Question

A flask contains a mixture of neon Ne, krypton Kr, and radon Rn gases. (Hint: The molar mass of the is Ne 20.180 g/mol, of the Kr is 83.80g/mol, and of the Kr 222g/mol)
(A) Compare the average kinetic energies of the Ne and Kr.
(B) Comparethe average kinetic energies of the Kr and Rn.
(C) Compare the average kinetic energies of the Rn and Ne.
(D) Compare the root-mean-square speeds of the Ne and Kr.
(E) Compare the root-mean-square speeds of the Kr and Rn.
(F) Compare the root-mean-square speeds of the Rn and Ne.

Answer 1

Answer and Explanation: Average Kinetic Energy of Gases is calculated by the formula: KE = $\frac{3}{2}RT$

where:

R is the ideal gas constant;

T is temperature in Kelvin;

So, kinetic energy is dependent only on temperature.

In the flask, the mixture of gases is in the same temperature, so kinetic energy will be the same.

The root mean square speed is:

$v_{rms} = \sqrt{\frac{3RT}{M} }$

where

M is the molar mass of the molecule.

Calculating root mean square speed for each element of the mixture:

Neon

$v_{rms} = \sqrt{\frac{3RT}{20.18} }$

$v_{rms} =\sqrt{\frac{1}{20.18} }.\sqrt{{3RT}}$

$v_{rms} = 0.222\sqrt{3RT}$

Krypton

$v_{rms} = \sqrt{\frac{3RT}{83.81} }$

$v_{rms} = 0.109.\sqrt{3RT}$

Radon

$v_{rms} = \sqrt{\frac{3RT}{222} }$

$v_{rms} = 0.067.\sqrt{3RT}$

Comparing the kinetic energies, it can be observed that Neon, which has the smaller molar mass, has the higher root-mean-square speed.

In conclusion, higher the molar mass, lower is the root-mean-square speed.