Question

Which of the following statements is not true ?

• A. The ratio of the mean speed to the rms speed is independent of the temperature.
• B. The square of the mean speed of the molecules is equal to the square of rms speed at a certain temperature.
• C. Mean kinetic energy of the gas molecules at any given temperature is not independent of the mean speed.
• D. The difference between rms speed and mean speed at any temperature for different gases diminishes as larger and larger molar masses are considered.

Answer 1

The correct option is B The square of the mean speed of the molecules is equal to the square of rms speed at a certain temperature.

(i) $V_{\text{av}}=\sqrt{\frac{8RT}{\pi M}}$

$V_{\text{rms}}=\sqrt{\frac{3RT}{M}}$

$\frac{V_{\text{av}}}{V_{\text{rms}}}=\frac{\frac{\sqrt{8}}{\sqrt{\pi }}}{\sqrt{3}}$
$\therefore$ independent of temperature.
$\therefore$ True.

(ii) $\frac{8RT}{\pi M}=\frac{3R{T}^{\prime }}{M}$

$\frac{T}{T^{{}^{\prime }}}=\frac{3\pi }{8}$

$T\ne T^{{}^{\prime }}$;
$\therefore$ False.

(iii) Mean kinetic energy is $\frac{3}{2}RT$
$V_{\text{av}}=\sqrt{\frac{8RT}{\pi M}}$

$V_{\text{av}}^2=\frac{8RT}{\pi M}$

or $\frac{3}{2}RT=V_{\text{av}}^2\times \frac{3\pi M}{16}$
SO, we can see it is not independent of mean or avgerage speed. They are both related via temperature.

(iv) True. This is because molecular weight(M) occurs in the denominator of the expression. And as M increases $u_{rms}-u_{av}$ will become smaller.