For molecules in an ideal gas, the root-mean- square (rms) speed ($v_{rms}$) and the most probable speed ($v_{mp}$) are related to the temperature (T) and molar mass (M) by the following formulas: $v_{rms} = \sqrt{\frac{3RT} {M}}$ $v_{mp} = \sqrt{\frac{2RT}{M}}$ The ratio of the rms speed to the most probable speed is therefore: $\frac{v_{rms}}{v_{mp}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{2RT} {M}}} = \sqrt{\frac{3RT/M}{2RT/M}} = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}$ Thus, the ratio $v_{rms} : v_{mp}$ is $\sqrt{3} : \sqrt{2}$.
Ratio of the rms speed and the most probable speed for molecules in an ideal gas is
Asked in WBPSC CS Prelims 2021 · Topic: Heat and Thermodynamics (Kinetic Theory of Gases (Assumptions, Pressure, Temperature, Ideal Gas Equation))
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