*Cp - Cv = R*

Total energy of an isolated system is constant, for a gas which is in a container.

When heat energy ΔQ is supplied to the gas system and the system does a work of ΔW and the internal energy of the system is increased by ΔU. let ΔT be the change in temperature. let there be μ number of moles of the gas.

Heat supplied =

*ΔQ = ΔU + ΔW *

*Let the heat be supplied at constant volume*, So that the final temperature be T2 = T+ΔT. and the pressure be P2 = P1+ΔP. Since work done ΔW = P * ΔV = 0, as volume is constant and the gas does not expand or compress. The heat energy absorbed into increasing internal energy.

ΔQ = μ Cv ΔT = ΔU => **Cv = 1/μ* ΔU / ΔT**

We also know that internal energy U depends only on temperature T and changes only wrt to T.

*Let the heat be supplied at constant pressure P2*, to the gas. Let the gas expand volume V2 from V1. The temperature changes from T1 to T2. Heat supplied is

ΔQ = μ Cp ΔT = ΔU + ΔW -- at constant pressure

μ Cp = ΔU/ΔT at constant P + ΔW at constant P

= ΔU/ΔT + Δ(P V) / ΔT

= μ Cv + P ΔV / ΔT we know PV = μ R T

= μ Cv + P Δ( μRT / P) / ΔT

= μ Cv + P ( μR / P * ΔT/ΔT

Hence, μ Cp = μ Cv + μ R

Cp = Cv + R for any ideal gas.