Question

For two moles of gas proove Cp - Cv = R

Answer 1

Dear student your question is a little wrong, since for 2 mol, the relation will be Cp-Cv=2R

Here is the proof,

Cp - Cv = R
This is a standard relation for one mole. For two moles the difference will be = 2 R.

In general Cp - Cv = n R ---- (0)
============
We know PV = n R T, V = (nR/P) * T --- (1)
At constant pressure, ΔV/ΔT = nR/P --- (2)

From first law of thermodynamics, we have that
ΔQ = ΔU + W = ΔU + P ΔV ---- (3)
We know that the internal energy U and the change in it ΔU, are entirely dependent on the temperature for the same amount of substance. So it does not depend on the pressure or volume.

Cp = [ ΔQ/ΔT ]_P=const
= [ ΔU/ΔT ]_P=const + P * [ ΔV/ΔT ]_P=const
= ΔU/ΔT + P * n R/P = ΔU/ΔT + n R

Cv = [ ΔQ/ΔT ]_V=const
= [ ΔU/ΔT ]_V=const + P [ ΔV/ΔT ]_V=const
= ΔU/ΔT + 0

Hence Cp = Cv + n R