 5

Smith et al., (2006)

Problem 11.19Problem 11.37Problem 12.32Problem 12.46Problem 12.59

Assignment 5Smith et al., (2006)

Problem 11.18Problem 11.25Problem 11.40Problem 12.30Problem 12.33