Question

An ideal gas having initial pressure $P$, volume $V$ and temperature $T$ is allowed to expand adiabatically until its volume becomes $5.66V$ while its temperature fall to $\frac{T}{2}$. How many degrees of freedom do the gas molecules have?

Answer 1

Initial pressure $=P$

Volume$=V$

Temperature $=T$

To expand adiabatically until its volume$=5.66V$

Temperature$=\frac{T}{2}$

Now, determination of the degree of freedom do the gas molecules have

$T{V}^{\gamma -1}=\left(\frac{T}{2}\right)(5.66V{)}^{\gamma -1}$

using Laplace correction

Taking log on both side we get

$(\gamma -1)log5.66=log2(\gamma -1)\gamma log5.66-log5.66=log2\gamma -log1\gamma =1+\frac{0.3010}{0.7525}\gamma =1+0.4=1.4$