Question

Calculate the Helmholtz free energy of a van der Waals fluid, up to an undetermined function of temperature as in $equation$$5.56$. Using reduced variables, carefully plot the Helmholtz free energy (in units of $NkTc$) as a function of volume for $T/T_c=0.8$Identify the two points on the graph corresponding to the liquid and gas at the vapor pressure. (If you haven't worked the preceding problem, just read the appropriate values off $Figure$$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 $F$.

Short answer

The Helmholtz free energy is:

$F=x{F}_l+(1-x)F_g=F_g-x\left(F_g-F_l\right)$

Step-by-step solution

Given information

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

Simplify

Terms used in this:

$P$in terms of $V$:

$F=-NkT\ln (V-Nb)+\frac{\left(NkT\right)\left(Nb\right)}{V-Nb}-\frac{2a{N}^2}{V}-PV+c\left(T\right)$

when we put the expression in terms of dimensionless variable

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

To plot this function we use

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

Then the total volume of the system is:

$F=x{F}_l+(1-x)F_g=F_g-x\left(F_g-F_l\right)$

On elimination x from equation ,we get:

$F=F_g-\frac{\left(V_g-V\right)}{\left(V_g-V_l\right)}\left(F_g-F_l\right).$