Question

The root mean square (rms) speed of the molecules of an ideal gas at a temperature T is proportional to

• A. $T$
• B. $T^{2}$
• C. $\sqrt{T}$
• D. $1/T$
• E. $1/\sqrt{T}$

Hint:

If you double the absolute temperature, the speed doesn't double—it only increases by a factor of $\sqrt{2}$ ($\approx 1.41$). To actually double the speed, you would need to quadruple (4x) the temperature!

Answer 1

The Correct Option is C

Concept:
Physics - Kinetic Theory of Gases.
Step 1: State the formula for rms speed.
The root mean square speed ($v_{rms}$) of gas molecules is given by the formula: $$ v_{rms} = \sqrt{\frac{3RT}{M}} $$ Where:

• $R$ is the universal gas constant.
• $T$ is the absolute temperature (in Kelvin).
• $M$ is the molar mass of the gas.

Step 2: Identify constant and variable terms.
For a specific gas, $R$, $3$, and $M$ are constants. Thus, the equation can be written as: $$ v_{rms} \propto \sqrt{T} $$
Step 3: Conclusion.
The rms speed of the molecules is directly proportional to the square root of the absolute temperature of the gas.