The root mean square speed of molecules of nitrogen gas at $27^{\circ} \mathrm{C}$ is approximately : (Given mass of a nitrogen molecule $=4.6 \times 10^{-26} \mathrm{~kg}$ and take Boltzmann constant $\mathrm{k}_{\mathrm{B}}=1.4 \times 10^{-23} \mathrm{JK}^{-1}$ )

To find the root mean square speed of molecules of nitrogen gas, we can use the formula: <br/><br/> $v_{rms} = \sqrt{\frac{3k_BT}{m}}$ <br/><br/> where $v_{rms}$ is the root mean square speed, $k_B$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $m$ is the mass of a nitrogen molecule. <br/><br/> First, we need to convert the temperature from Celsius to Kelvin: <br/><br/> $T = 27^{\circ}\mathrm{C} + 273 = 300\,\mathrm{K}$ <br/><br/> Now, substitute the given values of $k_B$, $T$, and $m$ into the formula: <br/><br/> $$ v_{rms} = \sqrt{\frac{3(1.4 \times 10^{-23}\, \mathrm{JK}^{-1})(300\,\mathrm{K})}{4.6 \times 10^{-26}\,\mathrm{kg}}} $$ <br/><br/> Simplify and calculate the root mean square speed: <br/><br/> $v_{rms} = 523\,\mathrm{m/s}$ <br/><br/> The root mean square speed of molecules of nitrogen gas at $27^{\circ}\mathrm{C}$ is 523 m/s.