June 2004
STATISTICAL MECHANICS & THERMODYNAMICS
PROBLEMS FROM THE
1. The partition
function for an ideal gas of N buckyball
molecules(C60, i.e., each molecule is made up of 60 carbon atoms) in
a volume V at temperature T can be written in the form
(a) If z
in turn can be written in the form ,
on what variables (i.e., ) does depend?
What is ?
Explain your answers!
(b) Derive the equation of state for the C60-buckyball
gas.
When Nl buckyballs
condense to form a liquid, the crudest approximation on can make is to treat
these molecules as if they still form a gas, with the additional provisos that
(i) each C60 molecule is assumed to have a potential energy due to interactions with the test of the
molecules, and (ii) each molecule is free to move in a volume of ,
where is a constant.
(c) Based on the above information, find the
partition function for a buckyball liquid consisting of Nl in terms of the function and relevant parameters given in this problem.
(You may want to use the approximation )
(d) Find an expression for the vapor
pressure of an ideal C60 gas in equilibrium with its liquid phase.
2. The entropy-temperature (S-T)
cycle of a “nameless” reversible engine is shown in the figure.
The highest and lowest temperatures Th and Tl
as well as the highest and lowest entropies Sh and Sl
are marked on the figure. The area of the loop in the ST-plane is A
in some units.Clearly S(T) as well as the inverse T(S)
are double-valued functions except at the extremes Th, Tl,
Sh, and Sl. Accordingly, these functions
are more properly written as S(T) = f±(T)
and T(S) = g±(S),
where the two signs + and - denote the
upper and lower branches of these double-valued functions, respectively.
For parts (b)-(d) of this problem
express your answers in terms of the above-defined variables and functions.
(a) Using an arrow, indicate on the figure
the direction of the S-T path traversed by the engine cycle when it does
work in each cycle. Explain this result.
(b) What is (i) the work done by the
engine in one cycle and (ii) the heat input to the engine over one
cycle?
(c) Suppose now that you have a Carnot engine
that operates between the same temperature extremes Th and Tl
as our “nameless” engine. Draw a representation on the ST-plot above, or
on your own reproduction of it, of this Carnot cycle, including the direction
of the path, assuming that the Carnot engine does the same amount of work per
cycle as the “nameless” engine. Describe and quantify your plot.
(d) Is the “nameless” engine more efficient
than, equal in efficiency to, or less efficient than the Carnot cycle you
described in part (c)? Prove this result.
3. Here are two
facts about a substance:
·
The entropy is the following function of T and
V:
·
At constant temperature T0 the
work the substance does on its surroundings as it expands from V0
to V is
.
(a) Find the Helmholtz free energy F,
assuming that it is zero at the state values specified by the subscript 0.
(b) Find the equation of state of this
stuff. Is there a point in parameter space where it is ideal?
(c) For this part we’ll simplify the algebra
by assuming that ln(V/V0) = 1 and also α = 1.
At what temperature T¢ would the system do twice as much work in going
from V0 to V as it does at T = T0?
4. You are
interested in learning about the p-V-T
(pressure-temperature-volume) relation as well as the energy U of a
system. [Feel free to use b = 1/kT if you like.] You have been able to
calculate the grand canonical partition function of your system.
(a) On what variables does depend?
(b) How can you express the pressure and the
energy in terms of ?
(c) What is for a gas of N noninteracting bosons,
expressed as an appropriate product (i.e., your system is in a small box and
the energies are discrete). As a second part to this, express as a discrete sum.
(d) Your system has some peculiar features:
The box is a μ-dimensional cube with sides of length L, and the
energy-momentum relation is e = aps, where a is a proportionality
constant. Using these facts, allow the box to get large and convert the sum
over allowed energies to an integral over a continuum of energies; express as an integral.
(e) Show that pV = (s/μ)U.
5. Consider a
large regular lattice of magnetic atoms in an external field .
The spin dependent part of the Hamiltonian is approximately
,
where the double sum is restricted to the q nearest neighbors of each
atom. For simplicity you can assume spin ½ and that the external magnetic field
is along the z-direction.
(a) Which sign of J leads to
ferromagnetism? Choose this sign for the rest of the problem.
(b) Using mean field theory (the Weiss
molecular-field approximation), obtain the self-consistent equation for the
mean value of each spin, or equivalently for the magnetization when there are N/V
atoms per unit volume.
(c) Find the Curie temperature Tc
in terms of J and q.
(d) For T < Tc
the onset of spontaneous magnetization is proportional to (Tc
-
T)b.
Obtain the value of b in this (mean-field) theory.
(e) Qualitatively, how are these predictions
of mean-field theory modified when fluctuations are taken into account? Are
fluctuations more important in 3-d or in lower dimensions? Is the true Tc
greater or smaller than that computed in part (b)? Is the true b
greater or smaller?
6. (a) Find the Bose-Einstein condensation
temperature TBEC for a large number, N, of
non-interacting atoms of mass M confined to a volume V. Assume
the atoms have spin 0, or have integer spin but the spin degeneracy is lifted
by an applied magnetic field. A derivation of the full result is required (8
points), but you can get partial credit by just answering such questions as:
What is the momentum of an atom in the condensate? What is the value of the
chemical potential for T £ TBEC? What can be said about the
value of TBEC by dimensional analysis alone?
(b) How can one create, in practice, such a
BE condensate? (1 point)
(c) Estimate the number density N/V
necessary to obtain TBEC = 1 mK if the atoms are Na23
(1 point).
Useful integral:
7. An electron
is confined to a plane. Ignore
interaction of electron spin with the magnetic field (i.e., imagine that the
magnetic moment is zero). Within the
plane it moves freely, except the motion is restricted to an area A. When a magnetic field B is applied
perpendicular to the plane, the Landau quantization of the electron orbits
produces energy levels
and wc
= eB/mc is the cyclotron frequency. The degeneracy of the l'th
quantum level is
(independent of l)
where F
= BA is the magnetic flux passing through the area A and F0
= hc/2e is the magnetic flux
quantum. Consider N electrons confined
to area A of the plane and neglect interactions between the electrons.
(a) Calculate the magnetic field dependence
of the Fermi energy m
in the limit of low temperature kBT<<
(b) Sketch the Fermi energy as a function of
magnetic field in the range
0.8B0<B<1.2B0, where
(c) Explain the physical origin of the Fermi
energy change when the field is close to B0.
(d) Electrons of density r = 1019cm-3
are confined to a 1 nm thick layer.
Estimate the strength of the magnetic field which will produce the shift
in Fermi energy near B0.
(e) Estimate the temperature at which the
Fermi energy effect of part (c) can be observed in the laboratory.
8. Consider the
(i) Isothermal compression from
volume Va to volume Vb at temperature T1,
(ii) Heating from temperature T1 to temperature T2 at fixed volume Vb,
(iii) Isothermal expansion from Vb to Va at temperature T2,
and
(iv) Cooling back to temperature T1 at fixed volume
What is the efficiency of the
9. Distinguishable
particles of spin 1/2 with magnetic moment are placed in an external magnetic field H (i.e. the energy of each particle is and its magnetization is ). The
number of particles is not fixed.
(a.) Find the grand canonical partition function at chemical potential m and temperature T
(with
(b.) Find the magnetization from the grand
canonical partition function.
10. Consider an
ideal gas of massless bosons in thermal equilibrium, where the number N of particles is a conserved quantity.
(a) Derive an expression for the number of
thermally excited particles (i.e. particles with momenta ).
(b) If the number-density is ,
find the critical temperature, below which Why does this critical temperature not occur
in a photon gas?
Useful information:
11. Consider a
system consisting of a liquid and a gas in phase equilibrium. Assume that the gas phase can be approximated
by an ideal gas and that the volume occupied by the liquid phase is negligible.
(a) Derive the Clausius-Clapeyron equation
which determines the slope of the phase boundary in the p – V
plane. The equation is
where q is the latent heat of vaporization per molecule, and is the change in volume when one molecule is
transferred from the liquid to the gas.
(b) Determine ,
the change in volume per particle of the gas with temperature along the phase
equilibrium curve. Express your answer
in terms of p, q, and kT.
(c) Determine the specific heat per molecule
of the gas ccurve along the phase equilibrium curve. This is defined as
where qg is the amount of heat per molecule added to the gas
only, and as the temperature is varied, the pressure and volume are varied to
keep the system along the coexistence curve.
Express your answer in terms of q and cp.
(d) Discuss the behavior of the expressions
derived in (a) and (b) at low temperature.
Give a qualitative physical explanation of the sign of these
expressions.
12. (a) A gas does work when it expands
adiabatically. What is the energy source
for this work? Consider an adiabatic
curve plotted for a gas on a P-V diagram.
Give a general argument, either physical or mathematical, to prove that
the adiabatic curve must always be steeper than the isothermal one where they
cross.
(b) Forty-four grams of CO2 taken
to be an ideal gas, are used to operate a Carnot heat engine between the
isotherms T1 = 273K and T2 = 373K, and the adiabatics
S1 = 198 Joules/K and S2 = 222 Joules/K. In one cycle, how much work does the engine
do, and what is it's efficiency? By what
factor does the volume change during either isothermal segment of the cycle.
13. Consider a
system of N distinguishable
non-interacting spins in a magnetic field B. Each spin has a magnetic moment of size m, and
each can point either parallel or antiparallel to the field. Thus, the energy of a particular state is
(a) Determine the thermodynamically defined
internal energy U of this system as a
function of b B,
and N
by employing an ensemble characterized by these variables.
(b) Determine the entropy S of this system as a function of b, B, and N.
(c) Determine the behavior of the energy U and the entropy S for this system in the limit T® 0.
14. (a) Correct the following Kelvin statement
of the Second Law: "No process exists which converts all of a given amount of heat into work".
(b) The following process appears to violate
the above incorrect statement:
A cylinder of gas is at a pressure p,
a temperature T and volume V. Heat DQ is added at constant
volume thus increasing the pressure to p
+ Dp and the temperature to T + DT. The gas is now allowed to
expand adiabatically and reversibly doing an amount of work equal to DQ and
thus converting the heat added into useful work.
Plot the p-V phase diagram.
Integrate the
combined First and Second Laws and show that DS > 0 for the process starting at the initial point and ending
at the final point.
15. An ideal
monatomic gas is contained in a thermally insulated cylinder. The gas is compressed by a piston moving with
a uniform velocity u, which is small compared to the average molecular
speed. Using kinetic theory, compute the
rate of change of the pressure with time.
Hence compute dP/dV and show
that the result is consistent with the adiabatic equation of state = constant.
Use the following standard symbols:
P, V, T = pressure, volume, temperature
A = area of piston
m = mass of molecule
n
= number of molecules per unit volume
vz = component of molecular velocity normal to
the piston
16. (a) What is the molar specific heat at
constant volume of a perfect diatomic gas at room temperature?
A rough estimate is sufficient. The
presence of molecular vibrations can usually be neglected. Explain why this is correct.
(b) What is the specific heat at constant
pressure of the same gas?
17. The surface
temperature of the sun is T0
(=5500K); its radius is R0(=
7 x 1010 cm); the radius of the moon is R; the mean distance between sun and moon is L (= 1.5 x 1013
cm). Assume that both the sun and the
moon absorb all electromagnetic radiation incident upon them.
(a) What is the total flux of energy from
the sun? It is sufficient to indicate
how this depends on Ro and T0.
(b) What is the incident flux of energy on
an element of the lunar surface, as a function of a properly defined angle q?
What is the flux of energy radiated by the same element?
(c) Neglecting the thermal conductivity of
the lunar soil, compute the temperature distribution on the lunar surface.
18. A thin-walled
vessel of volume V, kept at constant
temperature, contains a gas which slowly leaks out through a small hole of area
A. The outside pressure is very low
compared to the pressure in the vessel.
(a) Relate the flow of molecules through the
hole to the density and some average velocity of the gas molecules in the
vessel. It is not necessary to compute
explicitly numerical factors of order unity.
(b) Find the time required for the pressure
(or density) in the vessel to decrease to 1/e
of its initial value.
19. A neutron star
consists mainly of neutrons, but it also contains protons and electrons in
thermal equilibrium as a result of the reactions
.
(The antineutrino or neutrino on the right side of these reactions escapes the
star, hence do not reach equilibrium with the other particles, and their
chemical potentials can be set to zero.) In the model of a neutron star to be
treated here, the neutrons, protons, and electrons are all taken to be
ultrarelativistic [ ]. (Real life is more complicated.)
(a) Obtain an expression for the Fermi
energy EF as a function of the number density N/V
for an ultrarelativistic gas of identical fermions.
(b) Find the thermal equilibrium relation
between EFn, EFp, and
EFe for a neutron star, where the superscripts on EF
describe particle types.
(c) Using the results of the first parts of
the problem, find in our model the relative amounts of neutrons, protons, and
electrons.
20. A sample of
weight W0 hangs from an
elastic thread of cross-section Ao and length Lo (at
equilibrium). For a small change of the
weight, DW, the length of the thread changes by
an amount DL
proportional to DW, DL = C D W.
(a) How do you expect C to depend on A0 and L0?
(b) Write a “free energy” which is a
function of DW and a “free energy” which is a
function of DL.
Make an explicit analogy to the usual F and G functions.
(c) Write an expression for the root mean
square fluctuation in the position of the sample. Exhibit the dependence on temperature and on
the geometrical factors A0
and L0?
21. A solution of
heavy molecules in water is put in a centrifuge (a cylinder of radius R) and
spun with angular frequency w. Let the mass of
a molecule be M and the mass of the equivalent volume of water be m (M >
m). Suppose the initial concentration of
the solution to be co. Find the concentration at equilibrium in the
spinning centrifuge as a function of the radial coordinate r (r <R). Assume that the molecules in solution do not
interact with each other.
22. Consider a
dilute gas of identical molecules, each having (in addition to very tightly
bound electrons) one loosely bound electron with binding energy -Eb. As a function of temperature and of the
volume per molecule what fraction of the molecules are ionized?
23. A closed
cylinder is divided by partitions into 3 equal compartments of volume V which
each contain one mole of a different inert ideal gas. The gases are at the same temperature. Calculate the change in entropy which occurs
when the partitions are removed allowing the three gases to diffuse
isothermally to a uniform mixture. Use
either macroscopic thermodynamics or statistical thermodynamics.
24. (a) Derive an expression for the
Joule-Thompson coefficient m º (¶T/¶p)H for a general gas in terms of cp and
a.
(b) Prove that no cooling of an ideal gas
will result when it is forced through a porous plug.
25. A system
maintained at constant temperature T is compressed from pressure P1
to pressure P2. Find (a) the change in entropy and (b) the change in
internal energy.
26. a) Explain briefly the theorem of
equipartition of energy.
b) The lattice translational and
rotational modes for diatomic molecules in solids and liquids have frequencies of the order of 1012
Hz while stretching modes have frequencies of the order of 1013
Hz. On the basis of these numbers,
estimate the numerical value of the specific heat of liquid N2
around 100°K.
(h
= 6.6 x 10-27 erg, k = 1.38 x 10-16 ergs/°K)
27. Prove that the
maximum work which can be extracted from an arbitrary system held at constant
volume by cooling it from a temperature T1 to a temperature To
is
W
= N
Cv [ (T1 - T0) - T0 ln
(T1/T0) ]
28. Consider a
closed composite system made up of two sub-systems separated by a rigid,
impermeable, adiabatic partition. System
A on one side has NA particles of a monatomic ideal gas with entropy SA, internal energy UA
and volume VA. System B on
the other side has MB particles of a diatomic gas with corresponding properties SB, UB, VB.
Initially the temperature TA is greater than TB. The
partition is now made diathermic.
(a) Prove that when the systems achieve
equilibrium, the final temperatures are equal.
(b) Prove that the net flow of heat has
occurred from A to B.
(c) If VA = VB, and numerically NA = MB, will the final pressures be equal? Give a reason for your answer.
29. The
thermodynamic quantity Cp is measured as a function of temperature
under constant pressure conditions, yielding the experimental curve Cp
vs T. You wish to make a detailed comparison of this curve with theory; but all
you can find is a curve Cv vs T, derived theoretically under the
assumption of constant volume. You elect
to convert your experimental curve Cp vs T to constant-volume conditions.
(a) Derive the relation between Cp
and Cv at any particular temperature.
(b) Interpret physically any derivatives
appearing in your answer to a), and suggest whatcompanion experimental
measurements must be made before conversion can be carried out.
(c) At what point is Cp(T) = Cv(T)?
(d) Outline the steps you would follow to
convert the Cp vs T to an experimental Cv vs T curve.
30. The
fundamental relation in the entropy representation for black-body radiation is
S
= 4/3 (4s
VU3/c)1/4
where s
is the Stefan-Boltzmann constant.
(a) Find the equations of state for
black-body radiation.
(b) Evaluate Cp, Cv and
the ratio Cp/Cv for black-body radiation.
(c) Show that the Gibbs Free Energy, G, and hence the chemical potential m, are zero for black-body radiation and
explain why.
(d) Show that PV4/3 = const. in
adiabatic expansion.
31. A thermally
insulated container is divided into two parts by a thermally insulated
partition. Both parts contain one mole
of an ideal gas with a constant specific heat Cv. One part is at a temperature T1
and pressure p1 while the other is at a temperature T2 =
2T1 and a pressure P2 = 2p1. If the partition is removed, calculate
(a) The final temperature and pressure.
(b) The change in entropy if the gases are
different.
Hint: The entropy of N moles of an ideal gas in a volume V at a temperature T
may be written
where So is a constant.
32. Using Maxwell
relations, derive a general expression for the Joule-Kelvin coefficient, (¶T/¶p)H,
of a real gas in terms of easily measurable quantities. Show that this coefficient is 0 for an ideal
gas and has the value for a van der Waals gas when a and b are small. The equation of state for a Van der Waals gas
is (p + a/v2) (v - b) = RT.
33. A
thermodynamic system consists of a perfect monatomic gas confined in a cylinder
by a frictionless piston.
(a) What physical process would have to be
carried out to change the temperature at constant entropy S1 from an
initial value T1 to a greater final value T2 and what
physical conditions would have to be satisfied by the walls of the
container? Give a quantitative
expression for the physical change in terms of T1 and T2.
(b) What physical process would have to be carried
out to change the entropy at constant temperature T2, from an
initial value S1 to a small final value S2 and what
physical conditions would have to be satisfied by the walls of the
container? Give a quantitative
expression for the physical change in terms of S1 and S2.
34. The Gibbs free
energy of n moles of an ideal monatomic gas with reference to a standard state
(Po,To) is given by the expression:
G
- Go = nRT 1n (P/Po) - ncp T 1n (T/To)
+ ncp (T - To) - So (T - To)
Derive an expression for the entropy of n moles of a perfect gas as a function
of the temperature and the pressure.
Show that there is a change in entropy
when two volumes V1 and V2,
at temperatures T1 and T2, respectively, each at the same
pressure P and each containing the same number of moles n of the same monatomic
ideal gas are allowed to interdiffuse.
Assume that the thermal capacities of the adiabatic walls of the
containing vessels can be neglected.
35. Given the
adiabatic equation of state pVg = f(S,N) and the heat capacity at constant
pressure Cp = Nc, where g and c are constants and f(S,N) is a function of the
entropy S and the number of particles N, deduce the enthalpy H = U + pV of the
system as a function of S, N and p.
36. Show that if a
saturated vapor is expanded adiabatically it becomes supersaturated (liquid
should appear) if the following condition is satisfied
v2b2l>(v2-v1)cp2 ,
where v is the volume, cp the heat capacity at constant pressure, l
the latent heat (all per particle), b the thermal
expansion, and the indices 1 and 2 refer to the liquid and the vapor
respectively. Show that this is always
satisfied when the vapor can be considered to be a perfect gas.
37. Consider a
system of a large number N of distinguishable particles in which the energy of
each particle can assume only two distinct values, 0 and e >
0.
(a) Find the entropy of the system.
(b) Find the most probable values for the
occupation numbers n0 and n1 of the two levels.
(c) Find the temperature as a function of
the internal energy U and show that
it can be negative.
(d) Discuss what happens when a system at
negative temperature is allowed to exchange heat with a system at positive
temperature.
38. Consider a
thermodynamic system containing a constant number of particles. The fundamental expression dU = TdS - PdV
defines a function U(S,V).
(a) For a given volume, draw schematically
what U(S,V) must look like in order to satisfy the conditions T > 0, CV = T (¶S/¶T)V
> 0, and T = 0 for S = 0. (Nernst
Theorem).
(b) Do you know of any system for which T
could be both positive and negative?
(c) What should U(S,V) look like then in
order to continue to satisfy CV >
0 and the Nernst Theorem?
(d) Calculate the isothermal compressibility
c
= -(1/V) (¶V/¶P)T and the thermal expansion coefficient b =
(1/V) (¶V/¶T)P
in terms of the derivatives of U(S,V).
39. Consider a
classical system of non-interacting particles in equilibrium at temperature T.
The fraction of particles having values between (px, px
+ dpx); (py, py + dpy);
(pz + dpz) for the components of momentum,
is proportional to exp (-e/kT) dpx dpy dpz,
where e(px,
py, pz)
is the energy, not necessarily of the form p2/2m. Show that
<px
¶e/¶px>
= k T
where < > represents the average value.
Deduce as a particular case the classical theorem of equipartition of
energy, i.e., the case where e(px, py, pz) = p2/2m.
40. A perfect gas
initially at a pressure po and a certain temperature is allowed to
expand adiabatically and reversibly until the pressure p1 is
reached. At constant volume it is then
warmed-up slowly back to the original temperature. If the final pressure is P2, show
that
g = Cp/Cv
= ln(po/p1)/1n (po/p2).
41. Show that the
equation of an adiabatic for a substance with constant Cp is of the form Tf (e) = const. Unit mass of such a substance undergoes the
following cycle of changes:
(a) heating
at constant pressure from temperature T1 to T2,
(b) adiabatic expansion until its temperature falls to T3,
(c) cooling at constant pressure until its temperature falls to T4,
(d) adiabatic compression until its temperature rises to T1. If the
initial and final states are the same, show that the work done in the cycle is
Cp (T2-T1+T4-T3).
If T2 and
T4 are fixed, and T1 and T3 are variable, show
that the work done in the cycle is a maximum when and is thus equal to
42. Two identical perfect gases with equal temperatures T and equal numbers of molecules N, but with different pressures P1 and P2 and volumes V1 and V2, are contained in separate vessels. Knowing that the entropy of a perfect gas is S = +Nk ln P Nf(T) (where f(T) is a function of the temperature only), find the change in entropy when the vessels are connected.