## Alternative Form of Ideal Gas Equation

Suppose we have
$n$
mols of a gas of molecular mass
$m$
occupying a volume
$V$
at temperature
$T$
and pressure
$p$
. The gas obeys the idea; gas equation
$pV=nRT$
where
$R=8.324 J/mol K$
. This is sometimes not the most convenient form.
We can write
$R=N_A k$
where
$N_A=6.023 \times 10^{23}$
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
$\frac{3}{2}kT$
so
$kT=\frac{1}{3}m \bar{v^2}$
. With this substitution the equation becomes
$pV=nN_A \frac{1}{3}m \bar{v^2}$
.
Dividing by
$V$
gives
$p= \frac{nN_Am}{V} \frac{1}{3} \bar{v^2}= \rho \frac{1}{3} \bar{v^2}$
.
Now use again
$\frac{}{2}m \bar{v^2} = \frac{3}{2}kT \rightarrow \bar{v^2}=\frac{3kT}{m}$
to obtain
$p= \rho \frac{kT}{m}$
.