Partial molar volume of ideal gas and Gibb's theorem

[gfd43tg]

Hello,

I am working on the derivation that proves that the partial molar volume of an ideal gas is equal to the molar volume of an ideal gas.

I am following up to the point in the textbook where they set

(∂n/∂ni)nj = 1

where ni is the number of of moles of species i, and nj is the number of moles of species j ≠ i.

Attached is a photo of the derivation in the textbook

 

[gfd43tg]

 

[Chestermiller]

If n = n₁+n₂+n₃+n₄+n₅
and you take the partial derivative of n with respect to, say n₃ holding all the other subscripted n's constant, what do you get?

Chet