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

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1 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 - PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours Answer QUESTION ONE and TWO other questions No credit will be given for attempting any further questions Approximate part-marks for questions are given in the right-hand margin Only CASIO fx85wa Calculators or CASIO fx85ms Calculators are permitted PH610B/4 Royal Holloway and Bedford New College

2 GENERAL PHYSICAL CONSTANTS Permeability of vacuum µ 0 = 4π 10-7 H m -1 Permittivity of vacuum ε 0 = F m -1 1/4πε 0 = m F -1 Speed of light in vacuum c = m s -1 Elementary charge e = C Electron (rest) mass m e = kg Unified atomic mass constant m u = kg Proton rest mass m p = kg Neutron rest mass m n = kg Ratio of electronic charge to mass e/m e = C kg -1 Planck constant h = J s = h/π = J s Boltzmann constant k = J K -1 Stefan-Boltzmann constant σ = W m - K -4 Gas constant R = 8.31 J mol -1 K -1 Avogadro constant N A = mol -1 Gravitational constant G = N m kg - Acceleration due to gravity g = 9.81 m s - Volume of one mole of an ideal gas at STP = m 3 One standard atmosphere P 0 = N m - MATHEMATICAL CONSTANTS e.718 π 3.14 log e

3 ANSWER ONLY FIVE sections of Question One. page 1 PH610B You are advised not to spend more than 40 minutes answering Question One. 1. (a) Calculate the work done by 0.5 moles of a gas when it expands isothermally by a factor at 0 C. [4] (b) The Helmholtz free energy is defined as F = E TS. Find an expression for its differential, df, and hence show that pressure and entropy can be expressed as partial derivatives of F. [4] (c) An ideal gas is taken round a Carnot cycle. Draw a fully labelled sketch to illustrate this process on the p V indicator diagram. [4] Write down the Clausius-Clapeyron equation, defining the symbols used. The p T melting curve of liquid 3 He has a minimum at 0.3K; what fact can be inferred from this observation? [4] (e) Define the Fermi-Dirac distribution function n( ε ), and write down its form. Sketch n( ε ) at T = 0. [4] (f) Briefly discuss the phenomenon of Bose-Einstein condensation. [4]

4 page PH610B. (a) The Boltzmann distribution is written as N βε n( ε) = e. Z Define the symbols used and explain the physical significance of n( ε ). [6] (b) Discuss, briefly, the conditions under which it applies. [4] (c) A crystal is formed of atoms, which have a nuclear spin I = 1, and an associated magnetic moment 0.4µ N, where µ N = JT 1 is the nuclear magneton. State the possible values of the spin projection quantum number m I for a single nucleus and use the Boltzmann-Planck equation to write down the entropy of one mole in zero magnetic field. Discuss, without detailed calculation, the effect of applying a magnetic field of 10T at 5mK [5] It is discovered that, even in zero magnetic field, the quantum states m I are split according to the relation ε = ε + 0 [3 mi I( I 1)] Show that each particle has two energy levels, one of which is doubly degenerate. Find an expression for the single particle partition function. [3] 3 (e) If ε 0 = 5 10 K, find the ratio of the populations of these two levels at K. []

5 page 3 PH610B 3. (a) State the first law of thermodynamics. [4] (b) Prove that c c = R, for one mole of an ideal gas. [8] p v (c) The diagram below illustrates an experimental apparatus to measure γ = c / c. p v A vertical tube T contains a smoothly fitting ball S of mass m, free to slide along the tube with no friction and no gas escaping. The lower end of the tube is connected to a spherical vessel of volume V 0. When the ball is in its initial position the pressure inside the vessel is equal to atmospheric pressure. The ball is then allowed to drop, and initially comes to rest after falling a distance l. mgv0 Prove that γ =, where A is the cross sectional area of the bore of Apl 0 tube T, and p 0 is atmospheric pressure. Demonstrate that this result is dimensionally correct. [Assume adiabatic conditions, and that the gas is ideal]. [6] Describe qualitatively the subsequent motion of the ball, and suggest how it might be used to provide an alternative determination of γ. [] S T V 0

6 page 4 PH610B 4. (a) State the second law of thermodynamics, and briefly comment on its significance. [5] (b) (c) A system at a temperature 300 K, with a temperature-independent heat capacity of 00 JK 1 is heated reversibly by 10 J. Explain briefly the meaning of the term reversibly here. Evaluate the increase in temperature and entropy of the system. [5] Calculate the increase in entropy when one mole of an ideal gas undergoes a free expansion, without heat exchange with the surroundings, from an initial volume of 0 litre to a final volume of 30 litre. S p [You may assume the Maxwell relation = V T ]. [5] T V A vessel containing 0.5 litre of hot water at 80 C is immersed in 1litre of cold water at 0 C. Calculate the final temperature in thermal equilibrium and the increase in total entropy of the system. [Assume that the system is otherwise isolated from the rest of the world, and neglect the heat capacity of the containers. The specific heat capacity of water is 4. x 10 3 JK 1 kg 1. The density of water is 10 3 kgm 3 ]. Explain how, in principle, thermal equilibrium between two bodies could be achieved with no associated increase in entropy. [5]

7 page 5 PH610B 5. (a) Discuss briefly the distinguishing features of a particle, which determine whether it is a boson or a fermion. [4] (b) Define the Bose-Einstein distribution function n( ε ), and write down its form, defining the symbols used. [4] (c) State and explain the value of the chemical potential of a gas of photons. And explain why ε = ck for photons (in the usual notation). [3] Describe how this model accounts for the spectrum of black body radiation. 3 V ω E( ω) = 3 π c e ω / kt B. 1 [A full derivation is not required. In 3D the density of states in k-space is gkdk ( ) = V kdk]. π Briefly outline the properties of this spectrum. [5] (e) Waves propagating over the surface of liquid helium are quantised as a two dimensional gas of mass-less bosons (called ripplons ). The energy 3/ and wavenumber are related by the equation ε = ak. 7/3 Show that the internal energy of this gas ET ( ) T. A [In D the density of states in k-space is gkdk ( ) = kdk] [4] π END

DO NOT TURN OVER UNTIL TOLD TO BEGIN

DO NOT TURN OVER UNTIL TOLD TO BEGIN UIVERSITY OF LODO MSci EXAMIATIO 007 For Students of the University of London DO OT UTIL TOLD TO BEGI PH411A: STATISTICAL MECHAICS Time Allowed: TWO AD A HALF hours Answer THREE questions only. o credit