# Chapter 5 Solution of Thermodynamics

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Chemical Engineering Thermodynamics Chapter 5 Solution of Thermodynamics: Theory and applications

Chapter Outline5.1 5.2 5.3 5.4 5.5 Fundamental Property Relation The Chemical Potential and Phase Equilibria Partial Properties of Solution Ideal Gas Mixture Fugacity and Fugacity Coefficient: Pure Species and Species in Mixture/Solution 5.6 Fugacity Coefficient of Gas Mixture from the Virial Equation of State 5.7 Ideal Solution and Excess Properties 5.8 Liquid Phase Properties from VLE data 5.9 Property Changes of Mixing 5.10 Heat Effects of Mixing Process

Multi-component gases and liquids commonly undergoes composition changes by separation and mixing processes. This chapter gives the thermodynamics applications of both gas mixtures and liquid solutions.

5.1 Fundamental Property RelationThe definition of the chemical potential of species i in the mixture of any closed system: xnG Qi | xni P ,T ,n

j

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

5.2 The Chemical Potential and Phase EquilibriaFor a closed system consists of 2 phase in equilibrium, the mass transfer between phases may occur. At the same P and T, the chemical potential of each species of multiple phases in equilibrium is the same for all species.

Qi ! Qi ! ....... ! QiE F

T

5.3 Partial Properties in SolutionPure-species properties,M i ! Vi , H i , S i ,i

In a mixture solution: 1) A solution properties, M ! V , H , S , G 2) The partial properties base on components in a solution, M ! V , H , S ,i i i i i

Partial molar property M i of species i in solution:Mi

x nM xn i

, ,n j

In a solution of liquids, its properties:! xi ii

! , , S, G M i ! Vi , H i , S i , Gi

For a binary solution, its properties:

M ! x1M1 x2M2Similarly, for separate x1 and x2;

M ! M1 x2 M1 M2

M ! x1M1 M2 M2

(See Example 11.3)

5.4 Ideal Gas MixtureIn a ideal gas mixture, partial molar properties of 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.ig

T , P !

ig i

T , p i

Partial pressure of a species i in ideal-gas mixture: yi R pi ! ig ! yi P V Hence, for enthalpy; Hig

! yi i

ig i

For entropy; S ig ! yi S iigi

R yi ln yi iintegration constant

For Gibbs energy;ig

G ! yi +i T RT yi ln yi P i i

5.5 Fugacity and Fugacity Coefficient: Pure Species and Species in Gas Mixture or Solution of LiquidsFor pure species in ideal-gas state;

Gi ! +i T RT ln Pig

For pure species in real-gas state;

Gi |

i

R

ln f i

fi R Gi Gi ! R ln ! Gi Pig

fi Ji ! P is called fugacity coefficient of pure species.where

For species i in a mixture of real gases or in a solution of liquids, in equilibrium;i E ! f i F ! .... ! f i T f

The fugacity of each species is the same in all phases. For vapor-liquid equilibrium, f i v ! fi l

For species in gas mixture or solution of liquids, Fugacity coefficient of species i in gas mixture; fi Ji ! yi P Fugacity coefficient of species i in solution; fi Ji ! xi P

iig ! 1 For species i in ideal-gas mixture, J

5.6 Fugacity Coefficient for Gas Mixture from the Virial Equation of Statek ! P Bkk 1 ln J RT 2i

yi y j H ik H ij 2 j

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

H ik | 2 Bik Bii Bkk H kk ! 0 H jj | 0 H ii | 0

H ij | 2 Bij Bii B jj

H ki | H ik , etc.

RTcij Bij Bij | Pcij

(Examples 11.7, B ! B [ B 11.8 & 11.9)0 ij ij ij

1

ij

Bij0 ! 0.083

0.4221 .6 rij

Bij1 ! 0.139

0.1724 .2 rij

[i [ j [ij ! 2

Tcij ! ciTcj kij T 11/ 2

Vci Vcj Vcij ! 2 1/ 3 1/ 3 cij

3

Zci Zcj Zcij ! 2

Z cij R Pcij ! Vcij

5.7 Ideal Solution and Excess PropertiesM is defined as the difference between theactual value of solution and value from ideal solution;E

M | M ME

id

5.8 Liquid Phase Properties from VLE dataIn a vapor which a gas mixture and a liquid solution coexist in vapor/liquid equilibrium, For species i in vapor mixture,

i v ! yiJiv P fSimilar for species i in solution,

i l ! yiJiv P f

In vapor-liquid equilibrium, vapor is assumed ideal gas, hence, iig ! 1 J

i l ! f i v ! yi P fThus, fugacity of species i (in both the liquid and vapor phases) is equal to the partial pressure of species i in the vapor phase.

f1 ! y1 P

f 2 ! y2 P

In an ideal solution,

i id ! xi f i fBy introducing a activity coefficient; fi fi Ki ! ! id xi f i f i

5.9 Property Changes of MixingThis is a mixing process for a binary system. The 2 pure species both at T and P initially separated by a partition, and then allow to mix.

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.

Property changes of mixing is given by;(M | M xi M ii

M ! V , H , S,G

Thus, the volume change of mixing, (V and the enthalpy change of mixing (H are found from the measured quantities (V and Q .t

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

For volume in binary system;(V t (V | V x1V1 x2V2 ! n1 n2

For enthalpy in binary system;(H | H x1 H1 x2 H 2 ! n1 n2

5.10 Heat Effects of MixingHeat of mixing per mole of solution;(H | H xi H ii

Solving for binary systems;H ! x1 H1 x2 H 2 (H

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

Heat of mixing are similar in many respect to 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 generally Much 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 of solute per mole of solution. Since, (H is the ~ heat effect of mixing per mole of solution, (H is the heat effect of mixing per mole of solute.~ (H (H ! x1

Mixing processes are presented by physicalchange equations, same like chemical-reaction equations. When 1 mol of LiCl(s) is mixed with 12 mol of H2O, the process;LiCl( s ) 12H 2 O(l ) p LiCl(12H 2 O)

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

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

Tutorial 5 Smith et al., (2006) Problem 11.19 Problem 11.37 Problem 12.32 Problem 12.46 Problem 12.59

Assignment 5 Smith et al., (2006) Problem 11.18 Problem 11.25 Problem 11.40 Problem 12.30 Problem 12.33