Question

Root mean square velocity of molecules of a diatomic gas is to be increased to 1.5 times, If the process is done adiabtically, the ratio of initial volume to the final volumes is :

• A. 2.25
• B. 4.6
• C. 5.6
• D. 7.5

Answer 1

The correct option is D 7.5

1. RMS velocity of a diatomic gas $=\frac{\sqrt{3RT}}{M}$

for diatomic gas $=C_p=\frac{7}{2}$

$C_v=5/2$ $\gamma =\frac{C_p}{C_v}=7/5$

Let initial and final velocities be $v_i$ & $v_f$

RMS velocity initial $=\frac{\sqrt{3T_i}}{m}$ ___(1)

RMS velocity final $=\frac{3T_f}{M}$ ___(2)

(2)/(1)

$\sqrt{T_f}/T_i=1.5$

$T_f/T_i=(1.5{)}^2=2.25$

By adiabatic equation

$T_i{V}_i^{\gamma -1}=T_f{V}_f^{\gamma -1}$

$\frac{T_f}{T_i}=(\frac{V_i}{V_f}{)}^{\gamma -1}$ $\gamma -1=\frac{7}{5}-1=2/5$

$\therefore v_i(1-5{)}^2={\left(\frac{V_i}{V_f}\right)}^{2/5}$

$\therefore \frac{V_i}{V_f}=(1.5{)}^5=7.5$