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Root mean square velocity for the molecules of a perfect gas (as $P \rightarrow 0$) is given by: $$c = \left( \frac{3RT}{M} \right) ^{\frac{1}{2}}$$

On dimensional analysis, RHS gives $\left( \frac{[L]^2}{[T]^2[mol]} \right) ^{\frac{1}{2}}$ which must be equal to $\frac{[L]}{[T]}$, is it dimensionally inconsistent?

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  • $\begingroup$ It is my mistake, I mistook M's units to be that of mass instead $Mass * mol^{-1}$ $\endgroup$
    – drake01
    Sep 10 '17 at 12:13
  • $\begingroup$ You should delete the question since your premise for asking is flawed and recognized. It's not really about physics. $\endgroup$
    – Bill N
    Sep 12 '17 at 21:39
  • $\begingroup$ I'm voting to close this question as off-topic because it was based on a mistake that was since recognized. $\endgroup$
    – Floris
    Sep 13 '17 at 0:48
  • $\begingroup$ Sorry, but I can't find an option to delete the question $\endgroup$
    – drake01
    Sep 13 '17 at 2:18
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I would imagine that you forgot that $M$ is the mass per unit mol, which would solve your problem. However, since mol is a dimensionless number, it arguably doesn't need to appear in dimensional analysis (though can be helpful).

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Working in SI units (equivalent to dimensions) $$\text{m s}^{-1}=\sqrt{\frac{[\text{J}\ \text{K}^{-1}\ \text{mol}^{-1}] \ \ [\text{K}]}{[\text{kg}\ \text{mol}^{-1}]}}.$$ Alright?

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