Question

The compression factor of a fluid is defined as the ratio PV/NkT; the deviation of this quantity from 1 is a measure of how much the fluid differs from an ideal gas. Calculate the compression factor of a Van der Waals fluid at the critical point, and note that the value is independent of a and b. (Experimental values of compression factors at the critical point are generally lower than the Van der Waals prediction, for instance, 0.227 for H22O, 0.274 for CO22, and 0.305 for He.)

Short answer

The compression factor of van der Waal fluid at critical point is $\frac{3}{8}$.

Step-by-step solution

Given 

The compression factor of a fluid is given by

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

where Z is compression factor.

van  der Waal's equation

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

Partial differentiation of the equation w.r.t V

we get

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

Again differentiating we get

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

At critical point

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

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

Finding Vc, Tc, Pc.

On equating the above equations we get

$V_c=3Nb$

Substituting the equation in equation

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

Substituting the above values in equation (2)

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

Substituting the values in Equation (1)

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

The value of the compression factor is less than 1