SURE answers

C.3 Consider a gas of photons confined to a two-dimensional cavity of typical size L. The allowed frequencies of the standing waves in the cavity are given by = c 2L q n2 x + n2 y , where nx and ny are quantum numbers. a) Calculate the density of states. Note that in 2D photons have only one possible polarisation. [5] b) Derive the spectral density per unit area as a function of the wavelength. [6] c) Find the wavelength max corresponding do the maximum of this distribution. You may assume that hc/maxkBT 1. [5] d) Using the result from c), calculate what colour our Sun would have in 2D. Assume the surface temperature of the Sun to be T = 6000K.

B.1 A fridge is powered by pressurized boiling water. The fridge exchanges heat with the atmosphere (300K), the boiler (400K) and the cold compartment (250K). No external work is provided or produced. (a) Define a sensible measure of efficiency for the fridge, explaining your choice. [3] The proposed fridge can be regarded as two Carnot engines: A heat engine E running between T=300K and T=400K produces work which is used to drive a refrigerator R running between T=250K and T=300K. (b) Draw a diagram of the fridge, defining heat and work flows, and write down the relationships between them corresponding to the First Law. [3] (c) What is the maximum efficiency of E and of R ? ,2 [2] (d) Assuming ideal efficiencies, if 200J of heat is extracted from the cool compartment, how much heat is dumped to atmosphere? [6] (e) What is the overall efficiency of the device according to your definition in (a), and how would it change if the water was not pressurised? [4] B.2 (a) Sketch the P T projection of the P V T surface for a simple substance with solid, liquid and vapour phases, and identify its main features. [6] (b) The chemical potential, , is defined by adding a term to the Central Equation dU = T dS P dV + dN In principle, this means that can be determined by measuring the change in internal energy when a particle is added to a system: explain why this process is difficult to realise experimentally. [4] (c) Show that for a single component system, is the specific Gibbs free energy, i.e. G = N. [4] (d) By using the fact that the chemical potentials of coexisting phases are the same, derive the Clausius-Clapeyron equation dp dT = l T(2 1) where 1 and 2 are the specific volumes of the two phases and l is the specific latent heat of the transition. [4] (e) What feature(s) of the P T diagram in section (a) can be calculated using the ClausiusClapeyron equation

Problem Each summer, when the grass grows to a certain height, the groundsman goes to the uniform cow shop and rents some uniform cows to graze on it, until it reaches a particular level. From the previous 2 years the groundsman knows that it took: Year 1: 6 cows, 4 days to do the job Year 2: 3 cows, 9 days to do the job This year only 1 uniform cow is available. How many days will it take for the cow to do the job?

A.1 A cup containing 0.5 kg of water at 50C is left to cool to the temperature of its surroundings: 15C. Is this a reversible process? [1] What is the entropy change of the water and of the surroundings? [4] (cP for water = 4.2kJ kg1 K1 .) A.2 500g of ice at 5 oC is added to 500g of water at 20oC. What is the final ratio of ice to water in the system? [3] What is the difference in specific internal energy, u, between ice and water? [2] (assume atmospheric pressure throughout, specific heat capacities water cP =4.2Jg1K1 ; ice cP =2.1Jg1K1 , latent heat of melting 334Jg1 .) A.3 Define the coefficient of thermal expansion, . The Third Law states that the change in entropy in any process tends to zero as T 0. Show that this implies that thermal expansion becomes zero as T 0. ,4 [1] A.4 Two balls are taken at random from a bag containing 10 red and 5 blue balls. What is the probability that a) both balls are red? [2] b) both balls are of the same colour? [2] c) the two balls are not of the same colour? [1] A.5 a) Give the definitions of a microstate and a macrostate. [2] b) Give the expression for the Fermi-Dirac distribution. Explain the meaning of all the symbols you write down. [3] A.6 Consider N non-interacting atoms in contact with a heat bath at temperature T. Each atom can be either in the ground or excited state with energies 0 or kBT, respectively. a) What is the probability for a single atom to be in the excited state? [3] b) On average, how many atoms are going to be in the excited state?

B.3 This question pertains to an imaginary material (Tedium Boride, TB) with properties defined for mathematical tractability. Liquid TB has Helmholtz Free Energy F(V, T) = Fl aT3/2 bT + B(V Vl) 2/2 Solid TB has Helmholtz Free Energy F(V, T) = Fs aT3/2 + B(V Vs) 2/2 (a) Consider the constants Fls = Fl Fs and Vls = Vl Vs. What sign would you expect these quantities to have, and why? [2] (b) Write down expressions for the pressure and entropy of liquid TB as a function of T and V , [2] (c) evaluate the heat capacities Cp and Cv, thermal expansivity and bulk moduli Ks and KT for both phases [5] (d) What are the Gibbs free energies G(V, T) of the two phases? Taking P = 0, sketch the Gibbs free energy G vs T for each phase on the same graph. ,2 [2] (e) Are expressions for S and P consistent with the third law of thermodynamics? Justify your answer. [3] (f) Find an expression for the melting temperature Tm and latent heat of melting at P = 0 in terms of Vls, Fls, a, b and B. ,1

C.1 Consider four spins arranged on a ring with each spin pointing either up or down. Only neighbouring spins are assumed to interact and their interaction energy is given by Jsisj , where the spin variable si is +1 if the spin i is pointing upwards and 1 otherwise. a) How many microstates does this system have? [2] b) What are the macrostates of this system and how many microstates correspond to each macrostate? [4] c) Write down the expressions for the free energy corresponding to the macrostates with the total interaction energy 0 and 4J. [6] d) Give explicitly the temperature range where the system on average will be in a state with all spins pointing in the same direction. [4] e) Discuss qualitatively how this temperature range changes when the number of spins on the ring is increased to infinity and its bearing on phase transitions in 1-dimensional systems. [4] C.2 The partition function of a semi-classical ideal gas of N particles is given by Z = 1 N! V 3 N , where = h 2mkBT is the thermal wavelength. a) Use equipartition to calculate the total energy of the gas. Why is the usage of equipartition justified? [4] b) Calculate the pressure of the gas. [3] c) Using the total energy from a), calculate the entropy of the gas assuming N 1 and ln(N!) = N ln N N. [6] d) A container is divided by a partition into two equal regions of volume V , each holding N particles of the same gas at temperature T. Calculate the change in entropy after the partition is removed. Assume that the temperature did not change after the partition is removed. Explain the results. [7] Printed: Monday 8th December, 2014 P

C.3 Consider a gas of photons confined to a two-dimensional cavity of typical size L. The allowed frequencies of the standing waves in the cavity are given by = c 2L q n2 x + n2 y , where nx and ny are quantum numbers. a) Calculate the density of states. Note that in 2D photons have only one possible polarisation. [5] b) Derive the spectral density per unit area as a function of the wavelength. [6] c) Find the wavelength max corresponding do the maximum of this distribution. You may assume that hc/maxkBT 1. [5] d) Using the result from c), calculate what colour our Sun would have in 2D. Assume the surface temperature of the Sun to be T = 6000K.

C.3 Consider a gas of photons confined to a two-dimensional cavity of typical size L. The allowed frequencies of the standing waves in the cavity are given by = c 2L q n2 x + n2 y , where nx and ny are quantum numbers. a) Calculate the density of states. Note that in 2D photons have only one possible polarisation. [5] b) Derive the spectral density per unit area as a function of the wavelength. [6] c) Find the wavelength max corresponding do the maximum of this distribution. You may assume that hc/maxkBT 1. [5] d) Using the result from c), calculate what colour our Sun would have in 2D. Assume the surface temperature of the Sun to be T = 6000K.

C.1 Consider four spins arranged on a ring with each spin pointing either up or down. Only neighbouring spins are assumed to interact and their interaction energy is given by Jsisj , where the spin variable si is +1 if the spin i is pointing upwards and 1 otherwise. a) How many microstates does this system have? [2] b) What are the macrostates of this system and how many microstates correspond to each macrostate? [4] c) Write down the expressions for the free energy corresponding to the macrostates with the total interaction energy 0 and 4J. [6] d) Give explicitly the temperature range where the system on average will be in a state with all spins pointing in the same direction. [4] e) Discuss qualitatively how this temperature range changes when the number of spins on the ring is increased to infinity and its bearing on phase transitions in 1-dimensional systems. [4] C.2 The partition function of a semi-classical ideal gas of N particles is given by Z = 1 N! V 3 N , where = h 2mkBT is the thermal wavelength. a) Use equipartition to calculate the total energy of the gas. Why is the usage of equipartition justified? [4] b) Calculate the pressure of the gas. [3] c) Using the total energy from a), calculate the entropy of the gas assuming N 1 and ln(N!) = N ln N N. [6] d) A container is divided by a partition into two equal regions of volume V , each holding N particles of the same gas at temperature T. Calculate the change in entropy after the partition is removed. Assume that the temperature did not change after the partition is removed. Explain the results.