Cp is the specific heat capacity at constant pressure and Cv is the specific heat capacity at constant volume.Consider one mole of an ideal gas in a closed cylinder with a movable piston.Suppose an amount of heat is given at constant volume so that the temperature increases by Δt
ΔQ=Cv × Δt
Here ΔW= 0 since volume is constant then ΔQ = ΔU let the gas be now heated at constant pressure then ΔQ' = Cp × Δt also ΔQ' = ΔU ×PΔv so from above equation we get ΔQ' = ΔQ+ PΔv combing above equations we get Cp× Δt= Cv × Δt PΔv By differentiating ideal gas equation for one mole we get PΔv =RΔt
substituting this value we get Cp-Cv= R