Question

The root mean square velocity of molecules of gas is

• A. Inversely proportional to square root of temperature $\left(\sqrt{\frac{1}{T}}\right)$
• B. Proportional to square root of temperature $(\sqrt{T})$
• C. Proportional to square of temperature $\left(T^2\right)$
• D. Proportional to temperature (T)

Answer 1

The Correct Option is B

The question asks about the relationship between the root mean square (RMS) velocity of gas molecules and temperature. To solve this, let's examine the formula for the root mean square velocity of gas molecules:

The root mean square velocity, \(v_{\text{rms}}\), of molecules in a gas is given by the formula: \(v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\)

Where:

• \(k\) is the Boltzmann constant.
• \(T\) is the absolute temperature in Kelvin.
• \(m\) is the mass of a single molecule of the gas.

This formula shows that \(v_{\text{rms}}\) is directly proportional to the square root of the temperature \(T\). Therefore, as the temperature increases, the RMS velocity also increases, proportional to the square root of the temperature.

Let's analyze each of the options given:

• Inversely proportional to the square root of temperature \(\left(\sqrt{\frac{1}{T}}\right)\): This is incorrect because the formula shows a direct proportionality to \(\sqrt{T}\), not an inverse relationship.
• Proportional to the square root of temperature \((\sqrt{T})\): This is correct as explained by the formula \(v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\).
• Proportional to the square of temperature \(\left(T^2\right)\): This is incorrect because the formula exhibits a square root, not a square relationship with the temperature.
• Proportional to temperature (T): This is incorrect because the proportionality involves the square root of temperature, not a direct linear relationship to \(T\).

Thus, the correct answer is: Proportional to square root of temperature \((\sqrt{T})\).