A10429W1

SECOND PUBLIC EXAMINATION

Honour School of Physics Part A: 3 and 4 Year Courses

Honour School of Physics and Philosophy Part A

A1: THERMAL PHYSICS

TRINITY TERM 2019

Wednesday, 19 June, 9.30 am – 12.30 pm

Answer all of Section A and three questions from Section B.

For Section A start the answer to each question on a fresh page.

For Section B start the answer to each question in a fresh book.

A list of physical constants and conversion factors accompanies this paper.

The numbers in the margin indicate the weight that the Examiners expect to
assign to each part of the question.

Do NOT turn over until told that you may do so.

1
 Section A

1. According to simple kinetic theory, the viscosity of a gas of density ρ is given by

1
η ' ρλhvi,
3
where λ and hvi are the mean free path and mean speed of the molecules, respectively.
Estimate the molecular diameter of benzene given that the viscosity of the vapour at a
temperature of 293 K is 7.6×10−6 N m−2 s and the molar mass is 78 g mol−1 . [6]

2. State the Clausius inequality, and use it to show that thermodynamic entropy is
a function of state. [4]

3. One kilogram of water is compressed reversibly and isothermally at 300 K by an

increase in pressure from 1 bar to 20 bar. Calculate the heat ejected from the water. [6]
 
[Density of water = 997 kg m−3 , thermal expansivity βp = 1
V
∂V
∂T p = 2 × 10−4 K−1 .]

4. The Clausius–Clapeyron equation, which describes the boundary between two

phases in equilibrium, may be written
dp L
= ,
dT T (ρ2 − ρ−1
−1
1 )

where ρ1 and ρ2 are the densities of the two phases and L is the specific latent heat. A
skater has a mass of 75 kg, and the blade of her ice skate has an area in contact with the
ice of 10 mm2 . On the assumption that the ice under the skate must melt for effective
skating to take place, estimate the lowest temperature the ice can have to allow this. [6]

[Specific latent heat of fusion of ice, L = 3.34 × 105 J kg−1 ,

density of water = 1000 kg m−3 , density of ice = 917 kg m−3 .]

5. (a) A manufacturer claims that a bag of jelly beans contains an average of 312
sweets. A student buys 6 bags, counts the number of beans in each bag, and finds 309,
314, 310, 307, 311 and 312. Do these figures cast doubt on the manufacturer’s claim? [4]
(b) Road accidents in a certain area occur at an average rate of one every two
days. Calculate the probability of 0, 1, 2, . . ., 6 accidents in a given week in the area.
What is the most likely number of accidents per week? What is the expectation value
for the number of accident-free days per week? [5]

[The Poisson distribution is P (n) = µn e−µ /n! ]

6. Show that for an ideal quantum gas of fermions at T = 0 the mean speed of the
particles is 43 vF , where vF is the Fermi velocity. [4]

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 7. Define the coefficient of thermal conductivity κ. A sphere of radius a is embedded
in a medium of thermal conductivity κ. The surface of the sphere is held at a tem-
perature Ts , and the temperature in the medium at large distance from the sphere is
T∞ . Show that in the steady state, the rate of loss of heat from the sphere by thermal
conduction is given by
h = 4πκa(Ts − T∞ ). [5]

Section B

8. A paramagnetic substance contains N non-interacting spin- 21 ions, each carrying a

magnetic moment µ. The substance is placed in a magnetic field of flux density B which
induces a magnetic moment m. The work done on the substance due to an infinitesimal
change in B is −m dB.
(a) Write down the partition function for a temperature T , and hence show that

µB
m = N µ tanh( ).
kB T [5]

(b) Show that the magnetic internal energy of the substance is U = −mB. [4]

The substance is taken around the thermodynamic cycle OXY shown in the dia-
gram. OX and YO are isotherms at temperatures T1 and T2 , respectively, and m has a
constant value m1 along XY. All processes are reversible.
(c) Show that the entropy is constant along XY. [3]
(d) Assuming m1 /N µ  1, calculate Wtot , the total work done on the substance
in one cycle, and Qin , the heat absorbed by the substance along YO. [6]
(e) Hence, or otherwise, show that if the cycle is used for refrigeration the coeffi-
cient of performance η = Qin /Wtot depends only on T1 and T2 . [2]

[You may use the approximation tanh x ' x for x  1.]

A10429W1 3 [Turn over]

 9. (a) What is meant by the term critical point as applied to a real (non-ideal) gas?
Sketch the p–V phase diagram for a typical real gas, showing isotherms that pass above,
below and through the critical point. What is the physical significance of the region
below the critical point on the p–V diagram? [6]
(b) The Berthelot equation of state for one mole of a real gas may be written
 a 
p+ (V − b) = RT.
TV 2
Obtain expressions for Tc , Vc and pc , the critical temperature, volume and pressure,
respectively, in terms of the parameters a, b and R. Show that pc Vc /(RTc ) = Z, where
Z is a dimensionless constant which should be determined. [8]
(c) A Berthelot gas undergoes a Joule expansion from a volume V1 to a volume V2 .
Obtain the change in squared temperature ∆(T 2 ) of the gas in terms of a, V1 , V2 and
the constant-volume heat capacity CV . Why is ∆(T 2 ) independent of b? [6]

10. (a) Explain the terms distinguishable and indistinguishable as applied to a collec-
tion of particles. [4]
(b) A perfect gas of atoms each of mass m is confined to a surface of area A at
temperature T . Assuming non-relativistic speeds, show that the single-particle partition
function Z1 describing translational motion of the atoms is given by

Z1 = αAT,

where α is a constant which should be determined. [5]

(c) Show that the entropy S of such a gas containing N indistinguishable atoms
is given by
S = N kB {2 + ln(Z1 /N )} ,
and hence establish that under adiabatic conditions T ∝ A−1 . [6]

[You may use Stirling’s approximation ln N ! ' N ln N − N for large N .]

(d) Another perfect, non-relativistic gas confined in two dimensions consists of a

mixture of nA atoms of type A and nB atoms of type B, where nA + nB = N . The
A atoms are distinguishable from the B atoms, but both types of atom have the same
mass m. Show that the entropy of this gas is higher than that in part (c) by

∆S = −N kB (PA ln PA + PB ln PB ) ,

where PA = nA /N and PB = nB /N . What is the physical origin of ∆S? [5]

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 11. (a) The energy density u(ν) per unit frequency ν associated with electromagnetic
radiation in a cavity at temperature T is given by

8πhν 3 1
u(ν) = .
c3 ehν/kB T − 1
Show that the total energy E radiated each second from unit area of a black body at
temperature T is given by
E = σT 4 ,
and find an expression for the Stefan–Boltzmann constant σ. [4]
h Z ∞ x3 dx π4 i
= .
0 ex −1 15

(b) Suggest an experiment which could be performed to measure σ. [4]

A B
R1 R1
Rs
T1 T1
R2 R2 Ts
T2 T2

(c) A long cylinder of outer radius R1 is held at a fixed temperature T1 and

suspended co-axially inside another long cylinder (diagram A). The outer cylinder has
internal radius R2 and is held at a fixed temperature T2 . Both cylinders are ideal black-
body radiators, and the space between the cylinders is evacuated. Find the fraction of
radiation from the outer cylinder which reaches the inner cylinder, and hence show that
the rate of heat transfer per unit length from outer to inner cylinder is given by

W21 = 2πR1 σ(T24 − T14 ). [4]

(d) Liquid nitrogen at T1 = 77 K flows down the inner cylinder, which has radius
R1 = 5 mm. The outer cylinder is held at T2 = 300 K. A thin-walled cylindrical shield
of radius Rs = 10 mm is suspended co-axially in the evacuated space between the two
original cylinders (diagram B). The shield is also an ideal black-body radiator. Calculate
the temperature Ts of the shield when the system reaches steady state, and find the
factor by which the shield reduces the rate of heat transfer. [8]

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