Results of Midterm 1

1
Results of Midterm 1
of students
10
20
30
40
50
60
70
80
90
0
points
Grade Points
A 80-95
B 70-79
B 55-69
C 45-54
C 30-44
D,F lt30
2
Problem 1
One mole of a monatomic ideal gas goes through a
quasistatic three-stage cycle (1-2, 2-3, 3-1)
shown in the Figure. T1 and T2 are given. (a)
(10) Calculate the work done by the gas. Is it
positive or negative? (b) (20) Using two methods
(Sackur-Tetrode eq. and dQ/T), calculate the
entropy change for each stage and for the whole
cycle, ?Stotal. Did you get the expected result
for ?Stotal? Explain. (c) (5) What is the heat
capacity (in units R) for each stage?
(a)
1 2
V ? T ? P const (isobaric process)
2 3
V const (isochoric process)
3 1
T const (isothermal process)
3
Problem 1 (cont.)
Sackur-Tetrode equation
(b)
V
3
V2
2
V1
1
1 2
V ? T ? P const (isobaric process)
T
T1
T2
2 3
V const (isochoric process)
3 1
T const (isothermal process)
as it should be for a quasistatic cyclic
process (quasistatic reversible), because S is
a state function.
4
Problem 1 (cont.)
(b)
V
3
V2
2
1 2
V ? T ? P const (isobaric process)
V1
1
T
T1
T2
2 3
V const (isochoric process)
3 1
T const (isothermal process)
5
Problem 1 (cont)
(c)
Lets express both Q and dT in terms of dV
V
3
V2
2
1 2
V ? T ? P const (isobaric process)
V1
1
T
T1
T2
2 3
V const (isochoric process)
T const (isothermal process), dT 0 while dQ
? 0
3 1
6
Problem 2
1020 electrons form a two-state paramagnet. The
system is placed in an external magnetic field B
1T. The component of the electrons magnetic
moment along B is ? ?B ? 9.3x10-24 J/T.   (a)
(15) At T 300K, find the ratio N?/N? using
Boltzmann distribution. Calculate the entropy of
the system, make reasonable approximations. (b)
(15). Repeat the same for T 0.1K
(a)
- the high-T limit
Weve obtained the formula for S in this case
7
Problem 2 (cont.)
(b)
This is the high-T limit, we can follow two
paths
8
Problem 3
You are in possession of an Einstein solid with
three oscillators and a two-state paramagnet with
four spins. The magnetic field in the region of
the paramagnet points up and is carefully tuned
so that µB ?, where µB is the energy of a spin
pointing down, -µB is the energy of a spin
pointing up, and ? is the energy level
separation of the oscillators. At the beginning
of the experiment the energy in the Einstein
solid US is 4 ? and the energy in the paramagnet
UP is -4 ?. (a) (4) Using a schematic drawing of
the Einstein solid, give an example of a
microstate which corresponds to the macrostate US
4 ?. (b) (4) Using a schematic drawing of the
paramagnet, give an example of a microstate which
corresponds to the macrostate UP -4 ?. (c) (8)
Considering that the system comprises the solid
and the paramagnet, calculate the multiplicity of
the system assuming that the solid and paramagnet
cannot exchange energy. (d) (14) Now let the
solid and paramagnet exchange energy until they
come to thermal equilibrium. Note that because
this system is small, there will be large
fluctuations around thermal equilibrium, but
lets assume that the system is not fluctuating
at the moment. What is the value of US now? Draw
an example of a microstate in which you might
find the solid. What is the value of UP now? Draw
an example of a microstate in which you might
find the paramagnet.
9
Problem 3 (cont.)
Einstein solid
Two-state paramagnet
E2 ?BB
?
2?
E1 - ?BB
Most of the confusion came from the fact that we
usually measure the energy of an oscillator in
the Einstein solid from its ground state (which
is 1/2 ? above the bottom of the potential well),
whereas for the two-state paramagnet weve chosen
the zero energy in the middle of the energy gap
between spin-up and spin-down levels. The
avoid confusion, consider the number of energy
quanta ? available for the system.
(a)
US 4 ?
UP -4 ?
E2 ?BB
(b)
E1 - ?BB
(c)
10
Problem 3 (cont.)
(d)
In equilibrium, the multiplicity is maximum. The
two-state paramagnet can absorb only multiples of
2?. Two options 2? is transferred from S to P,
and 4? is transferred from S to P.
2? transfer S N 3, q 2, P N 4, N? 1
4? transfer S N 3, q 0, P N 4, N? 2
Thus, the equilibrium situation corresponds to
the transfer of 2? from the Einstein solid to the
two-state paramagnet
Example of one of the equilibrium microstates
E2 ?BB
?
2?
E1 - ?BB