Question

If for two gases X and Y with molecular weight $M_X$ and $M_Y$, it is observed that at a certain temperature T, the average speed of X is equal to most probable speed of Y, then:

• A. the root mean square speed of X can be made equal to root mean square speed of Y by keeping X at $\frac{4}{\pi }$ times temperature of Y
• B. the average speed of X can be made equal to average speed of Y by keeping X at $\frac{4}{\pi }$ times temperature of Y
• C. the average speed of X can be made equal to root mean square speed of Y by keeping X at $\frac{3}{2}$ times temperature of Y
• D. the average speed of Y can be made equal to root mean square speed of X by keeping X at $\frac{32}{3{\pi }^2}$ times temperature of Y

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A the root mean square speed of X can be made equal to root mean square speed of Y by keeping X at $\frac{4}{\pi }$ times temperature of Y
B the average speed of X can be made equal to root mean square speed of Y by keeping X at $\frac{3}{2}$ times temperature of Y
C the average speed of Y can be made equal to root mean square speed of X by keeping X at $\frac{32}{3{\pi }^2}$ times temperature of Y
D the average speed of X can be made equal to average speed of Y by keeping X at $\frac{4}{\pi }$ times temperature of Y

Given
$\sqrt{\frac{8RT}{\pi M_X}}=\sqrt{\frac{2RT}{M_Y}}$
$=>\frac{8RT}{\pi M_X}=\frac{2RT}{M_Y}$
$\frac{8}{\pi M_X}=\frac{2}{M_Y}$
$=>M_Y=\frac{\pi M_X}{4}$
$\therefore U_{rms}=\sqrt{\frac{3RT}{M}};U_{mp}=\sqrt{\frac{RT}{M}};U_{ave}=\sqrt{\frac{8RT}{\pi M}}$