Fugacity, from the virial equation of state

[2h2o]

Homework Statement

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

Homework 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.

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!

 

[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.

 

[Chestermiller]

Homework Statement

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

Homework 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.

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!

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.

 

[2h2o]

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.

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.

 

[Chestermiller]

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.

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.

 

[2h2o]

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.

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!