Q. Let $\gamma $ denote the ratio of specific heat for an ideal gas. Then choose the correct expression for number of degrees of freedom of a molecule of the same gas.
NTA AbhyasNTA Abhyas 2022
Solution:
For, $E$ = Total Energy, $f$ = Degree of freedom of molecule, We Know,
$C_{v}=$ Molar heat capacity at constant volume $=$ $\left(\frac{\partial U}{\partial T}\right)_{v}=\frac{f}{2}R$
$C_{p}$ = Molar heat capacity at constant pressure $=C_{v}+R=\frac{\left(\right. f + 2 \left.\right)}{2}R$
$\gamma $ = Ratio of specific heats $=\frac{C_{p}}{C_{v}}=\frac{f + 2}{f}$
$\Rightarrow f=\frac{2}{\gamma - 1}$
$C_{v}=$ Molar heat capacity at constant volume $=$ $\left(\frac{\partial U}{\partial T}\right)_{v}=\frac{f}{2}R$
$C_{p}$ = Molar heat capacity at constant pressure $=C_{v}+R=\frac{\left(\right. f + 2 \left.\right)}{2}R$
$\Rightarrow f=\frac{2}{\gamma - 1}$