Suppose the equation of state of a gas is
Beta p=rho/(1-b rho) - beta a rho^2
where beta=1/kb T, b is a constant and rho is the molecular density, N/V. The internal energy of this gas is given by
U=(5/2)N kb T - N a rho
Determine the final internal energy of the gas, initially at temperature T0, if it is expanded from a volume V0 to 1.1V0. Answer the question for the following three cases expressing your answer in terms of T0,V0, and N
A. adiabatically with zero external pressure
B. adiabatic while maximizing work done by the system
lots, mainly dU=dq+dw
The Attempt at a Solution
A. is fine: no change in internal energy (free adiabatic expansion)
B. I can't seem to get this. Maximizing work means making the process reversible, I believe. Lots of stuff talks about solving this this for an ideal gas, but trying in the same manner (dU=-pdV, substituting dU/dt dt=Cv dt for dU and an expression for p then integrating) does not seem possible because with the vdW equations, I can't seem to separate T from V to integrate. If someone has some insight, I would greatly appreciate it.
C. I haven't gotten here yet from working on B. but I'm fairly certain this should be fairly simple.