Question

A gas consisting of rigid diatomic molecules is expanded adiabatically. How many times has the gas to be expanded to reduce the r.m.s. velocity of the molecules to $\left(\frac{2}{3}\right)$ rd of initial value.

• A. $0.13$
• B. $7.6$
• C. $5$
• D. $3.8$

Answer 1

The correct option is A $0.13$

Let $v_{rms}$is the root mean square velocity of a diatomic gas having $\gamma =1.4$

$v_{rms}=\sqrt{\frac{3RT}{M}}$

$v_{rms}=\sqrt{T}$

To reduce rms velocity 2/3 times, temperature must reduce 4/9 times.

i.e. $T_2=\frac{9}{4}{T}_1$

for an adiabatic process,

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

${\left(\frac{V_2}{V_1}\right)}^{\gamma -1}=\frac{T_1}{T_2}=\frac{4}{9}$

$\frac{V_2}{V_1}={\left(\frac{4}{9}\right)}^{\frac{1}{\gamma -1}}={\left(\frac{4}{9}\right)}^{\frac{1}{1.4-1}}={\left(\frac{4}{9}\right)}^{2.5}$

$\frac{V_2}{V_1}=0.13$

$V_2=0.13V_1$

Hence, gas will expand 0.13 times.