L A N C A S T E R U N I V E R S I T Y

2010 EXAMINATIONS

Part II

PHYSICS - Paper 2.D

•Candidates should answer all those sections identiﬁed with the modules for which

they are registered.

•An indication of mark weighting is given by the numbers in square brackets following

each part.

•Use a separate answer book for each section.

PHYSICAL CONSTANTS

Planck’s constant h= 6.63 ×10−34 J s

¯h= 1.05 ×10−34 J s

Boltzmann’s constant kB= 1.38 ×10−23 J K−1

Mass of electron me= 9.11 ×10−31 kg

Mass of proton mp= 1.67 ×10−27 kg

Electronic charge e= 1.60 ×10−19 C

Speed of light c= 3.00 ×108m s−1

Avogadro’s number NA= 6.02 ×1023 mol−1

Permittivity of the vacuum ǫ0= 8.85 ×10−12 F m−1

Permeability of the vacuum µ0= 4π×10−7H m−1

Gravitational constant G= 6.67 ×10−11 N m2kg−2

Bohr magneton µB= 9.27 ×10−24 J T−1(or A m2)

Bohr radius a0= 5.29 ×10−11 m

Gas constant R= 8.31 J K−1mol−1

Acceleration due to gravity g= 9.81 m s−2

1 standard atmosphere = 1.01 ×105N m−2

Mass of Earth = 5.97 ×1024 kg

Radius of Earth = 6.38 ×106m

Density of iron = 7.6×103kg m−3

please turn over

1

Section A: Module 233: Thermal Properties of Matter

(The time allocated for this section is 80 minutes.

Candidates should answer question A1 and

either question A2 or question A3.)

Compulsory question:

A1. (a) State the First, Second and Third Laws of Thermodynamics. [8]

(b) (i) Describe the assumptions of the ideal gas model. Sketch a set of isotherms

for three temperatures, where T1> T2> T3, for an ideal gas on a P-V

diagram.

(i) Describe the assumptions of the Van-der-Waals gas model. Sketch a similar

set of isotherms for the Van-der-Waals gas on a P-Vdiagram. Clearly

identify an isotherm corresponding to temperature below which the Van-

der-Waals gas could be liquiﬁed. [8]

(c) State the equipartition theorem. Using the equipartition theorem, calculate

the internal energy and heat capacity at room temperature for one mole of i)

a monoatomic gas and ii) a diatomic gas. [8]

(d) Explain the distinction between ﬁrst order, second order and lambda phase

transitions. Illustrate your answer with examples (one for each transition) and

sketches showing how the speciﬁc heat varies with temperature. [8]

(e) The speciﬁc heat of paramagnetic solids show a Schottky anomaly at low tem-

perature. Describe brieﬂy what is a Schottky anomaly and how it can be

explained in terms of the microscopic properties of the system. [8]

Answer one of the following two questions:

A2. (a) Give deﬁnitions of entropy in the i) thermodynamical and ii) statistical frame-

works. Explain how the irreversibility of a process is linked to entropy. [16]

(b) (i) Calculate the number of microstates for a 2-level system that consists of

N-elements assuming that the temperature of the system is such that the

probabilities of occupying both energy levels are equal. Using Stirling’s

approximation ln n!≃nln n−nand the statistical framework, calculate

the entropy of the system. [14]

(ii) Calculate the entropy of the same system when only the lowest energy level

is occupied. Discuss the temperature of the system in this case. Explain

whether the system is least ordered in this case or in case (i), when both

levels are occupied equally. [10]

2

A3. (a) Deﬁne two of the four thermodynamic potentials for a hydrostatic system and

any quantities used in their deﬁnition. Describe why each of these potentials

is useful. [16]

(b) Write down the diﬀerential form of the internal energy U, or the central equa-

tion of thermodynamics, for a hydrostatic system.

Hence show that

T= ∂U

∂S !V

;P=− ∂U

∂V !S

where P,V,Sand Tare pressure, volume, entropy and temperature respec-

tively.

Derive the corresponding Maxwell relation. [14]

(c) The internal energy of the ideal gas is given by

U=U(S, V ) = αN kBN

V2/3

e2S/3NkB

where αis a constant, kBis the Boltzmann’s constant and Nis number of

atoms. Show that the equation of state for the ideal gas follows from this

equation. [10]

please turn over

3

Section B: Module 235: Nuclei & Particles

(The time allocated for this section is 40 minutes.

Candidates should answer question B1 and

either question B2 or question B3. )

Compulsory question:

B1. (a) The mass of the atom 20

10Ne is 19.9924 u. Find the nuclear binding energy in

MeV. [mn=1.0087 u, MH=1.0078 u, 1 u = 931.494 MeV/c2.] [1]

(b) The Homestake Experiment measured the ﬂux of solar neutrinos by measuring

the rate of the neutrino capture interaction given by νe+37

17Cl→37

18Ar++e−.

(i) What is the minimum energy in MeV that the neutrinos must have in

order to participate in this interaction?

[M(37

17Cl)=36.965903 u, M(37

18Ar)=36.966776 u, me=0.000549 u,

1 u = 931.494 MeV/c2.]

(ii) Neutrinos produced via the p-p reaction (the ﬁrst step in the proton chain)

in the Sun have a maximum energy of 0.42 MeV. Was Homestake sensitive

to these neutrinos? Justify your answer. [3]

(c) Consider a family of 5 nuclei with even mass number A, and another family of

5 nuclei with odd mass number A′.

(i) For each case, sketch the variation of the atomic mass M(Z,A) as predicted

by the Semi-Empirical Mass Formula (SEMF), as a function of Z. Put a

data point for each of the 5 nuclei on your plot.

(ii) Explain why these two sketches are diﬀerent, making reference to the ap-

propriate term in the SEMF.

(iii) On the sketch with even mass number A, indicate which nuclei are likely to

undergo β−decay, β+decay, or electron capture, and which nucleus(nuclei)

is(are) stable against βdecays.

(iv) Explain why some nuclei can undergo electron capture, but not β+decay. [10]

(d) Thorium 228 (228

90Th) is an α-emitter with a half-life of 1.91 years. If the

current activity of a thorium 228 sample is 32.0 Bq, calculate the number of

years needed for the activity to be reduced to 1.00 Bq. [4]

(e) Nuclear reactions may be initiated by bombarding a target with nucleons. For

each of the interactions below, determine the missing nucleon and A,Z for the

resulting unstable nucleus.

(i) 35

17Cl+ ? →?→32

16S+4

2He

(ii) 10

5B+ ? →?→7

3Li+4

2He [2]

4

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