Question

Cp and Cv are the specific heats of $1$ gm mole of a gas constant pressure and constant volume respectively. The ratio of adiabatic and isothermal elasticities will be:

• A. Cp/Cv
• B. Cv/Cp
• C. CpCv
• D. (Cp - Cv) / Cv

Answer 1

The correct option is A Cp/Cv

Adiabatic elasticity $\left(K_{\varphi }\right)$: Elasticity under adiabatic conditions.

Under adiabatic conditions,

$P{V}^{\gamma }=constant$ where $\gamma =\frac{Cp}{Cv}$

$P\gamma V^{\gamma -1}+V^{\gamma }\frac{dp}{dV}=0$

$\Rightarrow \frac{P\gamma }{V}+\frac{dP}{dV}=0$

$\Rightarrow \gamma P=-\frac{dP}{dV/V}$ (Volume elasticity)

$\therefore K_{\varphi }={\gamma }_P$

Isothermal elasticity $\left(K_T\right)$: Elasticity under isothermal conditions (Temperature constant).

Under isothermal conditions,

$PV=constant$

$\Rightarrow P+V\frac{dP}{dV}=0$

$\Rightarrow P=-\frac{dP}{dV/V}$ (volume elasticity)

$\Rightarrow K_T=P$

So, $Ratio=\frac{K_{\varphi }}{K_T}=\frac{{\gamma }_P}{P}=\gamma =\frac{Cp}{Cv}$