# Ideal Gas Model - Thermal Properties of Matter - Exam, Exams for Thermal Analysis. Allahabad University

## Thermal Analysis

Description: This is the Exam of Thermal Properties of Matter which includes Thermodynamic Quantity, Einstein’s Theory, Ehrenfest Classification, Phase Transitions, Thermodynamic Potential, Clausius-Clapeyron Equation, Molar Volume of Solid etc. Key important points are: Ideal Gas Model, Laws of Thermodynamics, Van-Der-Waals Gas Model, Equipartition Theorem, Lambda Phase Transitions, Specific Heat of Paramagnetic Solids, Schottky Anomaly, Definitions of Entropy
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Paper2D.dvi

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 h̄ = 1.05× 10−34 J s

Boltzmann’s constant kB = 1.38× 10 −23 JK−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× 108 ms−1

Avogadro’s number NA = 6.02× 10 23 mol−1

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

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

Gravitational constant G = 6.67× 10−11 Nm2 kg−2

Bohr magneton µB = 9.27× 10 −24 JT−1 (or Am2)

Bohr radius a0 = 5.29× 10 −11 m

Gas constant R = 8.31 JK−1 mol−1

Acceleration due to gravity g = 9.81m s−2

1 standard atmosphere = 1.01× 105 Nm−2

Mass of Earth = 5.97× 1024 kg Radius of Earth = 6.38× 106 m Density of iron = 7.6× 103 kgm−3

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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 -V diagram. 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 lnn! ≃ n lnn − n and 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 , S and T are 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 ) = αNkB

(

N

V

)2/3

e2S/3NkB

where α is a constant, kB is the Boltzmann’s constant and N is number of atoms. Show that the equation of state for the ideal gas follows from this equation. [10]

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 2010Ne is 19.9924 u. Find the nuclear binding energy in MeV. [mn=1.0087 u, MH=1.0078 u, 1 u = 931.494 MeV/c

2.] [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(3717Cl)=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 β decays.

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

(d) Thorium 228 (22890Th) is an α-emitter with a half-life of 1.91 years. If the current activity of a thorium 228 sample is 32.0Bq, calculate the number of years needed for the activity to be reduced to 1.00Bq. [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) 3517Cl+ ? → ? → 32 16S+

4 2He

(ii) 105 B+ ? → ? → 7 3Li+

4 2He [2]

4

Answer one of the following two questions:

B2. (a) In the Standard Model of Particle Physics all matter is composed of elemen- tary fermions: leptons and quarks. List all elementary fermions (ignoring antifermions) that can participate in:

(i) electromagnetic interactions,

(ii) strong interactions,

(iii) weak interactions. [6]

(b) An important background process in the T2K experiment is the production of neutral pions in the detector via the process ∆+(uud) → p + π0.

(i) Draw a tree-level Feynman diagram for this interaction, clearly labelling all fundamental particles and initial and final states.

(ii) The ∆+ resonance has a typical mass peak width of Γ ≈ 118 MeV. Calcu- late its approximate lifetime and state the force responsible for its decay. [h̄ = 6.582× 10−16 eV· s]

(iii) How far can light travel during this time? [6]

(c) The quark weak eigenstates D′ are given by D′ = UD, where D are flavour eigenstates, and U is the CKM matrix:

D =

d

s

b

 D′ =

d′

s′

b′

 U =

0.974 0.226 0.004 0.226 0.973 0.041 0.009 0.041 0.999

(i) Determine the mixture of flavour eigenstates which makes up the weak eigenstate of the strange quark.

(ii) Use your answer to part (i) to determine which of the following interactions occurs more readily: D+(cd̄) → K̄0 + e+ + νe or D

+(cd̄) → π0 + e+ + νe. [8]

5

B3. (a) For each of the reactions below, list all fundamental boson(s) which may me- diate the interaction.

(ii) e+e− → ? → τ+τ−

(iii) e+e− → ? → νν̄ [6]

(b) Consider the interaction e+e− →hadrons.

(i) Draw a tree-level Feynman diagram for e+e− → qq̄, clearly labelling all fundamental particles and initial and final states.

(ii) Sketch the cross-section for e+e− →hadrons as a function of centre-of-mass energy between 0 and 120 GeV.

(iii) The cross-section is proportional to the square of the matrix element. Ex- plain the structure of the sketch in terms of the propagator term in the matrix element.

(iv) Imagine that the photon is not massless, but rather has a mass of Mγ = 10GeV. Redraw your sketch accordingly.

(v) Explain why there is no scattering tree-level diagram for the process e+e− → qq̄ in the Standard Model. [14]

End of Paper

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