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)

Ratio of $v_{av}/v_{rms}$ remains constant.
Average speed is the arithmetic mean of the speeds of molecules in a gas at a given temperature, ie,
$v_{av}=\left(v_1+v_2+v_3+\dots \right)/N$
and according to kinetic theory of gases,
$v_{av}=\sqrt{\frac{8RT}{M\pi }}...(i)$
Also, rms speed (root mean square speed) is defined as the square root of mean of squares of the speed of different molecules, ie,
$v_{rms}=\sqrt{\left(v_1^2+v_2^2+v_3^2+\dots \right)/N}$$<br/><br/>=\sqrt{(\bar{v}{)}^2}$
and according to kinetic theory of gases,
$v_{rms}=\sqrt{\frac{3RT}{M}}...(ii)$
From Eqs. (i) and (ii), we get
$v_{av}=\sqrt{\left(\frac{8}{3\pi }\right)}{v}_{rms}=0.92v_{rms}...(iii)$
Therefore, $\frac{v_{av}}{v_{rms}}=$ constant
Hence, root mean square-velocity is also become $4$ times.