Help with chemical potential formula in a mixture

[Tryhard_]

TL;DR Summary

Why does the usual proof for expression of chemical potential for pure substances not work when they are in a mixture ? where does it fail and need to change ?

Hello, I've been struggling with understanding how the molar fraction term appears in the expression of chemical potential as a fonction of pressure when the element is in a mixture,
here is the proof given in my textbook for the expression of chemical potential of a pure substance (pure gas)

My problem isn't with this proof but rather why it doesn't work the same when the element is in a mixture because every step seems to work the same when the element is in a mixture, and the partial molar volume for a gas or a liquid in a mixture is the same as when same liquid is pure (it's the change in volume of the system per change in moles of the element right ?)
for example for a gas in a mixture of two gases let's say we have n1 + n2 = ntot => V = ntot*RT/P = (n1 + n2)*RT/P => dV/dn1 = RT/p where is the mistake ?
the text book uses partial pressure and molar fraction when elements are in a mixture but never gives a reason as to why:

(this is the formula the textbook gives when a liquid is in a mixture)

Thanks a lot in advance for any help!

 

[sbrothy]

This is above my pay grade but since no one else seems to be answering... may I guess that the answer has to do with the thermodynamics of the mixture?

 

[Chestermiller]

TL;DR Summary: Why does the usual proof for expression of chemical potential for pure substances not work when they are in a mixture ? where does it fail and need to change ?

Hello, I've been struggling with understanding how the molar fraction term appears in the expression of chemical potential as a fonction of pressure when the element is in a mixture,
here is the proof given in my textbook for the expression of chemical potential of a pure substance (pure gas)
View attachment 360833

My problem isn't with this proof but rather why it doesn't work the same when the element is in a mixture because every step seems to work the same when the element is in a mixture, and the partial molar volume for a gas or a liquid in a mixture is the same as when same liquid is pure (it's the change in volume of the system per change in moles of the element right ?)
for example for a gas in a mixture of two gases let's say we have n1 + n2 = ntot => V = ntot*RT/P = (n1 + n2)*RT/P => dV/dn1 = RT/p where is the mistake ?
the text book uses partial pressure and molar fraction when elements are in a mixture but never gives a reason as to why:
View attachment 360834 (this is the formula the textbook gives when a liquid is in a mixture)

Thanks a lot in advance for any help!

This is only valid for an ideal gas mixture where the partial molar volume of each species in the mixture is equal to RT\P, where P is the total pressure.