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.

One way of writing the equation of state for real gases is PV=RT[1+(B

1.	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.
Answer» Solution :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))`.