Biophysics: Problems III
[3.1] Evaluate the second virial coefficient in the following two cases:
(a) a hard sphere system
u(r) =
∞, r < σ
0,
r>σ
(b) a square well system



 ∞,
u(r) =
r<σ
−ε, σ < r < σ 

 0,
r > σ
[3.2] Consider the Van der Vaals equation written in the form (with v = V /N )
p=
kB T
a
− 2
v−b v
(a) Rewrite the Van der Vaals equation as a cubic in v. Using the fact that at the critical
point all roots of the cubic are the same, that is, (v − vc )3 = 0, equate coefficients of powers
of v to show that
vc = 3b,
pc = a/27b2 ,
kB Tc = 8a/27b2
(b) Introducing the dimensionless variables
π = (p − pc )/pc ,
φ = (v − vc )/vc ,
t = (T − Tc )/Tc
show that the Van der Vaals equation becomes
1+π =
8(1 + t)
3
−
2 + 3φ
(1 + φ)2
Note that this is independent of any parameters – a result known as the law of corresponding
states.
(c) Assuming that φ ∼ |t|1/2 , Taylor expand π to third order in |t|1/2 to obtain the approximate
equation of state
3
π = π(φ, t) ≡ 4t − 6tφ − φ3
2
10
Using the Maxwell construction for T < Tc , which takes the form
φg
φπ (φ)dφ = 0,
φl
deduce that φl = −φg . Combining this with the equilibrium condition π(φl , t) = π(φg , t),
show that
√
∆φ = φg − φl = 4 −t ∼
Tc − T
Tc
1/2
(d) Show that the isothermal compressibility κT at φ = 0 is given by κT = 1/6pc t.
(e) Using the Van der Vaals free energy
aN 2
F (V, T ) = N kB T log(N Λ3 /e) − log(V − N b) −
V
show that the specific heat
3
CV = N kB
2
which is the same as for an ideal gas. Hence, deduce from part (d) and problem [1.2] that
CP ∼ |Tc − T |−1 close to the bifurcation point.
[3.3] Consider a lattice model of a binary mixture with nearest neighbor interactions AA , AB ,
BB such that AA = BB (asymmetric case).
(a) Calculate within (Bragg-Williams) mean field theory the free energy per site f (φ, T ). Show
explicitly that this expression is not invariant under the substitution φ ↔ 1 − φ.
(b) Defining ∆ = AA − BB and = AA + BB − 2AB , obtain two equations for the critical
concentration φc and critical temperature Tc . Hint: The critical point is defined to be the
point at which ∂f 2 /∂φ2 = 0 and ∂ 3 f /∂φ3 = 0. Numerically solve these two equations for
∆/ = 0.1.
(c) Sketch f (φ, T ) as a function of φ for ∆/ = 0.00.1 and T < Tc . Compare with the
corresponding graph for ∆ = 0. Illustrate graphically how to determine the pair of volume
fractions for a binary mixture using the common tangent construction.
(d) Suppose that the symmetric system (∆ = 0) is close to the critical point T = Tc ,
φ = φc = 1/2. Taylor expand the free energy up to fourth order in ψ = φ − φc and show that
1/2
at equilibrium |ψ| ∼ TcT−T
for T < Tc .
c
11