PHYS09061 Thermal Physics

Section A: Answer ALL the questions from this Section

A.1 A block of metal at a pressure of 1 atmosphere ≈ 1 × 105 Pa is initially at a temperature of 20 °C. It is heated reversibly to 32 °C at constant volume. By breaking down the process into two steps, or otherwise, calculate the final pressure. If the heating had been carried out irreversibly, would this affect your answer? Why? (The material’s isobaric thermal expansivity and isothermal compressibility are, respectively, β = 5 × 10−5 K−1 and κT = 2 3 × 10−11Pa−1 .)

A.2 Under suitable conditions, the combined First and Second Law for a block of magnetic material of magnetisation M in a magnetic field B is given by dU = TdS + BdM,

where U, T and S have their usual meanings. Explain what is meant by saying that B is an intensive state variable and M is an extensive state variable.

Define the Gibbs free energy for this system, i.e. the thermodynamic potential for which (T, B) are the independent variables, and derive the associated Maxwell relation.

A.3 A Carnot cycle run in reverse may act either as a heat pump or a refrigerator, depending on the function to which it is put. Drawing on the analogy with a heat pump or otherwise, define the coefficient of performance (COP) for a Carnot refrigerator, and give an expression for the COP of such a device operating between hot and cold reservoirs at temperatures Th and Tc respectively.

A Carnot refrigerator operates between a room at temperature 300 K and a cold compartment at temperature T. Sketch a graph showing the COP as a function of T.

A.4 a) A tennis machine serves balls to hit random points of a tennis court. In a single serve, all points of the court have equal probabilities to be hit. Four patches, each of area of 1m2 , are marked on the court. What is the probability that a single ball would hit any of the patches, if the total area of the court is 200m2 ?

b) A random variable x takes values from the range (−π, π). The probability of a measured value of x to be in the range (x, x + dx) is given by p(x)dx = Cx2 cos x dx, where C is a constant. Write down the normalisation condition for p(x). Explicitly evaluate the integral in this condition, and determine C.

A.5 A system of N non-interacting atoms is subjected to an external magnetic field. Each atom has a magnetic moment of magnitude m that can point either parallel or antiparallel to the direction of the external field, given by a unit vector hˆ. Let M be the total magnetisation of the system defined as a vectorial sum of the magnetic moments of all atoms.

a) Calculate explicitly the number of microstates Ω(M) corresponding to a given vector M.

b) Use Stirling’s approximation to demonstrate that

A.6 A particle in equilibrium with a reservoir of energy at temperature T can occupy two microstates with energies ϵ0 and ϵ1. a) Calculate the average energy of the particle ϵ.

b) Let ϵ1 = ϵ0 + δ with δ ≪ kBT, where kB is the Boltzmann constant.

Determine ϵ to linear order in δ/kBT.

Section B: Answer ONE of the questions from this section

B.1 This question concerns phase transitions in and the phase diagram of the important mineral, calcite (CaCO3), which is the main component of limestone. The triangular carbonate ions in calcite occur in planar layers. At room temperature, all the carbonate ions in one layer point in the same direction (say, ‘left’), and all the carbonate ions in the next layer point in the opposite direction (say, ‘right’). This crystal structure is called ‘Calcite I’. At around 1100 K, it undergoes a phase transition to a structure known as ‘Calcite IV’ in which the orientation of the carbonate ions becomes disordered in some way. The relevant part the calcite P T phase diagram is shown below. This question focuses on the I → IV transition.

a) Explain what is meant by saying that a phase transition is discontinuous. Write down the Clausius-Clapeyron equation in terms of relevant molar quantities and explain what it means.

b) The above phase diagram shows two alternative experimentally-determined phase coexistence boundaries for the Calcite I → Calcite IV transition: Data Set 1 and Data Set 2. Computer simulations show that this transition has a positive change in molar volume, ∆v > 0, at all pressures. Given the nature of the transition as explained in the preamble, explain why Data Set 1 is almost certainly erroneous.

c) Computer simulations return a value of ∆v = 0.68 cm3 mol−1 at atmospheric pressure (1 × 10−4 GPa). By taking approximate measurements from the given phase diagram, estimate ∆s and express this in units of the universal gas constant, R = 8.314 J mol−1 K−1 . [Hint: you may want to use the fact that 1 GPa = 1 × 109 Pa = 1 × 103 J cm−3 .]

d) There is a suggestion in the literature that in Calcite IV, each triangular carbonate ion randomly points either ‘left’ or ‘right’. Using the microscopic interpretation that the molar entropy of a state is s = R ln Ω, where Ω is the number of microscopic states (such as ionic orientations) that are available for the system given a set of macroscopic state variables (P, V, T), explain why the ‘left/right’ (or two-state) picture is unlikely to be correct, and suggest an alternative interpretation for the orientation of carbonate ions in phase IV.

e) One mole of calcite (= 100 g) is heated from just below to just above the I → IV transition at T ≈ 1100 K at the rate of 1 kW. Estimate how long it should take to complete the transition.

B.2 This question uses thermodynamics to explore the behaviour of a box of photons (a ‘photon gas’). Unless otherwise stated, the symbols have their usual meaning.

a) Show that for any fluid system,

taking care to justify your steps, including deriving any intermediate thermodynamic relation(s) that you may need to use.

b) The equation of state for a photon gas is P = 1/3 u,where u = U/V is the specific internal energy. Show that

A cylinder contains a photon gas confined by a piston. When the piston is released, it is found that the photon gas is able to move outwards against atmospheric pressure, P0 = 1 × 10^5 Pa. What can you say about the initial temperature of the photon gas?

c) A photon gas changes its volume quasi-statically by dV in a frictionless apparatus at constant temperature T. Show that the heat exchange in this process is given by

Write down the differential entropy change in this process.

d) Using the expression for the differential entropy change derived in the previous part, show that the entropy of a volume V of photon gas at temperature T is given by

taking care to explain why the Third Law is needed to obtain this result.

Arguing from this result, show that the pressure and volume of a photon gas undergoing reversible adiabatic expansion satisfies the relation

e) Argue from the equation of state given above, Eq. (∗), that the constant pressure heat capacity of a photon gas is formally infinite.