Specific heats of an ideal gas at constant pressure and volume are denoted by $C_p$ and $C_v$ respectively.…

Specific heats of an ideal gas at constant pressure and volume are denoted by $C_p$ and $C_v$ respectively. If $\gamma=\frac{C_p}{C_v}$ and $R$ is the universal gas constant then $\mathrm{C}_{\mathrm{v}}$ is equal to
  1. $\frac{(\gamma-1)}{(\gamma+1)}$
  2. $\frac{(\gamma-1)}{\mathrm{R}}$
  3. $\mathrm{R} \gamma$
  4. $\frac{\mathrm{R}}{(\gamma-1)}$

Solution

$\begin{aligned} & \gamma=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}} \text { and } \mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}=\mathrm{R} \\ & \mathrm{C}_{\mathrm{p}}=\gamma \mathrm{C}_{\mathrm{v}} \\ & \therefore \gamma \mathrm{C}_{\mathrm{v}}-\mathrm{C}_{\mathrm{v}}=\mathrm{R} \\ & \text { Or, } \mathrm{C}_{\mathrm{v}}=\frac{\mathrm{R}}{\gamma-1} \end{aligned}$

Asked in: MHT CET 2021 (21 Sep Shift 1)

Practice more Thermodynamics questions on Aicharya