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6. ⇒ (MHT CET 2021 21th September Morning Shift )

The molar specific heats of an ideal gas at constant pressure and volume are denoted by ' $C_{p}$ ' and ' $C_{v}$ ' respectively. If $γ = \frac{C_{p}}{C_{v}}$ and ' $R$ ' is universal gas constant, then $C_{v}$ is equal to

A. $\frac{R}{γ - 1}$

B. $γ R$

C. $\frac{1 + γ}{1 - γ}$

D. $\frac{γ - 1}{R}$

Correct Option is (A)

$\begin{aligned} γ = \frac{C_{p}}{C_{v}}, Also C_{p} - C_{v} = R \\ C_{p} = γ C_{v} \\ ∴ & γ C_{v} - C_{v} = R \\ ∴ & C_{v} (γ - 1) = R \\ ∴ & C_{v} = \frac{R}{γ - 1} \end{aligned}$

7. ⇒ (MHT CET 2021 20th September Morning Shift )

Specific heats of an ideal gas at constant pressure and volume are denoted by $C_{p}$ and $C_{v}$ respectively. If $γ = \frac{C_{p}}{C_{v}}$ and $R$ it's the universal gas constant then $C_{v}$ is equal to

A. $\frac{(γ - 1)}{(γ + 1)}$

B. $\frac{(γ - 1)}{R}$

C. $R γ$

D. $\frac{R}{(γ - 1)}$

Correct Option is (D)

$\begin{aligned} γ = \frac{C_{p}}{C_{v}}, Also C_{p} - C_{v} = R \\ C_{p} = γ C_{v} \\ ∴ & γ C_{v} - C_{v} = R ∴ C_{v} (γ - 1) = R ∴ C_{v} = \frac{R}{γ - 1} \end{aligned}$

⇒Kinetic Theory of Gases