Lecture 9 Overview Ch. 13

1
Lecture 9 Overview (Ch. 1-3)

• Format of the first midterm four problems with
  multiple questions. Total 100 points.
• The Ideal Gas Law, calculation of ?W, ?Q and dS
  for various ideal gas processes.
• Einstein solid and two-state paramagnet,
  multiplicity and entropy, the stat. phys.
  definition of T, how to get from the multiplicity
  to the equation of state.
• Only textbook and cheat-sheets (handwritten!) are
  allowed.
• No homework and lecture notes.
• DONT forget to bring your calculator!

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?
3
Problem 1 (cont.)
(a)
1 2
V ? T ? P const (isobaric process)
2 3
V const (isochoric process)
3 1
T const (isothermal process)
4
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.
5
Problem 1 (cont.)
(b)
- for quasi-static processes
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)
6
Problem 1 (cont)
Lets express both ?Q and dT in terms of dV
(c)
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 ?Q
? 0
3 1
At home recall how these results would be
modified for diatomic and polyatomic gases.
7
Problem 2
One mole of a monatomic ideal gas goes through a
quasistatic three-stage cycle (1-2, 2-3, 3-1)
shown in the Figure. Process 3-1 is adiabatic P1
, V1 , and V2 are given. (a) (10) For each
stage and for the whole cycle, express the work
?W done on the gas in terms of P1, V1, and V2.
Comment on the sign of ?W. (b) (5) What is the
heat capacity (in units R) for each stage? (c)
(15) Calculate ?Q transferred to the gas in the
cycle the same for the reverse cycle what would
be the result if ?Q were an exact
differential? (d) (15) Using the Sackur-Tetrode
equation, calculate the entropy change for each
stage and for the whole cycle, ?Stotal. Did you
get the expected result for ?Stotal? Explain.
8
Problem 2 (cont.)
(a)
1 2
P const (isobaric process)
2 3
V const (isochoric process)
3 1
adiabatic process
9
Problem 2 (cont.)
(c)
1 2
P const (isobaric process)
2 3
V const (isochoric process)
3 1
adiabatic process
For the reverse cycle
If ?Q were an exact differential, for a cycle ?Q
should be zero.
10
Problem 2 (cont.)
Sackur-Tetrode equation
P
(d)
1
2
P1
3
V1
V2
1 2
V ? T ? P const (isobaric process)
V
2 3
V const (isochoric process)
?Q 0 (quasistatic adiabatic isentropic
process)
3 1
as it should be for a quasistatic cyclic
process (quasistatic reversible), because S is
a state function.
11
Problem 3
Calculate the heat capacity of one mole of an
ideal monatomic gas C(V) in the quasistatic
process shown in the Figure. P0 and V0 are
given.
P
10
Start with the definition
P0
we need to find the equation of this process
VV(T)
20
30
0
V
V0
40
12
Problem 3 (cont.)
50
Does it make sense?
C/R
the line touches an isotherm
2.5
1.5
0
V/ V0
1
1/2
5/8
the line touches an adiabat
13
Problem 4
(10) The ESR (electron spin resonance) set-up can
detect the minimum difference in the number of
spin-up and spin-down electrons in a
two-state paramagnet N?-N? 1010. The
paramagnetic sample is placed at 300K in an
external magnetic field B 1T. The component of
the electrons magnetic moment along B is ? ?B
? 9.3x10-24 J/T. Find the minimum total number of
electrons in the sample that is required to make
this detection possible.
- the high-T limit
14
Problem 5
Consider a system whose multiplicity is
described by the equation
where U is the internal energy, V is the volume,
N is the number of particles in the system, Nf is
the total number of degrees of freedom, f(N) is
some function of N.

• (10) Find the systems entropy and temperature
  as functions of U. Are these results in agreement
  with the equipartition theorem? Does the
  expression for the entropy makes sense when T ?
  0?
• (5) Find the heat capacity of the system at
  fixed volume.
• (15) Assume that the system is divided into two
  sub-systems, A and B sub-system A holds energy
  UA and volume VA, while the sub-system B holds
  UBU-UA and VBV-VA. Show that for an equilibrium
  macropartition, the energy per molecule is the
  same for both sub-systems.

(a)
- in agreement with the equipartition theorem
When T ? 0, U ? 0, and S ? - ? - doesnt make
sense. This means that the expression for ? holds
in the classical limit of high temperatures, it
should be modified at low T.
15
Problem 5 (cont.)
(b)
(c)