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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.

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