- #1

Dazed&Confused

- 191

- 3

## Homework Statement

Two ideal van der Waals fluids are contained in a cylinder, separated by an internal moveable piston. There is one mole of each fluid, and the two fluids have the same values of the van der Waals constants [itex]b[/itex] and [itex]c[/itex]; the respective values of the van der Waals constant [itex]''a''[/itex] are [itex]a_1[/itex] and [itex]a_2[/itex]. The entire system is in contact with a thermal reservoir of temperature [itex]T[/itex]. Calculate the Helmholtz potential of the composite system as a function of [itex]T[/itex] and the total volume [itex]V[/itex]. If the total volume is doubled (while allowing the internal piston to adjust), what is the work done by the system?

## Homework Equations

The van der Waal Helmholtz potential is [tex]

F = cNRT - \frac{a}{V}N^2 - T\left ( NR \log(V/N-b) + cNR \log ( cRT) + S_0 \right)

[/tex]

## The Attempt at a Solution

So the composite Helmholtz potential will just be the sum of the two. The temperatures and pressures are the same. The negative of the change of the van der Waal potential is the work done. The problem is calculating the equilibrium volume. Equating the temperatures we find [tex]

\frac{RT}{V_1-Nb} - \frac{a_1}{V_1^2} = \frac{RT}{V_2-Nb} - \frac{a_2}{V_2^2}

[/tex]

which along with [itex]V = V_1+V_2[/itex] gives you quintic polynomials to solve for [itex]V_1[/itex] in terms of [itex]V[/itex]. I don't see a way of doing this.