Explanation Write the expression for the Gibbs-Duhem equation. ∑ i = 1 j n i d μ i = V d p − s d t (I) Here, chemical potential of species i in the mixture is μ i , number of the moles of species i in the mixture is n i , change in the pressure is d p , entropy of the mixture is s and change in the temperature is d t . Here, for the binary mixture value of j is 2 . Write the expression for the chemical potential at constant pressure and constant temperature. d μ i = ( d μ i d y i ) T , p , y i d y i + ( d μ i d p ) T , y + ( d μ i d T i ) d T (II) Here, mole fraction of the component i is y i . At constant pressure and temperature d T = d p = 0 Substitute 0 for d T , and 0 for d p in the Equation (II). d μ i = ( d μ i d y i ) T , p , y i d y i (III) Substitute 0 for d T , 0 for d p , and 2 for j d μ i in the Equation (I). ∑ i = 1 2 n i d μ i = 0 (IV) For the binary mixture, y 1 + y 2 = 1 y 1 = 1 − y 2 d y 1 = − d y 2 Write the expression for the difference between chemical potential i between a specified states of the multi-component system. μ i − μ i ∘ = R T ln ( f ¯ i f i ∘ ) μ i = μ i ∘ + R T ln ( f ¯ i f i ∘ ) (V) Here, chemical potential for reference mixture of species i is μ i ∘ . Substitute γ i y i for f ¯ i f i ∘ in the Equation (V). μ i = μ i ∘ + R T ln ( γ i y i ) μ i = μ i ∘ + R T ( ln ( γ i ) + ln ( y i ) ) (VI) Here, activity coefficient is γ i . Differentiate equation (VI) with respect to the y i . μ i d y i = R T [ ( d ln ( γ i ) d y i ) T , p + 1 y i ] d μ i = R T [ ( d ln ( γ i ) d y i ) T , p d y i + 1 y i d y i ] Substitute R T [ ( d ln ( γ i ) d y i ) T , p d y i + 1 y i d y i ] for d μ i in the Equation (IV)

Fundamentals of Engineering Thermodynamics

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ISBN: 9781118832301

Author: SHAPIRO

Publisher: JOHN WILEY+SONS,INC.-CONSIGNMENT

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Chapter 11.9, Problem 128P

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The following expression for a binary solution (i=1,2), relating the activity coefficient and at temperature T and pressure p, using Gibbs-Duhem equation.

(y1∂lnγ1∂y1)p,T=(y2∂lnγ2∂y2)p,T

The application of this expression.

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Chapter 11.9, Problem 127P

Chapter 12.9, Problem 1E

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Fundamentals of Engineering Thermodynamics

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The following expression for a binary solution ( i = 1 , 2 ) , relating the activity coefficient and at temperature T and pressure p , using Gibbs-Duhem equation. ( y 1 ∂ ln γ 1 ∂ y 1 ) p , T = ( y 2 ∂ ln γ 2 ∂ y 2 ) p , T The application of this expression.

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The following expression for a binary solution (i=1,2), relating the activity coefficient and at temperature T and pressure p, using Gibbs-Duhem equation. (y1∂lnγ1∂y1)p,T=(y2∂lnγ2∂y2)p,T The application of this expression.

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Chapter 11 Solutions

The following expression for a binary solution (i=1,2), relating the activity coefficient and at temperature T and pressure p, using Gibbs-Duhem equation.

(y1∂lnγ1∂y1)p,T=(y2∂lnγ2∂y2)p,T

The application of this expression.