Question

The equation of a certain gas can be written as $\frac{T^{7/5}}{P^{2/5}}=constant$. Its specific heat at constant volume will be :

• A. $\frac{3}{2}R$
• B. $\frac{5}{2}R$
• C. $\frac{7}{2}R$
• D. $2R$

Answer 1

The correct option is B $\frac{5}{2}R$

The equation of a certain gas can be written as

$\frac{T^{7/5}}{P^{2/5}}=const$

$P^{-2/5}.T^{7/5}=const$

$P.T^{\left(\frac{7/5}{-2/5}\right)}=const$

$P{T}^{-7/2}=const$. . . . . .(1)

Equation for any polytropic process,

$P{T}^{(\gamma /\gamma -1)}=const$. . . . .(2)

Compare equation (1) and equation (2), we get,

$\frac{\gamma }{\gamma -1}=-\frac{7}{2}$

$2\gamma =-7\gamma +7$

$5\gamma =7$

$\gamma =\frac{7}{5}$

The specific heat at constant volume will be given by

$C_V=\frac{R}{\gamma -1}$

$C_V=\frac{5R}{2}$

The correct option is B.