Problem $1(20$ points$)$

For a non$-$ideal gas obeying the Van der Waals equation

$(P+\frac{a}{\frac{?}{\text{b}\text{a}\text{r}}(v{)}^2})(\frac{?}{\text{b}\text{a}\text{r}}(v)-b)\frac{?}{b}=ar(R)T$

Using the fact that at the critical point must be

$(\frac{dP}{d(\frac{?}{\text{b}\text{a}\text{r}}(v))}{)}_{cr}=0,$ and $,(\frac{d^{2}P}{\frac{d}{b}ar(v{)}^2}{)}_{cr}=0$

Prove the following relations between the constants a e b and the critical pressure and

temperature

$P_{cr}=\frac{a}{27b^2},T_{cr}=\frac{8a}{\frac{27}{b}ar(R)b}$