Imperial College London BSc/MSci EXAMINATION May 2008 THERMODYNAMICS & STATISTICAL PHYSICS

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1 Imperial College London BSc/MSci EXAMINATION May 2008 This paper is also taken for the relevant Examination for the Associateship THERMODYNAMICS & STATISTICAL PHYSICS For Second-Year Physics Students Wednesday, 28th May 2008: 14:00 to 16:00 Answer ALL parts of Section A, ONE question from Section B and ONE question from Section C. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Write your CANDIDATE NUMBER clearly on each of the four answer books provided. If an electronic calculator is used, write its serial number in the box at the top right hand corner of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book. Hand in four answer books even if they have not all been used. You are reminded that Examiners attach great importance to legibility, accuracy and clarity of expression. c Imperial College London /P.2/2 1 Turn over for questions

2 SECTION A 1. (i) Enthalpy is defined as H = U + PV. By using the fundamental equation of thermodynamics, show that T = ( H S ) P and V = ( H P ) S. (ii) A box is partitioned into a part filled with an ideal gas initially at T = T o and a vacuum part. The partition is removed and the ideal gas experiences a free adiabatic expansion until a new equilibrium is reached [NB: No calculations are needed to answer the following questions]. (a) What is the change of internal energy of the gas in the process? (b) What is the temperature change of the gas? (c) Is the entropy of the gas constant in this experiment? [5 marks] (iii) A box of temperature T and fixed volume V is filled with a photon gas. The box and the photon gas are in equilibrium. By considering a reversible heat exchange between the box/photon gas system and a reservoir at a temperature T + dt (dt being very small), show that the entropy of the photon gas has a cubic dependence upon T. [Information: the heat capacity at constant volume of a photon gas is C V = 16σV T 3 /c, σ being the Stefan constant and c the speed of light.] [Total 12 marks] 2008/P.2/2 2 Please turn over

3 2. (i) Two systems A and B have the same temperature T A = T B but different chemical potentials µ A > µ B. What happens when you put them in contact so that they can exchange particles and energy? (ii) (a) Explain the meanings of the terms macrostate, microstate and single-particle state. Give an example of a possible macrostate, microstate and single-particle state in a system of your choice. (b) What is meant by the occupation number n r of a single-particle state r? What values can it have in a system of indistinguishable bosons? What values can it have in a system of indistinguishable fermions? [Total 8 marks] 2008/P.2/2 3 Please turn over

4 SECTION B 3. Carnot cycle. (i) On a PV diagram show the four stages of a Carnot cycle operating between a hot reservoir at T H and a cold reservoir at T C. Denote by A,B,C,D the intersection between the needed isotherms and adiabats. Interpret the area A PV defined by the cycle in terms of work done / received by the working substance of the Carnot Cycle. (ii) Redo question (i) for a T S diagram instead of a PV diagram. Indicate the position of A,B,C,D in this new coordinate system and interpret the area A T S defined by the cycle in terms of heating / cooling of the working substance. (iii) Show that A PV = A T S. (iv) Give, in words, the definition of the efficiency η of a heat engine. By using the T S diagram drawn in (ii) and the result (iii), show that the efficiency of a Carnot engine is η = 1 T C /T H. [Hint: you do not need to make any assumptions about the nature of the working substance (e.g., ideal gas). The algebra should be elementary and short.] (v) Application to the ocean circulation. The oceans are sometimes said to behave like a Carnot engine operating between the hot tropics and the cold polar regions. By acknowledging that heating and cooling of the ocean only occur at the sea surface, show that, in order for the ocean to operate as an engine, its surface pressure must be higher in the tropics than in polar regions. [Hint: use the PV diagram from (i)]. 2008/P.2/2 4 Please turn over

5 4. Fourier s law. Two cubes made of the same material but at initially different temperature (T 2 and T 1, with T 2 > T 1 ) are put in contact. We study the second law in action by using a simple model for the heat transfer between the two cubes, the so-called Fourier s law. We will consider the volume of the cubes to be constant and we will neglect any heat transfer between the cubes and the surroundings (thermally and mechanically isolated cubes). (i) In an infinitesimal time interval, the heat entering cube 1 is written as q. Denoting by U 1 the internal energy of cube 1, show that the first law applied to cube 1 can be written as du 1 / = q. Write a similar equation for the second cube. What is the rate of change of total internal energy d(u 1 +U 2 )/? (ii) By using the fundamental equation of thermodynamics show that the rate of change of entropy is given by ds 1 ds 2 = q T 1 (cube 1) (1) = q T 2 (cube 2) (2) (iii) Using (ii), write the equation for the total entropy change and discuss its sign. [NB: You are expected to discuss its sign qualitatively from the second law and also by forming the equation for d(s 1 + S 2 )/]. (iv) We assume that for small temperature difference the heat flux satisfies q = K(T 2 T 1 ) in which K is a constant heat transfer coefficient (Fourier s law). By writing T 1 = T o + θ 1 and T 2 = T o + θ 2 in which T o is a reference temperature and θ 1 and θ 2 are small deviations from it (θ 1 /T o 1 and θ 2 /T o 1), show that the rate of entropy change for cube 1 can be approximated by ds 1 K(θ 2 θ 1 )(T o θ 1 ) T 2 o [Hint: you might use the result that 1/(1+x) 1 x for small x.] (3) (v) Write a similar expression for cube 2 and hence prove that the total entropy change is d(s 1 + S 2 ) K(θ 2 θ 1 ) 2 To 2 (4) What sign must K have for Fourier s law to satisfy the second law? Does this represent a heat transfer from hot to cold? 2008/P.2/2 5 Please turn over

6 SECTION C 5. Consider a gas of free electrons at temperature T and chemical potential µ. Their equilibrium state is described by the Fermi-Dirac distribution n FD (ε) = 1 e (ε µ)/k BT + 1, where ε is the energy of a single-particle state and k B is Boltzmann s constant. (i) (a) Explain what is meant by a degenerate fermion gas and the Fermi energy. (b) Explain briefly the origin of the degeneracy pressure, and discuss its temperature dependence, comparing it with a classical ideal gas. (c) Give an example of a physical system in which electrons are degenerate. [1 mark] (ii) Let us assume that the particles are ultrarelativistic, so that ε = cp, where c is the speed of light and p is the momentum of the particle. The density of states is then f(ε) = 8πV h 3 c 3 ε2, where h is Planck s constant and V is the volume of the system. (a) Show that at zero temperature and a given fixed chemical potential µ, the number density n = N/V of electrons is n = 8πµ3 3h 3 c 3. (b) What is the Fermi energy ε F of the system as a function of n? (c) Calculate the internal energy density u = U/V at zero temperature, expressing it as a function of ε F. Show that in terms of n, it is u = 3 4 ( ) 3 1/3 hcn 4/3. 8π (d) How high does the internal energy density u have to be for the assumption that the electrons are ultrarelativistic to be valid? Convert this into mass density using E = mc 2, and give a numerical answer in units of kg/m /P.2/2 6 Please turn over

7 6. (i) Explain briefly the difference between distinguishable and indistinguishable particles, and how it affects the counting of microstates. Give an example of a situation in which particles are distinguishable and a situation in which they are indistinguishable. (ii) Consider N identical but distinguishable harmonic oscillators in thermal equilibrium with a heat bath at temperature T. The single-particle states are labelled by a non-negative integer r = 0,1,..., and have energies ε r = hωr. (a) Calculate the canonical partition function of a single oscillator. (b) Calculate the canonical partition function of the whole system. (c) Calculate the temperature at which half of the particles are in the ground state. (iii) Consider now a system of N 1 indistinguishable bosonic oscillators at temperature T and chemical potential µ. The equilibrium state is described by the Bose- Einstein distribution 1 n BE (ε) = e (ε µ)/kbt 1, where ε is the energy of a single-particle state and k B is Boltzmann s constant. (a) If T hω/k B, the sum over states with r > 0 can be replaced by an integral. Show that the number of oscillators is then given by N = n dε hω 0 e (ε µ)/kbt 1, where n 0 is the mean occupation number of the ground state. (b) Using 1 ( ) dε 0 e (ε µ)/kbt 1 = k BT ln 1 e µ/k BT, show that you can write N as N = n 0 + k BT hω ln(n 0 + 1). (c) Sketch the two terms on the right hand side of the above equation against n 0 at constant temperature. Which term dominates at low N, and which one at high N? (d) For given N, calculate the temperature at which half of the particles are in the ground state. Is the assumption T hω/k B valid at this temperature? 2008/P.2/2 7 End of examination paper

UNIVERSITY OF LONDON. BSc and MSci EXAMINATION 2005 DO NOT TURN OVER UNTIL TOLD TO BEGIN

UNIVERSITY OF LONDON. BSc and MSci EXAMINATION 2005 DO NOT TURN OVER UNTIL TOLD TO BEGIN UNIVERSITY OF LONDON BSc and MSci EXAMINATION 005 For Internal Students of Royal Holloway DO NOT UNTIL TOLD TO BEGIN PH610B: CLASSICAL AND STATISTICAL THERMODYNAMICS PH610B: CLASSICAL AND STATISTICAL THERMODYNAMICS