Question

At a given temperature the ratio of root-mean-square velocities of two different gases of molecular weights M1 and M2 respectively is: 

1. \(\sqrt{\dfrac{M_1}{M_2}}\)
2. \(\sqrt{\dfrac{M_2}{M_1}}\)
3. \(\dfrac{M_1}{M_2}\)
4. \(\dfrac{M_2}{M_1}\)

Answer 1

Correct Answer - Option 2 : \(\sqrt{\dfrac{M_2}{M_1}}\)

Concept:

• Root Mean Square Speed: It is defined as the square root of the mean of squares of the speed of different molecules.
  • The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material.

From the expansion of pressure,

\(P = \frac{1}{3}{\rm{\rho v}}_{rms}^2\)

\({v_{rms}} = \sqrt {\frac{{3P}}{\rho }} \)

\(V_{rms}= \sqrt {\frac{{3PV}}{{Mass\;of\;gas}}} = \sqrt {\frac{{3RT}}{M}} \)

\([\because \rho = \frac{M}{V}]\)

⇒ vrms ∝ T 

Where, R = universal gas constant, M = molar mass, P = pressure due to density,ρ = density.

Explanation:

As, the two gases molecular weights M₁ and M₂, given,

we have the formula,

\(V_{rms}= \sqrt {\frac{{3PV}}{{Mass\;of\;gas}}} = \sqrt {\frac{{3RT}}{M}} \)

\( {V_{rms}} = \sqrt {\frac{{3RT}}{M}}\)

 

\( \therefore {V_{rms}}\alpha \frac{1}{M} \)

\( \therefore \frac{{{V_1}}}{{{V_2}}} = \sqrt {\frac{{{M_2}}}{{{M_1}}}}\)