91Ó°ÊÓ

The rms speed is the root of the mean of the square of speed of molecules of the gas. Rms value does not represent the averageor the most possible value of speeds.Rms speed of a gas is given as,

${\mathbf{V}}_{\mathrm{rms}}\mathbf{=}\sqrt{\frac{3\mathrm{RT}}{m}}\mathbf{}$

Here,T is the temperature, m is the mass of the molecule of gas and R is the gas constant.

1. Temperature of hydrogen gas, $\text{T}=20°=293\text{‿é}$
2. The gases are hydrogen and helium
3. As per table 19.1, molar masses of hydrogen and helium are$2.02\text{ k²µ}/\text{mol}\&\text{ }4.0\text{kg}/\text{mol}$ respectively.

As we know that the rms speed of both molecules is equal

We can write,

$\sqrt{\frac{3RT}{M_{helium}}=}\sqrt{\frac{3RT}{M_{hydrogen}}}$

Squaring on both sides

$\begin{array}{rcl}\frac{3\text{ R°Õ}}{{\text{M}}_{helium}}& =& \frac{3\text{R}\left(293\text{‿é}\right)}{{\text{M}}_{hydrogen}}\\ \text{T}& =& \frac{\left(293\text{‿é}\right)\left({\text{M}}_{helium}\right)}{{\text{M}}_{hydrogen}}\\ \text{T}& =& \frac{\left(293K\right)(4.0\text{ k²µ}/\text{mol})}{2.02\text{ k²µ}/\text{mol}}\\ & & \begin{array}{c}\text{T}=580\text{ }\text{K}\\ =\text{ }307°\text{C}\end{array}\\ & & \end{array}$

The temperature of helium, at which the hydrogen and helium molecules have the same speed, is $307°\text{C}$.