SA Thermodynamics and Statistical Physics Qualifier Exam DISCLAIMER: This is a sample PhD qualifier exam to only demon- strate the typical level of question that may be posed by the GRASP. Students must not infer anything regarding the content of their exam based on these examples; questions may be drawn from the full range of the topic. Choose four out of the following five problems. All problems are weighted equally. Show and explain all your work for full credit. 1. An ice cube (mass 30 g) at 0 ◦ C is left sitting on the kitchen table, where it gradually melts. The temperature in the kitchen is 25 ◦ C. The specific heat of water is 4.187 kJ kg−1 K−1 . The latent heat of fusion of water is 334 kJ kg−1 . MP (a) Calculate the change in the entropy of the ice cube as it melts into water at 0 ◦ C. (Don’t worry about the fact that the volume changes somewhat.) (b) Calculate the change in entropy of the water (from the melted ice) as its temperature rises from 0 ◦ C to 25 ◦ C. (c) Calculate the change in the entropy of the kitchen as it gives up heat to the melting ice/water. (d) Calculate the net change in the entropy of the universe during this process. Is the net change positive, negative, or zero? Is this what you would expect? 2. Use the thermodynamic identity to derive the heat capacity formula ∂S CV = T , ∂T V LE which is occasionally more convenient than the more familiar expression in terms of U. Then derive a similar formula for CP , by first writing dH in terms of dS and dP , where the enthalpy H = U + P V . 3. A power plant produces 1 GW (109 watts) of electricity at an efficiency of 40%. 1 SA (a) At what rate does this plant expel waste heat into its environment? (b) Assume first that the cold reservoir for this plant is a river whose flow rate is 100 m3 /s. By how much will the temperature of the river increase? (Cw = 4186 J/◦ C). (c) To avoid this “thermal pollution” of the river, the plant could instead be cooled by evaporation of river water. At what rate must the water evaporate? What fraction of the river must be evaporated? (Assume L = 2400 J/g at room temperature). 4. Consider a low-density gas of N2 molecules. (a) Calculate the partition function for the internal energy of the N2 molecule. Assume that the only energy levels that exist are the allowed rotational transitions. Note that the allowed rotational energies are E(j) = j(j + 1)ǫ, MP where ǫ is a constant. (b) In the low temperature limit (kT ≪ ǫ), truncate sum over states and compute the average energy and heat capacity. 5. Consider the dissociation of pure water into H+ and OH− ions: H2 O ↔ H + + OH − . (a) Assuming this is a very dilute solution, calculate the equilibrium concentration of H+ ions at 25 ◦ C, expressed in molality (mH + ). Hint: the chemical potential of the solvent (say species 1) and a solute (say species 2) of a dilute solution can be calculated as follows: N2 kT , N1 µ2 = µ◦2 + kT ln(m2 ), µ1 = µ◦1 − LE where µ◦ is the chemical potential of the substance in its “standard state” – pure liquid for the solvent and a concentration of one mole per kilogram solvent for the solute. For a very dilute solution as considered in this problem, the second term of µ1 can be ignored. Remember that NA k = R where NA is Avogadro’s number. (R = 8.315 J/K). (b) What is the pH of this system where pH is defined as: pH ≡ − log10 (mH + ). 2

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