Find the $v_{rms}$ of nitrogen gas at $300 \ K$.

(A) The root mean square speed $(v_{rms})$ of an ideal gas is given by the formula:
$v_{rms} = \sqrt{\frac{3RT}{M}}$
Where $R = 8.314 \ J \ mol^{-1} \ K^{-1}$ is the universal gas constant,$T = 300 \ K$ is the temperature,and $M$ is the molar mass of nitrogen $(N_2)$.
For nitrogen gas $(N_2)$,the molar mass $M = 28 \ g/mol = 28 \times 10^{-3} \ kg/mol$.
Substituting the values into the formula:
$v_{rms} = \sqrt{\frac{3 \times 8.314 \times 300}{28 \times 10^{-3}}}$
$v_{rms} = \sqrt{\frac{7482.6}{0.028}}$
$v_{rms} = \sqrt{267235.7}$
$v_{rms} \approx 516.95 \ m/s$
Rounding to the nearest whole number,$v_{rms} \approx 517 \ m/s$.