Thermodynamics and Statistical Mechanics II - Home Exercise 2

Thermodynamics and Statistical Mechanics II - Home Exercise 2
1. Calculation of
dT
dp
dT
dp
for water Calculate from the vapor-pressure equation the value of
near p = 1 atm for the liquid-vapor equilibrium of water. The heat of vaporization
at 100◦ C is 2260Jg −1 . Express the result
Kelvin
.
atm
2. Water-vapor coexistence Consider a water ? vapor coexistence.
(a) Calculate the heat capacity of the water vapor along the coexistence curve. You
may treat the vapor as an ideal gas.
(b) Calculate the volume change of the vapor as a function of temperature for this
water-vapor equilibrium.
3. Discovering the density Consider a liquid that is contained within a column of
constant cross-section. The column is open from its upper side. When the material is
cooled to a temperature of −5◦ C, all the liquid under a certain height freezes. When
the temperature is dropped to −5.2◦ C, the interface between the liquid and the solid
is 40cm higher than the previous situation. The latent heat (per unit mass) is 2 cal
g
and the density in the liquid phase is 1 cmg 3 . What is the density of the solid phase?
Hint: notice that the pressure at the original height of the interface doesn’t
change (why?)
4. 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.
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(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.
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