91Ó°ÊÓ

The compression factor of a fluid is given by

$Z=\frac{PV}{NkT}$

where Z is compression factor.

$P=\frac{NkT}{(V-Nb)}-\frac{a{N}^2}{V^2}$

Partial differentiation of the equation w.r.t V

we get

role="math" localid="1646976952526" $\frac{\mathrm{δ}P}{\mathrm{δ}V}=-\frac{NkT}{{(V-Nb)}^2}+\frac{2a{N}^2}{V^3}$

Again differentiating we get

role="math" localid="1646977057779" $\frac{{\mathrm{δ}}^{2}P}{{\mathrm{δ}}^{2}V}=\frac{NkT}{{(V-Nb)}^3}-\frac{6a{N}^2}{V^4}$

$$\begin{gathered} \frac{\mathrm{δ}P}{\mathrm{δ}V}=0 \\ \frac{{\mathrm{δ}}^{2}P}{{\mathrm{δ}}^{2}V}=0 \end{gathered}$$

role="math" localid="1646977123022" $$\begin{gathered} ⇒\frac{Nk{T}_c}{{(V_c-Nb)}^2}=\frac{2a{N}^2}{{V_c}^3}and \\ \frac{Nk{T}_c}{{(V_c-Nb)}^3}=\frac{6a{N}^2}{{V_c}^4} \end{gathered}$$

On equating the above equations we get

$V_c=3Nb$

Substituting the equation in equation

role="math" localid="1646977903462" $$\begin{gathered} \frac{Nk{T}_c}{{(3Nb-Nb)}^2}=\frac{2a{N}^2}{{\left(3Nb\right)}^3} \\ ⇒k{T}_c=\frac{3}{27}\frac{a}{b} \end{gathered}$$

Substituting the above values in equation (2)

$$\begin{gathered} P_c=\frac{Nk({\displaystyle \raisebox{1ex}{$3a$}\!\left/ \!\raisebox{-1ex}{$27b)$}\right.}}{(3Nb-Nb)}-\frac{a{N}^2}{{\left(3Nb\right)}^2} \\ ⇒P_c=\frac{1}{27}\frac{a}{b^2} \end{gathered}$$

$$\begin{gathered} Z_C=\frac{P_c{V}_c}{Nk{T}_c}=\frac{1}{27}\frac{a}{b^2}×3Nb×\frac{1}{N}×\frac{27}{3}×\frac{b}{a} \\ =\frac{3}{8} \end{gathered}$$

The value of the compression factor is less than 1