Data Provided: A formula sheet and table of physical constants are attached to this paper.

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1 Data Provided: A formula sheet and table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester ( ) From Thermodynamics to Atomic and Nuclear Physics Paper A: Thermal Physics and Solids 3 HOURS There are three sections to this paper. Each section contains a compulsory question and two optional questions. Section A is worth 40 marks. Sections B and C are worth 30 marks each. You must attempt the compulsory question and one optional question from each section. Answers to different sections must be written in separate books, the books tied together and handed in as one. Please clearly indicate the question numbers on which you would like to be examined on the front cover of your answer book. Cross through any work that you do not wish to be examined. 1 TURN OVER

2 COMPULSORY QUESTION SECTION A THERMAL PHYSICS 1 1. (a) What is meant by an adiabatic process? 1.0 mol of an ideal monatomic gas at 25 C and atmospheric pressure is compressed adiabatically to a volume of 15 litres. What is the new temperature of the gas? [4] A Carnot engine takes 300 J of heat from a reservoir at 600 K and discards 200 J of heat to a reservoir at 400 K. What is the entropy change during the isothermal expansion stage of the cycle? [4] Explain why processes that apparently result in an increase in order, such as crystallization, are not forbidden by the second law of thermodynamics. [4] (d) Given that F(T,V) and df SdT pdv derive a Maxwell relationship linking changes in entropy to changes in pressure. [4] (e) The partition function for a simple harmonic oscillator is given by Z e 1 e 1 2. Use this information to find an expression for the internal energy, U, of a simple harmonic oscillator in terms of ω and β. [4] 2 CONTINUED

3 ANSWER EITHER QUESTION 2 OR 3 2. (a) 3.0 mol of an ideal monatomic gas are held in a 50 litre vessel at a temperature of 200 C. The gas expands adiabatically to a volume of 75 litres driving a frictionless piston. i) What is the new temperature of the gas? [2] ii) How much work is done during the expansion? [3] A shutter in the side of the vessel is opened that allows the gas to undergo an adiabatic free expansion to a total volume of 100 litres. i) Explain why there is still a change in entropy of the gas even though no heat has entered or left the system. [3] ii) What is the entropy change of the gas due to the expansion? [5] The gas in above is actually better described by the van der Waals equation of state. 2 NkBT an ' p. 2 V Nb ' V i) What is the physical significance of the terms a ' and b ' in this equation? [4] ii) Qualitatively, how would you expect the temperature of the gas in part above to change due to the free expansion, if it were a van der Waals gas? Explain your answer. [3] 3 TURN OVER

4 3. Two scientists at an Antarctic research station, Fred and Jean, are each given a 1 kg block of ice. They are told to increase the entropy of the ice as much as they can. Fred smashes the block up into small pieces with a hammer and scatters the pieces widely over the snow field. Jean melts the ice and puts the resultant water into a thermos flask which she neatly stores in her backback. (a) Which has increased the entropy of the ice the most? Explain your answer. [4] When Jean heated the ice it went through a solid-liquid phase transition. Assuming she did this in an open vessel, which thermodynamic potential can be minimised to find the equilibrium state of the water? [2] Given that du TdS pdv dn and G U pv TS, derive an expression for how pressure changes with temperature at a phase boundary, i.e. the Clausius-Clapeyron equation dp L, dt T V2 V1 where L is the specific latent heat and V1 and V2 are the specific volumes of the two substances. [6] (d) Find the temperature at which Jean s ice would have melted at the summit of Mt Vernon (pressure 0.5 atmospheres), Antarctica s highest peak. (Lfusion = J kg -1, Vwater Vice = m 3 kg -1, atmospheric pressure: Pa). [3] (e) Jean continues to heat the water until it boils. From the Clausius-Clapeyron equation derive an expression linking the pressure and temperature at which a gas evaporates. Assume that the specific volume of the vapour phase is much larger than that of the liquid and that the vapour obeys the ideal gas equation. Use this relationship to find the temperature at which water boils at the summit of Mt Vernon. (Lvapour = J mol -1 ). [5] 4 CONTINUED

5 SECTION B - Solids COMPULSORY QUESTION 4. (a) Discuss the dependence of Umklapp processes on temperature at low, intermediate and high temperatures. [5] Measurements have shown that selenium has a Debye temperature of 90 K. If its molar heat capacity is determined to be 3.33 J K -1 mol -1 at 20 K, what is its molar heat capacity at (i) 5 K and (ii) 300 K? [3] The Hall coefficient for copper is given by RH = m 3 C 1. What is the Hall field when a magnetic field of 8.0 mt is applied across a copper sample through which a charge flux of 750 C m 2 s 1 flows? The field is applied perpendicular to the current flow. [2] (d) (e) The Wiedemann Franz law is given by κ/σ = LT, where κ is the electronic contribution of the thermal conductivity, σ is the electrical conductivity, and T is the absolute temperature. What are the units of the Lorenz constant L? You should simplify your answer to use the minimum number of different units. [3] Derive an expression for the minimum speed required for an electron to escape from a surface when undergoing thermionic emission. [2] 5 TURN OVER

6 ANSWER EITHER QUESTION 5 OR QUESTION 6 5. (a) Describe the main differences between the Debye and Einstein models of the thermal behaviour of solids, highlighting the failures of the Einstein model. [4] The average vibrational energy of an atom according to the Einstein model is given by E, 2 exp / kt B 1 where T is the absolute temperature and ω is the angular vibration frequency. Obtain an expression for the molar heat capacity according to the Einstein model as the temperature is lowered towards absolute zero. [3] The density of states (in one dimension) in the Debye model is given by 2 V g, where is the speed of sound and V is the volume of the sample. kb D / i) Show that, where 3 2 D is the Debye temperature 6 N / V and N is the number of atoms in the lattice. [4] ii) iii) Bismuth has a Debye temperature of 119 K, an atomic mass of kg/mol and a density of 9780 kg/m 3. What is the speed of sound in bismuth? [2] The measured speed of sound in bismuth is 1.79 km/s. Supposing that this value corresponds to the longitudinal component and that your answer in (ii) above corresponds to the combination of both longitudinal and transverse components (an effective speed of sound), what would be the transverse speed, assuming that there is only one value for the transverse speed of sound? [2] 6 CONTINUED

7 6. (a) i) Starting from Schrödinger s equation for a particle in a box, show that the energy of a free electron of mass m e in a metal can be given by E 2 nx ny nz, where n x, n y, and n z are 2mL e integers, and L is the size of the crystal. [5] ii) How many electrons have energy E? [2] ml 2 2 e What is the Fermi energy, F? [2] The density of states in the free electron model for electrons of energy E is given by where V is the volume of the metal containing N free electrons., Show that. [3] (d) The molar heat capacity for potassium is given at low temperatures by p m C T T, where = 2.08 mj mol -1 K (p+1) and = 2.57 mj mol 1 K (m+1), and m > p. Explain how a measurement of the low temperature molar heat capacity of potassium can reveal the separate electronic and thermal (lattice vibration) components. What are the values of m and p? Below what temperature does the electronic component dominate the heat capacity? [3] 7 TURN OVER

8 COMPULSORY QUESTION SECTION C THERMAL PHYSICS 2 7. (a) Fick s first law can be written as c j D. z Explain in words what this equation means. [3] For a gas, the heat capacity at constant pressure is larger than the heat capacity at constant volume. i) Explain why this is the case. ii) For one mole of a diatomic ideal gas what is the heat capacity at constant volume? (Assume the gas is at high temperature). Explain your answer. iii) For the gas in (ii) what is the heat capacity at constant pressure? [4] (d) In statistical physics what is meant by the thermodynamic limit? What is meant by an ensemble? [4] In an Einstein model of a solid each atom is represented by three oscillators. Explain why this is. For an Einstein solid with only three oscillators what are the total number of ways in which three energy units can be distributed? [4] 8 CONTINUED

9 ANSWER EITHER QUESTION 8 OR 9 8. A thin plastic sheet is perforated with tiny holes that can be either open or closed depending on the conformation of the molecules in the sheet surrounding the hole. The energy of the open state is zero and the energy of the closed state is +. (a) Find an expression for the partition function of a hole. [3] Find an expression for the probability that the hole is closed as a function of temperature. [2] Show that the average energy of a hole is given by E. e 1 [3] (d) (e) In what state would you expect to find a hole when at high temperature? Explain your answer. [3] If = 500 kb, where kb is Boltzmann s constant, find the Helmholtz free energy of a hole at 20 C. [4] 9 TURN OVER

10 9. A kg coin drops from a pirate s pocket (temperature 32.0 C) into the ocean where it cools to 19.0 C (the temperature of the ocean). (a) i) What is the change in entropy of the gold coin during this process, given that the specific heat capacity of gold, cgold = 129 J kg -1 K -1? [3] ii) What was the entropy change of the Universe due to the process? [3] Given that the molar mass of gold is kg mol -1 find the number of gold atoms in the coin. [3] The pirate, not remembering the heat capacity of gold, decides to try to estimate the entropy of the coin by assuming it is an Einstein solid and that the number of different ways of organising the system,, follows N eq. N i) What do e, q and N represent in this equation? [3] ii) If q = 10 24, find the entropy of the coin. [3] END OF EXAMINATION PAPER 10 CONTINUED

UNIVERSITY COLLEGE LONDON. University of London EXAMINATION FOR INTERNAL STUDENTS. For The Following Qualifications:-

UNIVERSITY COLLEGE LONDON. University of London EXAMINATION FOR INTERNAL STUDENTS. For The Following Qualifications:- UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications:- B.Sc. M.Sci. Physics 1B28: Thermal Physics COURSE CODE : PHYSIB28 UNIT VALUE : 0.50 DATE