The given Z vs P curve for 1 mole of a gas at 400K starts at Z=1 & P=0.
The slope at the point when the curve again intersect Z=1 is 0.005. The critical temperature of the gas is 500K.
My approach:
From the van der Waals' equation,
$$(P+\frac{a}{V^2})(V-b)=RT$$
On expanding, $$\frac{PV}{RT}=Z=1-\frac{a}{VRT}+\frac{Pb}{RT}+\frac{ab}{V^2RT}$$
And $$ PV=ZRT $$ $-(1)$
By both equations, $$Z=1-\frac{a}{VRT}+\frac{Zb}{V}+\frac{ab}{V^2RT}$$ $-(2)$
Differentiating with respect to Z for Equation (1)
$$ PV'+VP'=RT$$
Equation (2)
$$ 1= \frac{aV'}{V^2RT}+\frac{b}{V}-\frac{bV'}{V^2}-\frac{2abV'}{V^3RT}$$
& $$T_c=\frac{8a}{27Rb}$$
Now I am stuck, I don't even know if I am moving in the right direction because things are becoming too complex to solve.
Where,
a, b = van der Waals' gas constants
V = Volume of gas when curve again intersects Z = 1
T = 400K
$T_c$ = Critical Temprature