Derive a relation between cp and cv

Relationship between CP and CV for an Ideal Gas

From the equation q = n C ∆T, we can say:

At constant pressure P, we have qP = n CP∆T

This value is equal to the change in enthalpy, that is, q_P = n CP∆T = ∆H

Similarly, at constant volume V, we have q

V = n CV∆T

This value is equal to the change internal energy, that is, q

V = n CV∆T= ∆U

We know that for one mole (n=1) of ideal gas,

∆H = ∆U + ∆(pV )

= ∆U + ∆(RT )

= ∆U + R∆T

Therefore, ∆H = ∆U + R ∆T

Substituting the values of ∆H and ∆U from above in the former equation,

CP∆T = CV∆T + R ∆T

Or CP = CV + R

Or CP – CV= R