Question

If average velocity becomes $4$ times then t-hat 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,
i.e., $v_{a v}=\left(v_{1}+v_{2}+v_{3}+\ldots\right) / N$
and according to kinetic theory of gases,
$v_{a v}=\sqrt{\frac{8 R T}{M \pi}}$... (i)
Also, rms speed (root mean square speed)
is defined as the square root of mean of squares of the speed of different molecules, i.e., $v_{r m s}=\sqrt{\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+\ldots\right) / N}$
$=\sqrt{(\bar{v})^{2}}$
and according to kinetic theory of gases,
$v_{r m s}=\sqrt{\frac{3 R T}{M}}$... (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 is also become $4$ times.