Question

Which of the following statement(s) is / are true?

• A. The ratio of the average speed to the rms speed is independent of the temperature
• B. The square of the average speed of the molecules is equal to the square of rms speed at a certain temperature
• C. Average kinetic energy of the gas molecules at any given temperature is independent of the average speed
• D. The difference between rms speed and average speed at any temperature for different gases diminishes as larger and yet larger molar masses are considered

Answer 1

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

The correct options are
A The ratio of the average speed to the rms speed is independent of the temperature
C Average kinetic energy of the gas molecules at any given temperature is independent of the average speed
D The difference between rms speed and average speed at any temperature for different gases diminishes as larger and yet larger molar masses are considered

i) We know that $V_{AV}=\sqrt{\left(\frac{8RT}{\pi M}\right)}{V}_{rms}=\sqrt{\left(\frac{3RT}{M}\right)}\frac{V_{AV}}{V_{rms}}=\frac{\sqrt{8/\pi }}{\sqrt{3}}$
$\therefore$ It is independent of temperature. So this is true.
ii) Here $\frac{8R{T}_{av}}{\pi M}=\frac{3R{T}_{rms}}{M}\frac{T_{av}}{T_{rms}}=\frac{3\pi }{8}$
$\therefore T_{av}$ isn't equal to $T_{rms}$.
So this is false.
iii) Mean kinetic energy is the $\frac{3KT}{2}=\frac{m{V}_{rms}^2}{2}$.
From the above formula we can conclude that Mean K.E. doesn’t depend on $V_{AV}$ rather depends on $V_{rms}$. Soit is true.
iv) $V_{AV}=\sqrt{\left(\frac{8RT}{\pi M}\right)}{V}_{rms}=\sqrt{\left(\frac{3RT}{M}\right)}$
As in both the equations the molecular weight is in denominator then the mean speed and rms speed decreases with increase in molecular weight