NAME:
PHGN341: Thermal Physics
Exam I - February 18, 2011
1. One of the major problems facing the large scale deployment of renewable energy technologies like wind and solar is
their inherent variability. Consider the Bonneville Power Authority load balancing data shown in the figure below.
Bonneville Power has approximately 3.5 GW of installed wind generating capacity, yet as the wind generation (green)
data shows, it rarely actually generates that much. It shows large fluctuations on the time scale of days. Fortunately,
Bonneville has a large hydro capacity that can quickly fill the gaps, and indeed generates a surplus which is sold. If one
were to rely on wind alone, one would need a backup capacity roughly equal to the desired total generating capacity, an
impractical situation. One proposed solution to this problem is to install excess capacity and when the wind generates
a surplus, store the excess energy for use later. One proposed method for large scale energy storage is Compressed Air
Energy Storage (CAES).
(a) (10) Consider a CAES system that compresses air rapidly so the
process may be treated as adiabatic. Calculate the energy stored when 1 m3
of air at standard conditions (P0 = 105 Pa, T0 = 300 K) is compressed
to 20 MPa.
(b) (5) After compression, what is the stored energy density (J/m3 )? (For comparison a lead-acid battery stores about
360 J/m3 .)
(c) (5) What volume of compressed air is needed to store 3 GW-days of energy?
(d) (5) What is the temperature of the compressed gas?
(e) (5) Based on your results, discuss the practical issues associated with adiabatic CAES.
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2. (40) Consider two different Einstein solids initially separated and thermally isolated. System A consists of four oscillators
with quantum of energy, h̄ωA . System B consists of three oscillators with a quantum of energy twice that of System A,
h̄ωB = 2h̄ωA . Let qA be the number of h̄ωA energy quanta in system A, and qB be the number of h̄ωB energy quanta
in system B. Suppose the total energy is fixed at 6 h̄ωA (or equivalently, 3 h̄ωB ).
(a) (20) It takes two energy quanta from system A to excite one quantum of energy in system B; therefore system A
can only exchange even multiples of its energy quanta with system B; so qA can only take on the values {0, 2, 4, 6}
corresponding to qB having the values {3, 2, 1, 0}. Complete the multiplicity tables below for the two systems.
qA
ΩA (4, qA )
qB
0
3
2
2
4
ΩB (2, qB )
1
6
0
(b) (10) Initially, suppose that system A has qA = 6 and system B has qB = 0. Now let the two systems be in thermal
contact, but are otherwise isolated so that the net energy in h̄ωA units is fixed at 6 = qA + 2qB . Use your multiplicity
tables to find the equilibrium value of qA . Explain your reasoning and show all work.
(c) (10) What is the change in entropy in units of Boltzmann’s constant for the process described in part (b)?
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3. (30) The Sackur-Tetrode expression for the entropy of N atoms of mass, m, confined to a two-dimensional area, A, is
A 2πmU
S2d (N, A, U ) = k ln Ω2d (N, A, U ) = kN ln
+2 ,
N h2 N
where U is the total energy and h is Planck’s constant.
(a) (10) From the entropic definition of temperature, derive an expression for the total energy as a function of the
temperature. Comment on your result.
Comment:
(b) (10) From the entropic definition of “pressure” in two dimensions (i.e. force per unit length, instead of force per
unit area), derive an expression for the force per unit length, λ, as a function of the temperature. Comment on your
result.
Comment:
(c) (10) Find an expression for the two-dimensional entropy as a function of number of atoms, temperature, and the
force per unit length, S2d (N, λ, T ).
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