# τ τ τ τ τ τ

```1. Binodal Points Far from Tc
In this problem we will derive an expression for the binodal curve n   for    c . We will
need the VdW free energy per unit volume fV 
 
 
F
n
2
 n log 
  1  an and

V
n
1

nb


  q
 
the
given
definition
of
the
binodal
f  nl   f  ng 
f
f
.


n n ng n n nl
nl  ng
curve
by
the
common
tangent:
a. Under the assumption that for    c the gas can be treated as an ideal gas, derive
 ng
f
  log 
n
n n  ng
 q



 
b. Under the same low temperature assumption, explain why f ng  f  nl  and
ng  nl
are
valid
assumptions.
Using
these
assumptions,
solve
f  nl   f  ng  f  nl 
f
explicitly, obtain a quadratic equation for nl and


n n nl
nl  ng
nl
solve it to leading order in  . Explain the leading order term.
c. Substitute this nl back to the equation
f  nl 
f

and solve for ng .
n n ng
nl
d. Discuss your result in comparison with what was derived from the Clausius-Clapeyron
equation. What can you say about the meaning of the latent heat in the    c
regime?
2. Van der Waals Gas
Given the Helmholtz VdW free energy FVdW
  nQ
  Na 2
,
  N log  V  Nb    1 
  V
 N
a. Derive the equation of state
b. Derive the entropy and the internal energy
c. Transform into the scaled parameters pˆ 
V

p
, ˆ 
and Vˆ 
. Explain the
Vc
c
pc
meaning of this scaled form of the equation of state
d. Derive and explain the Maxwell construction
e. Plot T-V phase diagram. Indicate clearly all regions and boundaries
```