The volume of a diatomic gas ($\gamma$=7 / 5) is increased two times in a poly to opic process with molar heat capacity C=R. How many times will the rate of collision of molecules against the wall of the vessel be reduced as a result of this process?

In the process $p{V}^x=$ constant C= $\frac{K}{\gamma -1}+\frac{K}{1-x}$,

Given,

and $\gamma$=7 / 5

Substituting these values, we get $x=\frac{5}{3}$

Now, $p{V}^{5/3}$= constant

or

$p\propto \frac{1}{(V{)}^{5/3}}$

By increasing volume to two times pressure will decrease (2)^{5 / 3} times.

Now C= $\sqrt{\frac{3p}{\rho }}$=Rms speed .$\Rightarrow \frac{C"}{C}=\sqrt{\frac{p"}{p}\frac{\rho }{\rho "}}=\sqrt{\frac{1}{2^{{\displaystyle \frac{5}{3}}}}} \sqrt{2}=\frac{1}{2^{{\displaystyle \frac{1}{3}}}}$

Now p$\infty nC$  $\Rightarrow \frac{n"}{n}=\frac{p"}{p}\frac{C}{C"}=\frac{1}{2^{{\displaystyle \frac{4}{3}}}}$