3. APLICATION OF GIBBS-DUHEM EQUATION
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1 3. APLICATION OF GIBBS-DUHEM EQUATION Gibbs-Duhem Equation When extensive property of a solution is given by; Q' = Q'(T,P,n,n...) The change in extensive property with composition was; dq Q n dn Q n dn T, P, n,.. n j T, P, n,.. n j () Partial molar value of an extensive property of component i was defined as; Q Qi T, P, n n j i () i Combination of equations () and () yields; dq' = Q dn + Q dn +... (3) But, Q' = n Q + n Q + (4) Differentiation of equation (4) gives dq' = Q dn+ Q dn ndq + ndq +... (5) Subtraction of equation (3) from equation (5) yields ndq + ndq +... = 0 In general, ni dq i = 0 (6) Dividing by n, total number of moles of all the components, gives i dq i = 0 (7) Equations (6) and (7) are equivalent expressions of the Gibbs- Duhem equation.
2 3.. Integration of Gibbs-Duhem Equation It is performed to calculate partial properties from each other. Consider a component system, say Q is given. Q is asked. Gibbs-Duhem equation: dq + dq = 0 equivalents can be written for mixing (relative molar) and excess properties. Rearrangement of Gibbs-Duhem equation yields: dq dq Integration of both sides Q ( at ) Q ( at ) dq Integration yields: - Q ( at ) Q ( at ) Q ( at ) Q ( at ) - dq Q ( at ) Q ( at ) dq The value of Q must be known as boundary condition to integrate this function. Usually properties of pure substances are easier to find. Say property at = is known. Then let ' as = and " to any ; Q ( at ) Q o - Q ( at ) Q ( ) dq The value of integral may be determined graphically or analytically (if analytical dependence of Q to composition is given). Analytical determination involves the following steps: i. Organize Q as a function of only one composition variable ( or ). ii. Take the derrivative of Q and replace it into the integral. iii. Organize the function inside the integral in such a way to leave only one variable. At the same time, change, if and when necessary,
3 the limits of the integral to make it in accord with the derrivative of the integral. Use the relationship d = -d when necessary. iv. Integrate the function, replace the limits to determine the value of the integral. Analytical integration with relative partial and partial excess properties are also possible. The same steps can be used, by replacing Q with Q M and/or Q in the case of relative partial molar properties and partial excess properties respectively. Graphical determination requires plot of / vs. Q. The value of the integral is the area under the curve between the specified limits. However there are some problems with this integration: i. the value of Q becomes plus or -, if Q has logarithmic composition terms. ii. / becomes, when becomes zero. Therefore, the area under the curve may not be bounded well with the given limits. To determine the area under the curve properly; either alternative limits may be given or ways to resolve the problems of tails to infinity should be found. Furthermore, Gibbs- Duhem integration with partial molar properties can only be used for entropy and volume. For other properties Gibbs-Duhem integrations involving relative partial and/or excess properties have to be used. Then with relative partial molar properties, Gibbs-Duhem integration is (in general form): M M Q ( at ) Q ( at ) - Q M ( at ) Q M ( at ) d Q By setting the lower limit ' as = and " to any, Q M (at =) is zero, then Q M (at ) = - Q M ( at ) Q M ( ) d Q M M
4 Therefore, area under / vs. Q M curve gives the value of integral. Two of the above problems are still valid in this case (for some properties), but this form of the integral may be used with any thermodynamic property. To illustrate the problem, consider the following data for Fe-Ni alloys at 600oC. Ni ani G Ni M cal/mol Ni/Fe Ni / Fe -G Ni M The value of G NiM at lower limit (i.e. Fe=) is -, results in an unbounded area under the curve. Therefore, to get precise value for G M Fe (at Fe); either G M Fe at a composition other than Fe= should be given (known) or other alternatives should be considered. Use of excess properties: (in general form) Gibbs-Duhem integration with excess properties are: Q ( at ) Q ( at ) - Q ( at ) Q ( at ) dq By setting the lower limit ' as = and " to any, Q (at =) is zero, then
5 Q ( at ) - Q ( at ) Q ( at ) dq Area under / vs. Q curve gives the value of integral. One of the above problems is solved; by using excess properties, a finite value is assigned to the value of Q (at =). Therefore the area is bounded from the lower end of the integral. The problem of tail to infinity for / at =0 is still valid in this case. Then, previous problem for Fe-Ni alloys at 600oC requires integration of; Ni ani G Ni RT ln Ni cal/mol Ni/Fe log Ni log Ni G Fe ( at ) - Fe G ( at Ni Fe ) G ( at Ni Fe ) Ni Fe dg Ni Ni / Fe RT ln o Ni G Ni This is good for values Fe= to Fe > 0. As Fe 0; Ni/Fe. Therefore, GFe( at Fe 0 ) is mathematically indeterminate with excess properties.
6 The use of -Function: Problems arising from / as 0 may be resolved using this function. For any component i, i is defined as: i Qi ( ) i For - binary solution from previous relationships Q ( at ) - Q ( at ) Q ( at ) dq From above relationship Q Then dq d d Replacing into the integral yields Q ( at ) - ( at ) ( at ) d - d Q ( ) - at ( at ) ( at ) d - d By virtue of identity d(xy) = y dx + x dy, the first integral ( at ) ( at ) Then, Q d = d( ) d( ) - ( at ) - d( ) + d( ) - d = - + d + d - d = - - ( ) d = - - d
7 Numerical value for - can be determined, then the value of the integral can be determined graphically or analytically (if an analytical expression for the composition dependence of or other properties that leads to determine given). Analytical integration with -function can be done by replacing into the equation and integrating it between the given limits. Graphical determination can be done from the area under vs. graph. Then, previous problem for Fe-Ni alloys at 600oC requires determination of; G Fe ( at ) = - Fe Ni Fe Ni Fe - Fe Ni dfe Ni Ni 0/ ln Ni G Ni RT ln Ni cal/mol Ni/Fe log Ni log Ni Ni Fe
8 .5.5 For Fe = 0.7 ln Fe = -0.3 x 0.7 (-0.76) - (0.3 x x 0.3 x 0.5) = ln Fe = ; Fe = 0.985; afe = Direct Calculation of the Integral Property Consider a component system, say Q is given. Q is asked. Indirect method involves determination of Q by Gibbs-Duhem integration, then computation of the integral property from Q = Q + Q but Q = Q + ( - ) (dq/d) multiply both sides by d and divide by, then Q d Qd ( ) dq Replacing ( - ) = and d = - d Q d Qd dq d Q ( ) Integration of both sides
9 Q Q ( at ) d( Q ) = 0 Q d Q Q d Q o = 0 Using relative partial molar properties and setting QM at = to zero; M M Q Q d = 0 or Q M = 0 Q d M Equivalent relationship from partial excess properties and setting Q at = to zero; Q = 0 Q d Values of integrals can be obtained either analytically (if analytical dependence of Q or Q M or Q to composition is given) or graphically. Analytical integration involves the following steps: i. The replacement of the property (Q or Q M or Q ) into the integral. ii. Organization of the function inside the integral in such a way to leave only one variable. iii. Integration and replacement of the limits. Graphical integration involves the determination of the area under the curve (Q M or Q or Q vs. between the limits
10 G G RT ln RT ln G
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