During an adiabatic process, the density of a gas is found to be proportional to cube of temperature

Question

During an adiabatic process, the density of a gas is found to be proportional to cube of temperature. The degree of freedom of gas molecule is.....

Professor PC Thomas Classes · Answer

For adiabatic process, $\mathbf{T V}^{-\mathbf{1}}=$ constant
$$
\begin{aligned}
& T\left(\frac{m}{ho}ight)^{\gamma-1}=	ext { constant } \
& \frac{T}{ho^{\gamma-1}}=	ext { constant } \
ho \propto T^{y /(\gamma-1)} \Rightarrow & \frac{1}{\gamma-1}=3 \Rightarrow \gamma=4 / 3 \
f= & \frac{2}{\gamma-1}=\frac{2}{\left(\frac{4}{3}-1ight)}=6
\end{aligned}
$$