One way of writing the equation of state for a real gas is $ P V = R T (1 + \frac{B}{V} + ….) $ Where B is a constant. Derive an approximate expression for B in terms of vander waal’s constant ‘ a ’ and ‘ b ’

Q: One way of writing the equation of state for a real gas is $\large P V = R T (1 + \frac{B}{V} + ….) $ Where B is a constant. Derive an approximate expression for B in terms of vander waal’s constant ‘ a ’ and ‘ b ’ .

Solution: $\large (P + \frac{a}{V^2})(V-b) = R T $

or , $\large P = \frac{RT}{V-b} – \frac{a}{V^2}$

Multiply by V then,

$\large P V = \frac{R T V}{V-b} – \frac{a V}{V^2}$

$\large P V = R T (\frac{V}{V-b} – \frac{a }{V R T}) $

$\large P V = R T [ (1 -\frac{b}{V})^{-1} – \frac{a }{V R T} ] $

$\large P V = R T [ (1 + \frac{b}{V} + (\frac{b}{V})^2 + ….) – \frac{a }{V R T} ] $

$\large P V = R T [ 1 + (b-\frac{a}{RT})\frac{1}{V} + (\frac{b}{V})^2 + …. ] $

$B = b -\frac{a}{RT} $