Question

If average velocity becomes $4$ times then what will be the effect on rms velocity at that temperature?

• A. 1.4 times
• B. 4 times
• C. 2 times
• D. $\frac{1}{4}$ times

Answer 1

Answer: (B)

4 times

Answer 2

Ratio of $v_{a v} / v_{r m s}$ remains constant. Average speed is the arithmetic mean of the speeds of molecules in a gas at a given temperature, ie, $v_{a v}=\left(v_{1}+v_{2}+v_{3}+\ldots . .\right) / N$ and according to kinetic theory of gases, $v_{a v}=\sqrt{\frac{8 R T}{M \pi}} \ldots(1)$ Also, rms speed (root mean square speed) is defined as the square root of mean of squares of the speeds of different molecules, ie, $v_{r m s}=\sqrt{\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+\ldots . .\right) / N}$ $=\sqrt{(\bar{v})^{2}}$ and according to kinetic theory of gases, $v_{r m s}=\sqrt{\frac{3 R T}{M}} \ldots$ (ii) From Eqs. (i) and (ii), we get $v_{a v}=\sqrt{\left(\frac{8}{3 \pi}\right)} v_{r m s}$ $=0.92 v_{r m s} \ldots$ (iii) Therefore, $\frac{v_{a v}}{v_{r m s}}=$ constant Hence, root mean square velocity also becomes $4$ times.