Rms Speed

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 $M=28\times 10^{-3}$ kg/mol, and using $R=8.314$ J/(mol K) with $T=300$ K, the numerator $3RT=3\times 8.314\times 300=7482.6$. 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.