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Physics Question on kinetic theory

If ′Cp′'C_p' and ′Cv′'C_v' are molar specific heats of an ideal gas at constant pressure and volume respectively. If ′λ′'\lambda' is the ratio of two specific heats and ′R′'R' is universal gas constant then ′Cp′'C_p' is equal to

A

R γγ−1\frac{R\,\gamma}{\gamma-1}

B

γ R\gamma\,R

C

1+γ1−γ\frac{1 +\gamma}{1-\gamma}

D

Rγ−1\frac{R}{\gamma-1}

Answer

R γγ−1\frac{R\,\gamma}{\gamma-1}

Explanation

Solution

Given, that CpCV=γ\frac{C_{p}}{C_{V}}=\gamma...(i)
As we know that from Mayer's relation,
Cp−CV=RC_{p}-C_{V}=R
where, RR = universal gas constant
Substitute the value of CpC_{p} from the above relation in E (i), we get
γ=CpCp−R\gamma =\frac{C_{p}}{C_{p}-R}
γ(Cp−R)=Cp\gamma\left(C_{p}-R\right) =C_{p}
⇒γCp−Cp=γR\Rightarrow \gamma C_{p}-C_{p} =\gamma R
Cp(γ−1)=γRC_{p}(\gamma-1) =\gamma R
⇒Cp=γRγ−1\Rightarrow C_{p} =\frac{\gamma R}{\gamma-1}
Hence, CpC_{p} is equal to γRγ−1\frac{\gamma R}{\gamma-1}