JEE Physics 2023 Question with Solution

Given: Statement I compares the RMS speeds at $-73^\circ \text{C}$ and $527^\circ \text{C}$. Statement II compares $PV$ with the translational kinetic energy of an ideal gas.

Find: Which statement is true.

For an ideal gas, the root mean square speed satisfies

$$v_{\text{rms}} \propto \sqrt{T}$$

Convert the temperatures to kelvin:

$$T_1 = -73^\circ \text{C} = 200 \, \text{K}, \qquad T_2 = 527^\circ \text{C} = 800 \, \text{K}$$

Now,

$$\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{800}{200}} = \sqrt{4} = 2$$

So, the RMS speed doubles. Therefore, Statement I is true.

For an ideal gas,

$$PV = nRT$$

The translational kinetic energy of the gas molecules is

$$\text{Translational KE} = \frac{3}{2} nRT$$

Hence,

$$PV \neq \frac{3}{2} nRT$$

So, Statement II is false.

The solution concludes that the correct option is D, which corresponds to: Statement I is false but Statement II is true. This contradicts the actual working shown above. Based on the working, the correct option should be B.

Given: Two independent statements from kinetic theory.

Find: The truth value of each statement.

1. Statement I

• RMS speed depends on absolute temperature, not Celsius temperature directly.
• After converting to kelvin, the temperatures become $200 \, \text{K}$ and $800 \, \text{K}$.
• Since

$$\frac{800}{200} = 4$$

the RMS speed ratio is

$$\sqrt{4} = 2$$

• Therefore, Statement I is true.

2. Statement II

• From the ideal gas equation,

$$PV = nRT$$

• Total translational kinetic energy of $n$ moles is

$$\frac{3}{2} nRT$$

• Therefore $PV$ is not equal to the translational kinetic energy; rather,

$$\text{Translational KE} = \frac{3}{2} PV$$

• Therefore, Statement II is false.

So the correct interpretation of the working is: Statement I is true but Statement II is false, hence the correct option is B.