91Ó°ÊÓ

The square root of the average of the squares of the speeds ofmoleculesis known as root-mean-square speed.

The root-mean-square speed is given as follows:

\({v_{{\rm{rms}}}} = \sqrt {\frac{{3RT}}{M}} \) … (i)

Here, R is the universal gas constant, T is the temperature, and M is the molecular mass.

The rms speed is directly proportional to the square root of temperature and inversely proportional to the square root of molecular mass.

\(v \propto \sqrt {\frac{T}{M}} \)

Since the temperature is the same for both the gas molecules,

\(\begin{aligned}{c}\frac{{{v_{\rm{1}}}}}{{{v_2}}} &= \frac{{\sqrt {\frac{T}{{{M_1}}}} }}{{\sqrt {\frac{T}{{{M_2}}}} }}\\\frac{{{v_{\rm{1}}}}}{{{v_2}}} &= \sqrt {\frac{{{M_2}}}{{{M_1}}}} \end{aligned}\)

Thus, it is proved that the ratio of rms speeds is equal to the inverse ratio of the square root of their molecular masses, \(\frac{{{v_1}}}{{{v_2}}} = \sqrt {\frac{{{M_2}}}{{{M_1}}}} \).