Question

When plotting graphs and performing numerical calculations, it is convenient to work in terms of reduced variables, Rewrite the van der Waals equation in terms of these variables, and notice that the constants a and b disappear.

Short answer

$p=\frac{8t}{3v-1}-\frac{3}{v^2}$

Step-by-step solution

Given information 

$$\begin{gathered} P=p{P}_c \\ T=t{T}_c \\ V=v{V}_c \end{gathered}$$

and

role="math" localid="1646979368068" $P=\frac{NkT}{(V-Nb)}-\frac{a{N}^2}{V^2}$ (van der Waal's equation)

 Substituting the values of P, V and T in the van der Waal equation.

$$\begin{gathered} P=\frac{NkT}{(V-Nb)}-\frac{a{N}^2}{V^2} \\ p{P}_c=\frac{Nkt{T}_c}{(v{V}_c-Nb)}-\frac{a{N}^2}{{\left(v{V}_c\right)}^2} \end{gathered}$$

 Substituting the values of Pc, Vc and Tc in equation. 

$p\left(\frac{1}{27}\frac{a}{b^2}\right)=\left(\frac{N}{(3Nbv-Nb)}\frac{8}{27}\frac{a}{b}t\right)-\left(\frac{a{N}^2}{{\left(3Nbv\right)}^2}\right)$

further solving the equation we get

$p=\frac{8t}{3v-1}-\frac{3}{v^2}$

The van der Waal's equation is independent of constants a and b when represented in reduced variables