Whenever we have to find the RMS velocity of gas we use the formula √3RT/M, where R is the ideal gas constant, T is the absolute temperature and M refers to the molar mass of that particular gas. However, sometimes, the formula √3KT/M, where K is Boltzman constant, is also used. I just don't know when this second form is used. Please help me clear that out.
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The second form is equivalent to the first form, but $M$ (perhaps better written as $m$) in the second form refers to the mass of an individual atom. You can recover the second form from the first form by using $R=N_Ak_B$, where $N_A$ is Avogadro's number and $k_B$ is Boltzmann's constant: $$ \frac{R}{M} =\frac{N_Ak_B}{M} =\frac{k_B}{M/N_A} =\frac{k_B}{m} $$ since $$ \left[\frac{M}{N_A}\right] =\frac{\mbox{kg}}{\mbox{mol}}\times\frac{\mbox{mol}}{\mbox{atom}} =\frac{\mbox{kg}}{\mbox{atom}} =\left[m\right] $$ and $[\cdot]$ means "the units of".
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$\begingroup$ So, please tell me if I am getting it right that M in the first form is the molar mass of gas ( let's take an example of oxygen gas) is 32 grams, whereas in second form it's 32 amu or 32 * 1.67 * 10^(-24) grams. $\endgroup$ Commented Nov 22 at 17:11
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$\begingroup$ @Physics_enthus That’s correct, just remember to convert both masses to kilograms or use units of the ideal gas constant and Boltzmann constant that are consistent with a mass measured in grams. $\endgroup$– CW279Commented Nov 22 at 17:34