Question

How many times a diatomic gas should be expanded adiabatically so as to reduce the root mean square velocity to half?

• A. $64$
• B. $32$
• C. $16$
• D. $8$

Answer 1

The correct option is B $32$

As, $V_{rms}=\sqrt{\frac{3RT}{M}}=\frac{(V_{rms}{)}_1}{(V_{rms}{)}_2}=\sqrt{\frac{T_1}{T_2}}$
Now, $T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$
$\frac{T_1}{T_2}={\left(\frac{V_2}{V_1}\right)}^{\gamma -1}$
Thus, $\frac{(V_{rms}{)}_1}{(V_{rms}{)}_2}={\left(\frac{V_2}{V_2}\right)}^{\frac{\gamma -1}{2}}$
$\Rightarrow \frac{v}{\frac{v}{2}}={\left(\frac{V_2}{V_1}\right)}^{\frac{\frac{7}{5}-1}{2}}$

$\Rightarrow 2={\left(\frac{V_2}{V_1}\right)}^{\frac{2}{5}\times \frac{1}{2}}$
$\Rightarrow 2={\left(\frac{V_2}{V_1}\right)}^{\frac{1}{5}}$
$\Rightarrow \left(\frac{V_2}{V_1}\right)=2^5=32$