Question

An ideal gas having initial pressure $P$, volume $V$ and temperature $T$ is allowed to expand adiabatically until its volume becomes $5.55V$ while its temperature falls to $\frac{T}{2}$. How many degrees of the freedom do the gas molecules have?

Answer 1

$T{V}^{\gamma -1}=$ constant (Adiabatic change)

$T_1{V}_1^{\gamma -1}=$$T_2{V}_2^{\gamma -1}$

$\frac{T_1}{T_2}=(\frac{V_2}{V_1}{)}^{\gamma -1}$

$T_2=\frac{T_1}{2}$

$V_2=5.55V_1$

$2=(5.55{)}^{\gamma -1}$

Taking log on both sides we get;

$\gamma -1=\frac{\log \left(2\right)}{\log \left(5.55\right)}=0.4$

$\gamma -1=\frac{2}{f}=0.4$ $(\gamma =1+\frac{2}{f})$

$\therefore f=5$