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MHT CET · Physics · Thermodynamics

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. A \(\frac{(\gamma-1)}{(\gamma+1)}\)
  2. B \(\frac{(\gamma-1)}{\mathrm{R}}\)
  3. C \(\mathrm{R} \gamma\)
  4. D \(\frac{\mathrm{R}}{(\gamma-1)}\)
Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{\mathrm{R}}{(\gamma-1)}\)

Step-by-step Solution

Detailed explanation

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