Rms Speed

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Consider a tank of nitrogen molecules each of molar mass 28 grams per mole maintained at 300 K, so what is the approximate root mean square speed of these molecules?

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Determining molecular speed relies on the root mean square speed formula from NCERT Class 11, Chapter 13 (Kinetic Theory), written as $v_{rms} = \sqrt{\frac{3RT}{M}}$, where $R$ is the universal gas constant, $T$ the absolute temperature and $M$ the molar mass in kilograms per mole. Converting the molar mass gives $M = 28 	imes 10^{-3}$ kg/mol, and using $R = 8.314$ J/(mol K) with $T = 300$ K, the numerator $3RT = 3 	imes 8.314 	imes 300 = 7482.6$. Dividing by $0.028$ gives $267236$, and the square root is approximately $517$ m/s. The option 300 m/s numerically echoes the temperature but has no physical basis. The option 1034 m/s is exactly double the correct value, arising from forgetting the square root and doubling instead. The option 173 m/s comes from dropping the factor of three in the formula. The rms speed itself is defined as the square root of the average of the squares of molecular speeds, a quantity that appears naturally in the kinetic pressure derivation. A magnitude check confirms the answer: gas molecular speeds at room temperature are typically several hundred metres per second, comparable to the speed of sound in the gas, so 517 m/s is entirely reasonable for nitrogen.

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