91Ó°ÊÓ

The root-mean-square speed of the atoms of an ideal gas is proportional to the square root of the temperature of the gas.

The root-mean-square speed is expressed as-

$V_{ms}=\sqrt{\frac{3RT}{M}}$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦(1)

Here, $R$is gas constant, $M$is the molar mass of the gas, $T$is the absolute temperature.

Convert the equation In terms of $V_{ms}$

Squaring the equation (1),

${v^2}_{ms}=\frac{3RT}{M}$

Thus, the temperature becomes

$T=\frac{M×{v^2}_{ms}}{3R}$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.(2)

Increasing the RMS (root mean square) speed by two times, by multiplying the variable $v_{ms}$ by 2, and denoting the new temperature as $T_{new}$,

$$\begin{gathered} T_{new}=\frac{M×{\left(2v_{ms}\right)}^2}{3R} \\ =4×\frac{M×\left({v^2}_{ms}\right)}{3R} \end{gathered}$$

From (2), substituting $\frac{M×{v^2}_{ms}}{3R}=T$, we get

$T_{new}=4T$

Thus, the temperature needs to be set four times the initial value, to increase the value of RMS speed by two times.