Here is the question as posed by the book: Calculate the Helmholtz free energy o
ID: 1672813 • Letter: H
Question
Here is the question as posed by the book:Calculate the Helmholtz free energy of a van der Waals fluid, up toan undetermined function of temperature as in eqn 5.56(G=-NkTln(V-Nb)+(NkT)(Nb)/(V-Nb)-2aN^2/V+c(T)). Using reducedvariables, carefully plot F (in units of NkTc) as a function ofvolume for T/Tc=0.8. Identify the two points on the graphcorresponding to the liquid and gas at the vapor pressure. Then prove that F of a combination of these to states can berepresented by a straight line connecting these two points on thegraph. Explain why the combination is more stable, and soon.
The second half is easy enough to explain, I am 99% sure that Iknow what I am supposed to get as an answer, but whenever I try, Iend up with alphabet soup.
My general strategy has been to use F=G+pV to find an equation forF as a function of p, v, and t. (These are reduced variablesfor the van der Waals equation; t=T/Tc, p=p/pc, v=V/Vc and Vc=3Nb,pc=a/(27b^2) and kTc=8a/(27b))
When I plug in all this garbage, I end up with:
F/(NkTc)=(3/8)pv-t[ln(3v-1)+ln(Vc/3)]+t/2-54/(24v)+c(T)/(NkTc)
Which I am not entirely sure I believe. (Note that all thevariables, v, p, and t are reduced.) I don't like having thatVc term in there. Also, plotting this is a rather unfortunateaffair...
Can anyone double check my work please?
Explanation / Answer
... That looked awful. Here's a (hopefully) reformatted version:G = -n k T Log[V - n b] + n k T n b/(V - nb) -2 a n2/V +c[T] Then F = G +PV. And make the folowing substitutions for the unreducedvariables: V=v3 n b,
T=t Tc,
P=p a/(27 b2),
Tc=8a/(27 k b) & I get : F=-((2 a n)/(3 b v))+(a n p v)/(9 b)+(8 a n2 t)/(27 (-bn+3 b n v)) + c[(8 a t)/(27 b k)]-(8 a n tLog[-b n+3 b n v])/(27b)
One simplification Mathematica came up with is: 1/27 (27 c[(8 a t)/(27 b k)]+a n (-((8 t)/(b-3 bv))+(3 (-6+p v2))/(b v)-(8 t Log[b n (-1+3v)])/b)) I wonder if you made the substitutions withreduced variables correctly, i.e. you should replace the unreducedvariables in the expression with the equivalent reducedexpressions? I'm thinking that might generate critical valuesin your finalexpression. I wonder if you made the substitutions withreduced variables correctly, i.e. you should replace the unreducedvariables in the expression with the equivalent reducedexpressions? I'm thinking that might generate critical valuesin your finalexpression.