Critical Compressibility Factor (\({Z}_c\))
The critical Compressibility factor \({Z}_c\) of a van der Walls gas is given by,
\({Z}_c=\frac{{P}_c{V}_c}{R{T}_c}\) (1)
As we know,
$${P}_c=\frac{a}{27{b}^2}$$
$${V}_c=3b$$
$${T}_c=\frac{8a}{27Rb}$$
By putting the value of \({P}_c, {V}_c\) and \({T}_c\) in equation 1 :
$$\Rightarrow{Z}_c=\frac{(\frac{a}{27{b}^2})(3b)}{{R}[\frac{8a}{27Rb}]}$$
$$\Rightarrow{{Z}_c=\frac{3}{8}}$$
$$\Rightarrow\fbox{\({Z}_c\)=0.375}$$
Note-
We can test whether a gas behaves as a van der Waals gas by seeing whether its Compressibility factor is equal to 0.375
