Chapter 5: Q. 5.54 (page 185)

Calculate the Helmholtz free energy of a van der Waals fluid, up to an undetermined function of temperature as in $e q u a t i o n$ $5.56$ . Using reduced variables, carefully plot the Helmholtz free energy (in units of $N k T c$ ) as a function of volume for $T / T_{c} = 0.8$ Identify the two points on the graph corresponding to the liquid and gas at the vapor pressure. (If you haven't worked the preceding problem, just read the appropriate values off $F i g u r e$ $5.23$ .) Then prove that the Helmholtz free energy of a combination of these two states (part liquid, part gas) can be represented by a straight line connecting these two points on the graph. Explain why the combination is more stable, at a given volume, than the homogeneous state represented by the original curve, and describe how you could have determined the two transition volumes directly from the graph of $F$ .

Expert verified

The Helmholtz free energy is:

$F = x F_{l} + (1 - x) F_{g} = F_{g} - x (F_{g} - F_{l})$

Step by step solution

We have been given $F = G - P V$ for any type of system, we can use gibbs free energy.

Terms used in this:

$P$ in terms of $V$ :

$F = - N k T \ln (V - N b) + \frac{(N k T) (N b)}{V - N b} - \frac{2 a N^{2}}{V} - P V + c (T)$

when we put the expression in terms of dimensionless variable

$= - t \ln (3 v - 1) - \frac{9}{8 v} + f (t)$

To plot this function we use

$Plot [- . 8 * \log [3 v - 1] - 9 / (8 v), {v, . 4, 6}]$

Then the total volume of the system is:

$F = x F_{l} + (1 - x) F_{g} = F_{g} - x (F_{g} - F_{l})$

On elimination x from equation ,we get:

$F = F_{g} - \frac{(V_{g} - V)}{(V_{g} - V_{l})} (F_{g} - F_{l}) .$

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