Question

At 400K, the root mean square (RMS) speed of a gas X (molecular weight = 40) is equal to the most probable speed of gas Y at 60 K. The molecular weight of the gas Y is__________ .

Answer 1

Hint: The formula to calculate the value of root mean square speed is given as ${{V}_{rms}}=\sqrt{\dfrac{3RT}{M}}$ and also here we have to calculate the value of most probable speed so the formula is ${{V}_{mps}}=\sqrt{\dfrac{2RT}{M}}$. Now equate both the speeds and then calculate the molecular weight of Y as asked in the question.

Complete answer:
From your chemistry lessons you have learned about the root mean square speed and the most probable speed of a gas.
Root mean square speed of a gas is defined as the root square of the squared average velocity of a gas molecule at a particular temperature. The formula through which we find the room mean square speed of a gas is
\[{{V}_{rms}}=\sqrt{\dfrac{3RT}{M}}\]
Where, T is the absolute temperature,
R is the gas constant and
M is the molar mass of the gas
Most probable speed is defined as the speed acquired by the maximum no. of gas molecules and the formula to find most probable speed is
\[{{V}_{mps}}=\sqrt{\dfrac{2RT}{M}}\]
Now let us take two gas one is X (rms speed of gas)and other is Y, so the Molecular mass and temperature of gas will be ${{M}_{X}},{{T}_{X}}$ and the molecular mass and temperature for gas B (mps of gas)will be ${{M}_{Y}},{{T}_{Y}}$.
So, in the question we have asked to equate both root mean square speed of a gas which is taken as 'X' gas and most probable speed which is 'Y' gas
So, ${{V}_{rms}}X={{V}_{mps}}Y$
\[\sqrt{\dfrac{3R{{T}_{X}}}{{{M}_{X}}}}=\sqrt{\dfrac{2R{{T}_{Y}}}{{{M}_{Y}}}}\]
Here root under and R will get canceled out from both the side and we get,
\[\dfrac{2{{T}_{Y}}{{M}_{X}}}{3{{T}_{X}}}={{M}_{Y}}\] ………..(1)
Now in the question the value of ${{M}_{X}}=40,{{T}_{X}}=400\,and\,{{T}_{Y}}=60$ is given, put all the values in equation (1),
\[{{M}_{Y}}=\dfrac{2\times 60\times 40}{3\times 400}=4\]

Thus, the molecular weight of gas Y is 4.

Note:
If in the question instead of molecular mass given mass is given the do not put the value of given mass in the place of molecular mass. SI units of temperature should be changed in Kelvin and formulas of room mean square speed and most probable speed is somehow similar so do not get confused between them.