# Homework Help: Fugacity, from the virial equation of state

1. Oct 8, 2012

### 2h2o

1. The problem statement, all variables and given/known data

Find the fugacity from the virial equation of state, where B is a constant.

2. Relevant equations

$Z=\frac{PV}{RT}=1+\frac{B}{V}$

Don't know how to do underbars in TeX, but the V terms are on a per-mol basis. B is a constant and no further expansions of the EOS are needed. We'll be calculating this at some constant temperature T and at a discrete pressure P.

3. The attempt at a solution

Omitting the derivation of fugacity,

$f=P*exp(\frac{1}{RT}\int^{0}_{P}{V-\frac{RT}{P}dp})$

But solving the VEOS explicitly for V, to substitute into fugacity, isn't going to work. Solving for P doesn't seem like it would be helpful. I'm presently trying to figure out how to do this with an iterative method, but the integral is throwing me off.

Any insights?

Thanks!

2. Oct 9, 2012

### 2h2o

Ok. Going further. Gave up on iterative solutions. Now trying to derive fugacity now in terms of dV.

$f = P*exp[ (G - G^*) / (RT) ]$and recognizing $(G - G^*)$ as a departure function:

$$(G - G^*) = [H - H^*] - T[S - S^*]$$

$H^* = 0, Sd = - (BR) / (PV)$ (using the virial EOS)

and get to the point:

$f = P*exp[ [ H / (RT) - T * ( -BR / ( PV) ) ]$

substituting $1/Z = (RT / (PV),$ simplifies to

$f = P*exp[ H / (RT) + B/Z]$

But I know nothing of V, which was the whole point of confusion earlier and the reason for trying this route. I must have made a conceptual mistake somewhere but I'm not seeing it.

Last edited: Oct 9, 2012
3. Oct 10, 2012

### Staff: Mentor

Either of the two methods you mentioned above ought to work. Why don't they? Is it that the integration is difficult analytically? That's not really a reason.

4. Oct 10, 2012

### 2h2o

No, the integral itself isn't the problem. My problem was the V term inside the integral, which comes from the virial equation of state (not the ideal EOS). The V-based virial EOS I was working cannot be solved explicitly for V and therefore cannot be meaningfully substituted into the integral.

Which leads me to what the problem was: my choice of the virial EOS. I hadn't remembered that there is a pressure-based virial EOS. (Z = 1 + B'P) where B' = B/(RT) Using that EOS makes this a trivial calculation.

5. Oct 11, 2012

### Staff: Mentor

Who says it can't be solved explicitly for V? Just multiply both sides of the equation by V, and then solve the resulting quadratic equation.

6. Oct 11, 2012

### 2h2o

I said it can't be solved explicitly, that's who. I am also an idiot for not seeing that even when it is, retrospectively, so obvious. So thank you for pointing that out. No wonder I was running in circles.

Cheers!