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Current Question (ID: 11149)
Question:
$\text{The Earth's atmosphere contains both oxygen and nitrogen. The mass of an oxygen molecule is greater than that of a nitrogen molecule. On a certain day, the temperature of air in a room is 300 K.}$ $\text{Consider the following statements regarding the motion of oxygen and nitrogen molecules:}$ $\text{(A) Both gases have the same mean square velocity } (v^2).$ $\text{(B) Nitrogen molecules have a greater mean square velocity } (v^2) \text{ than oxygen molecules.}$ $\text{(C) Nitrogen molecules have a greater mean kinetic energy than oxygen molecules.}$ $\text{(D) Oxygen molecules have a greater mean kinetic energy than nitrogen molecules.}$ $\text{Choose the correct option from the options given below:}$
Options:
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1. $\text{(A) only}$
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2. $\text{(A) and (C) only}$
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3. $\text{(B) and (D) only}$
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4. $\text{(B) only}$
(Correct)
Solution:
$\text{Hint: } v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$ $\text{Step 1: Compare their mean square velocities.}$ $\text{RMS velocity is given by:}$ $v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$ $\text{Therefore, the mean square velocity is given by:}$ $\overline{v^2} = \frac{3RT}{M}$ $\Rightarrow \overline{v^2} \propto \frac{1}{M}$ $\Rightarrow \overline{v^2}_{\text{N}_2} > \overline{v^2}_{\text{O}_2} \quad [\because M_{\text{O}_2} > M_{\text{N}_2}]$ $\text{Therefore, nitrogen molecules have a greater mean square velocity than oxygen molecules.}$ $\text{Step 2: Compare their kinetic energies.}$ $\text{Mean kinetic energy is given by:}$ $K_{\text{avg}} = \frac{3}{2}kT$ $\text{Both gases are at the same temperature, so the mean kinetic energy of both gases will be the same.}$ $\text{Hence, option (4) is the correct answer.}$
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