Thermodynamics and Statistical Mechanics II - Home Exercise 5

Thermodynamics and Statistical Mechanics II - Home Exercise 5
1. Gas-solid equilibrium
Consider the gas-solid equilibrium under the extreme assumption that the entropy of the
solid may be neglected over the temperature range of interest. Let ?ε0 be the cohesive
energy of the solid, per atom. Treat the gas as ideal and monoatomic. Make the
approximation that the volume accessible to the gas is the volume V of the container,
independent of the much smaller volume taken by the solid.
(a) Show that the total Helmholtz free energy of the system is given by- F = FS +Fg =
−NS ε0 + Ng τ [ln(ng λ3T ) − 1], where the total number of atoms, N = Ng + NS is
constant.
(b) Find the minimum of the free energy with respect to Ng ; show that in the equilibrium condition Ng = nq V e−βε0 .
(c) Find the equilibrium vapor pressure.
2. Entropy,energy and enthalpy of van der Waals gas
(a) Write down the free energy of a VdW gas. Explain the meaning of the constants
a and b. From this, derive the VdW equation of state.
h i
nq (V −N b)
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(b) Show that the entropy of the VdW gas is σ = N ln
+2 .
N
(c) Show that the energy is U = 32 N τ −
N 2a
.
V
(d) Show that the enthalpy is H(τ, V ) = 52 N τ +
N 2 bτ
V
2
− 2 NV a or H(τ, P ) = 25 N τ +
N bP − 2 NτaP .
All results are given to first order in the van der Waals correction terms a and b.
(e) Draw qualitatively the temperature versus volume phase diagram for a VdW gas.
Indicate clearly the following boundaries and regions on the diagram: liquid, gas,
single phase fluid, critical temperature, critical volume, metastable regions, unstable region, spinodal curve, and binodal curve.
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3. Numerical VdW calculations(1)
(a) The VdW constants for N2 are: NA2 a = 0.136P a · m6 mol−2 , NA b = 3.85 ×
10−5 m3 mol−1 . How accurate is the assumption that Nitrogen can be considered
as an ideal gas at normal P and T ?
(b) For Argon, the critical point occurs at a pressure PC = 4.83M P a and temperature
TC = 151K. Determine values for the VdW constants a and b for Ar and estimate
the diameter of an Ar atom.
(c) Look up values of VdW constants for H2 , N2 , CO and O2 (state your source!).
Plot, for each of the gasses,
PV
nT
(in units of Jmol−1 k −1 ) as a function of P (in
units of atm). Take the pressure range 0 < P < 20atm.
4. Regular binary solutions
Metals A and B form a regular binary solution with a positive heat of mixing (i.e.,
Ω > 0) so that the A − B phase diagram contains a miscibility gap.
(a) Based on the approach discussed in the previous homework, write down the full
expression for G(XB ).
(b) Starting from the equation in part (a), derive an equation for
d2 G
2 ,
dXB
assuming that
GA = GB = 0.
(c) Use the above equation to calculate the temperature at the top of the miscibility
gap, TC , in terms of Ω.
(d) Plot qualitatively the miscibility gap for this system in the phase diagram; indicate
the equation(s) that you would use for a numerical determination.
(e) Similarly, plot on the same diagram as in (d) the chemical spinodal; indicate the
equation(s) that you would use for a numerical determination.
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