PHGN341: Thermal Physics
Exam II - April 14, 2010
NAME:
1.
2
P
1
3
V
Consider an engine cycle that operates in the shape of a right triangle as shown. The engine operates in the clockwise
direction and uses Helium gas as the medium. Analyze this engine by filling out the table below assuming the following
conditions: P1 = P3 = 1.0 bar, V1 = 1.0 m3 , P2 = 5.0 bar, V2 = V3 = 4.0 m3 . From your table find the efficiency of the
triangle engine under these operating conditions. Show all work and use the back if you need more space.
Step
Work
∆U
1→2
2→3
3→1
1
Q
Substance
Calcite
Aragonite
2.
∆f H (kJ)
-1206.9
-1207.1
∆f G (kJ)
-1128.8
-1127.8
S (J/K)
92.9
88.7
V (cm3 )
36.93
34.15
Calcium Carbonate (CaCO3 ) can exist in two crystalline forms: calcite and aragonite. The thermodynamic properties
of one mole of these forms at 298 K and 1 bar of pressure are given in the table. Assume that the temperature of the
earth increases by 16 C per kilometer of depth while the pressure increases by 32 MPa per kilometer of depth. Assume
that at ground level the temperature is 25 C (298 K) and the pressure is 1 bar (105 Pa). At what depth is Aragonite
the stable form? Show all work.
3. From the Classisus-Clapeyron relation and the ideal gas law, the liquid-vapor saturation pressure as a function of
temperature is well approximated by:
L
Ps (T ) = P0 e− RT ,
where L is the molar latent heat of vaporization, R = 8.315 J/K is the molar gas constant, and T is the temperature.
For the water-steam phase transition one measures Ps = 0.1234 bar at T = 50 C (323 K), with L = 42.92 kJ/mol.
a. From these values find P0 .
b. Suppose the temperature is 30 C and the relative humidity is 50%. What is the dew point (temperature)? (Recall
that the relative humidity is the ratio of the actual water vapor pressure to the saturation pressure.)
2
4. Consider a system of N spin-3/2 paramagnetic atoms at temperature, T, in an external magnetic field, B0 . The energy
of such an atom in an external magnetic field is U = −µ0 B0 mj , where mj is the quantum number for the z-component of
the angular momentum and µ0 is the Bohr magneton. (Recall that a spin-3/2 atom can have four possible m-substates:
mj = {−3/2, −1/2, +1/2, +3/2}.
a. Find the single atom partition function.
b. Find the average energy of one atom.
c. What is that partition function of the N-atom system assuming the atoms are indistinguishable?
d. What is the Helmholtz free energy, F , for this system? (Use Sterling’s approximation, N ! ' N N e−N .)
e. What is the chemical potential?
3