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. 2 R

Answer 1

Answer: (A)

The equation of a certain gas can be written as
$\frac{ T ^{7 / 5}}{ P ^{2 / 5}}=$ const
$P ^{-2 / 5} \cdot T ^{7 / 5}=$ const
$P . T ^{\left(\frac{7 / 5}{-2 / 5}\right)}=$ const
$P T ^{-7 / 2}=$ const. $\ldots$ (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
$C _{ V }=\frac{ R }{\gamma- 1 }$
$C _{ V }=\frac{ 5 R }{ 2 }$