Alternative Form of Ideal Gas Equation

Suppose we have  
  mols of a gas of molecular mass  
  occupying a volume  
  at temperature  
  and pressure  
. The gas obeys the idea; gas equation  
\[R=8.324 J/mol K\]
. This is sometimes not the most convenient form.
We can write  
\[R=N_A k\]
\[N_A=6.023 \times 10^{23}\]
  is Avogadro's constant, the number of particles in a mol and  
\[k=1.38 \times 10^{-23}J/K\]
  is Boltzmann's constant.
The ideal gas equation becomes  
\[pV=nN_A kT\]
For an ideal gas the average kinetic energy  
\[\frac{1}{2}m \bar{v^2}\]
  is also equal to  
\[\frac{3}{2}kT \]
\[kT=\frac{1}{3}m \bar{v^2}\]
. With this substitution the equation becomes  
\[pV=nN_A \frac{1}{3}m \bar{v^2}\]
Dividing by  
\[p= \frac{nN_Am}{V} \frac{1}{3} \bar{v^2}= \rho \frac{1}{3} \bar{v^2}\]
Now use again  
\[\frac{}{2}m \bar{v^2} = \frac{3}{2}kT \rightarrow \bar{v^2}=\frac{3kT}{m}\]
  to obtain  
\[p= \rho \frac{kT}{m}\]

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