One way of writing the equation of state for real gases is
`PV=RT[1+(B)/(V)+….]`
where `B` is a constant. Derive an approximate expression for `B` in terms of the van der Waals constants `a` and `b`.

Explanation

The van der Waals equation for 1 mole of a gas is
`(p+(a)/(bar(V)^(2)))(bar(V) - b) = RT`
where `bar(V)` is the molar volume.
or `(p+(a)/(bar(V)^(2))) = (RT)/((bar(V) - b))`
or `(p bar(V) + (a)/(bar(V))) = (bar(V))/((bar(V) - b)) RT`
or `(p bar(V))/(RT) + (a)/(bar(V)RT) = (bar(V))/(bar(V) - b)`
`(p bar(V))/(RT) = (bar(V))/(bar(V) - b) - (a)/(RT bar(V))`
`= (1)/((1-(b)/(bar(V)))) -(a)/(RT bar(V)) = (1-(b)/(bar(V)))^(-1) - (a)/(RT bar(V))`
At low pressures, `(b)/(bar(V)) lt lt 1` so that we can expand the first term using `(1-x)^(-1) = 1 + x + x^(2) +`........... . This yields the virial equation in terms of volume :
`(pbar(V))/(RT) = [1+(b)/(bar(V)) + ((b)/(bar(V)))^(2) +.....] - (a)/(RTbar(V))`
`= 1 + (b-(a)/(RT))(1)/(V) + ((b)/(bar(V)))^(2) +.....`
or `p bar(V) = RT [1 + (b-(a)/(RT)) (1)/(bar(V)) + ((b)/(bar(V)))^(2) +....]`
Comparing with the given equation, we have
`B = (b-(a)/(RT))`.