Question

The RMS speed of an ideal gas is:

• A. Directly proportional to density $ d $
• B. Inversely proportional to density $ d $
• C. Inversely proportional to $ \sqrt{d} $
• D. None of the above

Hint:

Remember: - RMS speed depends on temperature and molar mass. - Since density is related to molar mass, the RMS speed is inversely proportional to the square root of density.

Answer 1

The Correct Option is C

Step 1: Recall the formula for RMS speed
The RMS speed ($ v_{\text{rms}} $) of an ideal gas is given by: $$ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} $$ where:
$ R $ is the universal gas constant
$ T $ is the absolute temperature
$ M $ is the molar mass of the gas
Step 2: Relate density to molar mass Density ($ d $) of a gas is defined as: $$ d = \frac{M}{V_m} $$ From the ideal gas law, $ PV_m = RT $, so: $$ V_m = \frac{RT}{P} $$ Substitute into the density equation: $$ d = \frac{M}{\frac{RT}{P}} = \frac{MP}{RT} \quad \Rightarrow \quad M = \frac{dRT}{P} $$ Step 3: Express RMS speed in terms of density
Substitute $ M = \frac{dRT}{P} $ into the RMS speed formula: $$ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3RT}{\frac{dRT}{P}}} = \sqrt{\frac{3P}{d}} $$ This simplifies to: $$ v_{\text{rms}} \propto \frac{1}{\sqrt{d}} $$ So, the RMS speed is inversely proportional to the square root of the density.
Step 4: Analyze each option (a) Directly proportional to density $ d $ — Incorrect
(b) Inversely proportional to density $ d $ — Incorrect
(c) Inversely proportional to $ \sqrt{d} $ — Correct
(d) None of the above — Incorrect, since (c) is correct
Step 5: Conclusion
The correct relationship between RMS speed and density is: $$ (c) Inversely proportional to \sqrt{d} $$