Root Mean Square Speed of Gas

Question

The RMS speed of particles in a gas is

$v_{rms} = \sqrt{\frac{3RT}{M}}$

where $M$ = molar mass; according to this Wiki entry: http://en.wikipedia.org/wiki/Root-mean-square_speed

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1. The gas laws state that $pV = nRT$ where $n$ = the number of moles of gas.

2. Further more, the kinetic theory of gases gives the following equation $pV = \frac{1}{3}Nm(v_{rms})^2$ where $N$ = number of particles and $m$ = mass of gas sample.

Combining 1 and 2 gives:

$nRT = \frac{1}{3}Nm(v_{rms})^2$

which simplifies to:

$v_{rms} = \sqrt{\frac{3nRT}{Nm}}$

As $n = \frac{N}{N_{A}}$:

$v_{rms} = \sqrt{\frac{3RT}{N_{A}m}}$

Also $m = Mn = \frac{MN}{N_{A}}$. Therefore, $N_{A}m = MN$ Substituting this in:

$v_{rms} = \sqrt{\frac{3RT}{MN}}$

However the RMS equation from Wikipedia contains no $N$ or reference to the number of particles.

Why does this happen?

Answer 1

I think you have an error in assumption 2. If $N$ is the number of molecules, then the mass of the sample would be $N$ multiplied by the mass per molecule, not $N$ multiplied by the total mass of the sample. You are kind of "overcounting" mass. If you take $m$ to be the mass per molecule (molecular mass), then I believe it works out.