Question

derive the relationship Cp - Cv = R.

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Answer 1

Answer:

Derivation of the equation Cp-Cv =R is explained below:

According to the first law of thermodynamics:ΔQ = ΔU + ΔW where, ΔQ is the amount of heat that is given to the system, ΔU is the change in internal energy and ΔW is the work done.We can write –ΔQ = ΔU + PΔV, as ΔW = PΔV Since ΔQ = nCpp_{p}ΔT and ΔU = nvv_{v}CΔT Therefore, nCpp_{p}ΔT = nCvv_{v}ΔT + PΔV ------------------(3)We know that PV = nRT  At T1 Kelvin:  PV₁ = nRT₁ ------------(a)  At T2 Kelvin: PV₂ = nRT₂ ------------(b)  Subtracting (a) from (b):PV₂ - PV₁ = nRT₂ - nRT₁  P(V₂ - V₁) = nR(T₂ - T₁) Where V₂ - V₁ = ΔV and T₂ - T₁ = ΔT Therefore, PΔV = nRΔTPutting the value of PΔV in equation (3):nCpp_{p}ΔT = nCvv_{v}ΔT + nRΔT nCpp_{p}ΔT = nΔT(Cvv_{v} + R)Cpp_{p} = Cvv_{v} + Ror Cpp_{p} - Cvv_{v} = R  The following relationship can be given considering the ideal gas behaviour of a gas. here R has been called universal gas constant