For ideal gases, the canonical partition function can be related to the molecular partition function such that $Q=\frac{q^N}{N!}.$ To account for intermolecular interactions, J$.$D$.$ van der Waals proposed that some correction terms must be included. Firstly, the non$-$zero volume of the molecules implies that they are restricted to move in a smaller volume, $V-N{b}^{\text{'}}$ instead of $V.$ Then, the canonical partition for attractive forces is $Q^{\text{a}\text{t}\text{t}}=$ exp$(\frac{a!N^2}{VkT}).$

$($i$)$ Obtain the total canonical partition function for this system.

$($ii$)$ If pressure, $P=kT(\frac{dellnQ}{\text{d}\text{e}\text{l}\text{V}}{)}_T,$ prove that $P=\frac{nRT}{V-nb}-\frac{a{n}^2}{V^2}.$