8.10. Consider a Maxwell relation in amulticomponent system based on the identityaN2 ON1/ T,PON1 ON2/ T,Ps(a) For a binary mixture, show that this implies a relation in terms of the chemicalpotentials of the components and their dependence(b) Repeat part (a) for anon the mole fractions.arbitrary system of C components.

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Asked Jun 10, 2019
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This problem is (8.10) from a book  "Thermodynamics and Statistical Mechanics An Integrated Approach by M. Scott Shell"

8.10. Consider a Maxwell relation in a
multicomponent system based on the identity
aN2 ON1/ T,P
ON1 ON2/ T,P
s
(a) For a binary mixture, show that this implies a relation in terms of the chemical
potentials of the components and their dependence
(b) Repeat part (a) for an
on the mole fractions.
arbitrary system of C components.
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8.10. Consider a Maxwell relation in a multicomponent system based on the identity aN2 ON1/ T,P ON1 ON2/ T,P s (a) For a binary mixture, show that this implies a relation in terms of the chemical potentials of the components and their dependence (b) Repeat part (a) for an on the mole fractions. arbitrary system of C components.

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Expert Answer

Step 1

For a system of variable composition (open system), the differential change in molar Gibbs free energy for 1 mol of a mixture/solution is written as:

dG (sdT)+(vdP)+u^dx1 + /t2dx2
(1)
_
At constant T and P equation (1) is written as:
(2)
7 хр^m+ 1 xp11- гоР
Chemical potential of component i Partial molar Gibbs free energy of I
aGIx = G
(3)
7 хр^н + 1xp-1- гор
dGr[aG/ x1]12 dx, +[aG/ x]r,,1 dx
(4)
2
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dG (sdT)+(vdP)+u^dx1 + /t2dx2 (1) _ At constant T and P equation (1) is written as: (2) 7 хр^m+ 1 xp11- гоР Chemical potential of component i Partial molar Gibbs free energy of I aGIx = G (3) 7 хр^н + 1xp-1- гор dGr[aG/ x1]12 dx, +[aG/ x]r,,1 dx (4) 2

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Step 2

Applying Euler reciprocity relation to equation (4), we get,

dZ = MdxNdy
(ам /д)х %3D (ӘN /a)у
(аn dx )- (@uzdx,)2
(5)
Equation (5) shows the dependence of chemical potential on composition
Applying equation (3) for components 1 and 2
-[oG/TP
(6)
42oGITp.
(7)
Substitution (6) and (7) in (5), we get
(82GITP(2GI,
Т.Р
Он бк), р (ануом)т.Р
Answer: (4/x,)
Т.Р
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dZ = MdxNdy (ам /д)х %3D (ӘN /a)у (аn dx )- (@uzdx,)2 (5) Equation (5) shows the dependence of chemical potential on composition Applying equation (3) for components 1 and 2 -[oG/TP (6) 42oGITp. (7) Substitution (6) and (7) in (5), we get (82GITP(2GI, Т.Р Он бк), р (ануом)т.Р Answer: (4/x,) Т.Р

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Step 3

(b)  Equation (5) is modified and rearranged as follows:

...

о-р+ р
Typ-- Чp
-(аи дӑх, ) - (өм дӑк)
It implies that, 2
(7)
For a system of C number of components equation (5) may be written as:
(8)
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о-р+ р Typ-- Чp -(аи дӑх, ) - (өм дӑк) It implies that, 2 (7) For a system of C number of components equation (5) may be written as: (8)

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