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The ratio of root mean square speed of two gases of molecular weights M1 and M2 at a temperature T will be - 
1. \(\sqrt{\dfrac{M_2}{M_1}}\)
2. \(\sqrt{\dfrac{M_1}{M_2}}\)
3. \(\dfrac{M_1}{M_2}\)
4. \(\left(\dfrac{M_2}{M_1}\right)^{1/3}\)

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Correct Answer - Option 1 : \(\sqrt{\dfrac{M_2}{M_1}}\)

CONCEPT

  • Root Mean Square Speed 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.
    • The RMS speed of any homogeneous gas sample is given by:

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

Where R = universal gas constant, T = temperature and M = molar mass

CALCULATION:

Let V1 be the RMS speed of gas of molecular weight M1;

V2 be the RMS speed of gas of molecular weight M2;

Now at a temperature, the ratio of speeds will  be 

\(\frac {V_1}{V_2} = \frac {\sqrt {\frac {3RT}{M_1}}}{\sqrt {\frac {3RT}{M_2}}}\)

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

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