Question 99: Thermodynamics & KTG

Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A :
If the average kinetic energy of $\mathrm{H}_2$ and $\mathrm{O}_2$ molecules, kept in two different sized containers are same, then their temperatures will be same.
Reason R :
The r.m.s speed of $\mathrm{H}_2$ and $\mathrm{O}_2$ molecules are same at same temperature.
Choose the correct answer from the options given below

Accepted Answer

For an ideal gas, the average translational kinetic energy of one molecule is
$$ \overline{K} = \frac{3}{2}kT $$
where $k$ is Boltzmann constant and $T$ is absolute temperature.
So, if the average kinetic energy of $\mathrm{H}_2$ and $\mathrm{O}_2$ molecules is the same, then from
$$ \frac{3}{2}kT_{\mathrm{H}_2} = \frac{3}{2}kT_{\mathrm{O}_2} $$
we get
$$ T_{\mathrm{H}_2} = T_{\mathrm{O}_2} $$
Hence, Assertion A is true.
Now, the r.m.s. speed is given by
$$ v_{\mathrm{rms}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}} $$
This depends on the mass $m$ of one molecule or molar mass $M$.
Since $\mathrm{H}_2$ and $\mathrm{O}_2$ have different molar masses, at the same temperature their r.m.s. speeds will not be the same.
In fact,
$$ v_{\mathrm{rms}} \propto \frac{1}{\sqrt{M}} $$
So lighter $\mathrm{H}_2$ molecules move faster than heavier $\mathrm{O}_2$ molecules at the same temperature.
Therefore, Reason R is false.
So, the correct option is:
$$ \boxed{	ext{Option C: A is true but R is false}} $$

Answer

Both $\mathbf{A}$ and $\mathbf{R}$ are true and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$

Answer

Both $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$

Answer

A is true but $\mathbf{R}$ is false

Answer

A is false but $\mathbf{R}$ is true

Thermodynamics & KTG

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