Answer ALL SIX questions from Section A and ANY THREE questions from

Section B

The numbers in square brackets in the right-hand margin indicate a provisional allocation

of maximum possible marks for difffferent parts of each question.

The following may be assumed, if required:

Speed of light in vacuo

c

2.998 × 108 m s−1

Planck’s constant

~

1.054 × 10−34 m2 kg s−1 (=J s)

Gravitational constant

G

6.67 × 10−11 m3 kg−1 s −2

Boltzmann’s constant

k

1.381 × 10−23 m2 kg s−2 K−1 (=J K−1 )

Stefan-Boltzmann constant

σ

5.670 × 10−8 W m−2 K−4

Electron mass

me

9.109 × 10−31 kg

Proton mass

mp

1.67 × 10−27 kg

Mass of of the sun

M

1.989 × 1030 kg

P ∞ j=0 z j =

1

1−

z , for |z| < 1 .

PHAS2228/2018 1

CONTINUEDSection A

(Answer ALL SIX questions from this section)

1. (a) State the four laws of thermodynamics.

[4]

(b) Wet lettuce is put into a spin dryer. Explain why the entropy of the system

[2]

comprised of salad and water is reduced by operating the spinner. How is this

fact reconciled with the second law of thermodynamics?

2. In this question, the symbols E, S, V , p, T stand for, respectively, the internal

energy, entropy, volume, pressure and temperature of the system under scrutiny.

(a) Express the fifirst law for a generic transformation of a gas with a fifixed number

[1]

of particles, in terms of the change in internal energy ∆E, the heat received

from a reservoir ∆Q, and the work done by the gas p∆V .

(b) Let ∆S be the change in entropy of the gas and ∆Sr be the change in entropy

[2]

of the reservoir during the transformation. Express the second law as an

inequality involving ∆S and ∆Sr.

(c) Show that, if the entropy and volume are kept constant, the second law is

[3]

equivalent to minimising the internal energy E.

3. (a) Defifine the terms microstate and macrostate.

[2]

(b) Let pj be the probability of the microstate j in a system with Ω microstates

[4]

and entropy S = −k P Ω j=1 pj ln pj . Show that, if the Ω microstates are all

equally likely, the entropy S of the system is the Boltzmann entropy k ln Ω.

(c) Consider a reservoir with the number of accessible states at energy E given

[2]

by Ω(E) = eβE. By adopting the expression for the Boltzmann entropy given

above, and the defifinition of temperature, prove that β =

1

kT

, where T is the

temperature of the reservoir.

4. (a) Describe the physical circumstances that the grand canonical ensemble is de-

[2]

signed to represent. The grand partition function ZG at temperature T =

1/(βk) and chemical potential µ of a generic system with microstates labelled

by j (for j = 0, 1, 2, . . . ,∞), with energy Ej and number of particles Nj in

each microstate, is given by

ZG = X

j

e −β(Ej−µNj ) .

(b) Derive an expression for the average number of particles h Ni as a derivative

[4]

of a function of ZG.

PHAS2228/2018 2

CONTINUED5. (a) State a symmetry property satisfified by the wavefunction of many identical

[2]

bosons and the corresponding property satisfified by the wavefunction of many

identical fermions.

(b) Describe what is meant by a degenerate Fermi gas.

[2]

(c) Describe what is meant by a Bose-Einstein condensate.

[2]

6. (a) Write down the occupation number f(ε) of an electron at energy ε in a semi-

[2]

conductor at temperature T and chemical potential µ.

(b) Brieflfly describe what is meant by a “hole” in a semiconductor, and explain

[4]

why a hole can be regarded as a particle of positive charge +e, with an occu

pation number equal to 1 − f(ε) in the valence band.

(c) The density of conduction electrons n in an undoped semiconductor is given

by n = α e β(µ−εg) , where εg is the energy band gap, β = 1/(kT) and α is a

constant; the density of holes in the same semiconductor is given by α e −βµ .

Apply the equation of charge neutrality to determine the value of the chemical

[2]

potential µ.

PHAS2228/2018 3

CONTINUEDSection B

(Answer ANY THREE questions from this Section)

Note: only three Section B answers will be marked

7. A complex physical system with n microstates labelled by j, with energies {Ej},

is in contact with a bath at a temperature T. A generic state of the system is

described by the set of probabilities {pj}, for j = 1, . . . , n.

The thermodynamic internal energy E of the system may be represented by the

average

E =

n

X

j=1

pjEj ,

whilst the system’s entropy S is given by

S = −k

n

X

j=1

pj ln pj .

(a) The system undergoes an infifinitesimal transformation, whereby the energy

Ej of each microstate is changed by an amount dEj and its probability pj is

changed by an amount dpj . Write down an expression for the infifinitesimal

[6]

change dE in the internal energy of the system. By using the fifirst law, identify

the quantities dQ (heat transferred to the system) and dW (work performed on

the system) writing them down in terms of {Ej}, {pj}, and their infifinitesimal

increments.

(b) Show that the infifinitesimal variation dS in the entropy of the system satisfifies

[4]

dS = −k

n

X

j=1

ln(pj )dpj .

(c) Write down an expression for the total entropy change dStot (including the

[4]

environment) during the transformation.

(d) What should dStot be equal to when the system attains equilibrium? Show

[6]

that, at equilibrium at temperature T = 1/(βk), the system must follow the

canonical distribution

pj =

e −βEj

P n

l=1 e −βEl

.

PHAS2228/2018 4

CONTINUED8. (a) Two non-interacting electrons in a quantum dot may be described as a system

of two indistinguishable spin 1/2 fermions that can occupy two single-particle

states, with energy 0 and ε respectively. Identify all microstates of the system.

[5]

Evaluate the number of microstates with energies 0, ε and 2ε. (Hint: the spin

of the particles is relevant to distinguish microstates).

(b) Calculate the canonical partition function of the system at temperature T.

[2]

(c) Derive an expression for the entropy when the system is in thermal contact

[6]

with a bath at a temperature T. What is the entropy at T = ε/[k ln(2)]?

(d) A magnetic fifield is turned on, which shifts the energy of electrons with spin

s = +1/2 by +ε, and the energy of electrons with spin s =

−1/2 by −ε.

Determine the new energy of each microstate and the canonical partition

[3]

function in the presence of the magnetic fifield.

(e) Determine the entropy of the system in the presence of the magnetic fifield at

[2]

temperature T = ε/[k ln(2)].

(f) You should have noticed a decrease in the entropy when the magnetic fifield is

turned on. Why is this the case?

[2]

PHAS2228/2018 5

CONTINUED9. A sample of sodium chloride is contaminated with lithium and potassium impuri

ties, which replace sodium in the ionic lattice. Each sodium site has a reference

energy of 0 if it is occupied by sodium; replacing a sodium ion with a lithium or

potassium impurity shifts the energy from 0 to a positive value E. The chemical

potential of lithium impurities is µLi, whilst the chemical potential of potassium

impurities is µK.

(a) Write down the grand partition function ZG of a lattice site at temperature

[5]

T = 1/(βk).

(b) Assuming the sample has N total sites that may be populated by impuri-

[5]

ties, determine expressions for the average number of lithium and potassium

impurities in the sample.

(c) Show that the total entropy of the lattice is given by

[4]

S = N k ln(ZG) + N k  e −β(E−µLi

)β

(E − µLi)

Z

G

+

e −β(E−µK

)β

(E − µK)

Z

G

 .

(d) Determine the average number of lithium and potassium impurities, as well

[4]

as the entropy of the lattice for the following confifigurations of parameters:

i.

µLi = kT ln(10−4 ) + E and µK = kT ln(10−3 ) + E ,

ii.

µLi = µK = E ,

iii.

µLi → ∞ with µK = µLi ,

iv.

µLi → −∞ with µK = µLi .

(e) If nothing were known about the preparation of the sample, other than the

possibility for lithium and potassium impurities to replace sodium ions, what

would the average numbers of impurities and the entropy of the sample be?

[2]

PHAS2228/2018 6

CONTINUED10. (a) Explain why the grand potential Φ of a generic gas satisfifies Φ = −pV , where

[3]

p and V are, respectively, the pressure and volume of the gas.

(b) The density g(ε) of single-particle energy levels of a gas of free spin 1/2

fermions reads

g(ε) =

2 V

π 2

 2 ~m

2  3/2 √ ε ,

where m is the mass of the fermions and V is the volume of the gas. Show

[3]

that the Fermi energy of a gas of N spin 1/2 fermions in the volume V is

given by

εF =

~ 2

2m

 3π 2N

V  2/3

.

(c) Determine the grand potential h Φi = h Ei−T S−µh Ni at complete degeneracy [6]

(as T

by → 0), and hence show that the pressure at complete degeneracy is given

p =

~ 2

5m

(3π 2 ) 2/3  N

V  5/3

.

(d) Describe a mechanism whereby an old star, once nuclear fusion cannot be

[4]

fuelled any longer, can stabilise itself against gravitational collapse.

(e) An astronomical object is discovered with a radius of 1.4×104 km and a mass

equal to twice the mass of the sun. Knowing that the pressure due to gravity

at the centre of a sphere of mass M and radius R is given by 3GM2/(8πR4 ),

deduce the mass of the fermions within the star, and their likely nature.

[4]

PHAS2228/2018 7

CONTINUED11. A solid may be modelled as a continuous set of quantum harmonic oscillators with

density of modes in frequency space given by

g(ω) =

3V ω2

2π 2v 3

,

where V is the volume of the solid and v is its average speed of sound.

(a) Show that the canonical partition function of a single harmonic oscillator at

[4]

frequency ω and temperature T is given by

Z(ω) =

1

2 sinh(β~ ω/2)

,

where β = 1/(kT).

(b) Assume that the solid is comprised of a lattice of N atoms and introduce

[4]

a maximum cut-offff frequency ωD, above which there is no oscillation mode.

Show that ωD = v(6π 2n) 1/3 , where n = N/V .

(c) Explain why the canonical partition function of the solid Zvib satisfifies the

[2]

relation

lnZvib = Z 0 ωD

g(ω) lnZ(ω)dω .

(d) Relate the heat capacity at constant volume CV to the canonical partition

function Z, and show that

[6]

CV =

3V ~ 2

8v 3π 2kT2 Z 0 ωD

ω

4

[sinh (β

~

ω/2)]2 dω

(e) Show that, for kT  ~ ωD, CV is proportional to T 3 . Why does this model [4]

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