For this problem you might also need to know that 1 kcal = kj and that the specific heat capacity of water at 25 C is 1 kcal/(kg K).

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Gertrude Brown
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1 Umeå Universitet Department of Physics Peter Olsson Examination, Thermodynamics B, 6hp, , at 9:00 15:00. Allowed aids: Calculator, Beta, Physics Handbook, english dictionary, one A4-sized paper with notes, figures, and equations. Hand in each problem on a separate page. The calculations and the reasoning should be easy to follow. Good luck! 1 Energy conversions A person with weight 60 kg climbs a mountain that is 1500 m high. (1p) a) Assume that she converts the chemical energy from food into mechanical energy with an efficiency of 25% and that essentially all the mechanical work is used to climb vertically, roughly how many bowls of corn flakes(standad serving size 100 kcal) should the hiker eat before setting out? b) Assume that the rest is converted into thermal energy. If there (1p) were no way to dissipate this energy, by how many degrees would her body temperature increase? (You need to make a simple assumption about the properties of the body.) c) In fact the extra energy mostly goes into evaporating water from (1p) her skin. How many liters of water should she drink during the hike to replace the lost fluid? (At 25 C, which is a reasonable temperature to assume, the latent heat of vaporization of water is 580 kcal/kg, somewhat more than the value at 100 C.) For this problem you might also need to know that 1 kcal = kj and that the specific heat capacity of water at 25 C is 1 kcal/(kg K). 2 Rising bubbles Two identical bubbles of gas form at the bottom of a lake and then rise tothesurface. Becausethepressureismuchloweratthesurfacethanat the bottom, both bubbles expand as they rise. However, bubble A rises very quickly, so that no heat is exchanged between it and the water. Meanwhile, bubble B rises slowly (impeded by a tangle of seaweed), so that it always remains in thermal equilibrium with the water, which has the same temperature everywhere. Which of the two bubbles is larger by the time they reach the sur- (3p) face? Explain your reasoning carefully: Describe changes in temperature and/or heat flows in the processes.

2 Examination in Thermodynamics B, 6hp, Engine Consider an engine cycle consisting of three steps: (1) adiabatic expansion from volume V 1 to V 2 (2) isothermal compression at temperature T 1 back to the V 1, and (3) increase of pressure at constant volume back to the original state. a) Make a sketch of this cycle in a P-V diagram. Also show at which (1p) stages heat is entering and leaving the process. Put in arrows for Q h and Q c. b) DetermineQ h andq c andfindtheefficiency ofsuchanengine. As- (2p) sume that the working substance is a two-atomic gas with frozen vibrational degrees of freedom. c) Assume that this process could instead be run with a gas consist- (1p) ing of molecules with more atoms per molecule and that this can be done with the same V 1, V 2, and T 1. How would this change of working substance affect the efficiency? 4 Two state system A system S consists of three distinguishable particles that (due to interactions with the environment) each can be in two different states, say up and down. The energies of these one-particle states are ǫ and 2ǫ, respectively. a) Make a list of all microstates of S and their respective energies. (1p) b) Assume that all microstates are equally probable and calculate (1p) explicitly the average energy U. Assume that there is another system, S, which is identical with S with the exception that it has only two particles. Let the systems be able to exchange heat with each another. A macrostate is now described by the energies of the respective systems, U and U. c) Assume that the whole system S S is totally isolated with U + (1p) U = 7ǫ. Which macrostates are possible? d) Determine the entropy of the most probable macrostate. (1p) 5 Entropy Imagine a material with the heat capacity given by C V = at 3 where a (3p) is a constant. Find how the entropy of this material depends on U.

3 Examination in Thermodynamics B, 6hp, Refrigerator door You open a kitchen refrigerator and close it again. In this process a (4p) part of the cold air (volume fraction ǫ = 0.2) which was originally in the refrigerator is replaced by warm air of the room. After a short pause you try to open the refrigerator one more time and find that a much larger force is now needed. Calculate this force. Take room temperature T 1 20 C, temperature in the refrigerator T 2 0 C, and area of the refrigerator door A = 0.5 m 2. 7 Lead battery For a lead battery the total reaction can be written Pb + PbO 2 + 4H + + 2SO 2 4 2PbSO 4 + 2H 2 O. a) Use thedatainthetablebelowtodetermine thevaluesof H and (1p) Gforthisreactionpermoleofmetalliclead(=Pb). Assumethat the reaction takes place at room temperature and atmospheric pressure. b) Assuming ideal performance, how much electrical work can you (1p) get out of the cell for each mole of lead? Please be careful to explain the signs. c) How much heat is involved? Is the heat generated as waste heat (1p) or is it necessary to supply heat to get current out of the battery? Substance H kj G kj S J/K Pb (s) PbO 2 (s) H + (aq) SO 2 4 (l) PbSO 4 (s) H 2 O (l) Thermodynamic properties: Values for one mole of material at 298 K and 1 bar. Total number of points = 24

4 Examination in Thermodynamics B, 6hp, Topics in Thermodynamics, 1.5 hp 8 Mixtures of two types of molecules Consider a mixture where x is the fraction of B molecules and 1 x is the fraction of A molecules. If they are not mixed, the total free energy is just the sum of the separate free energies, G = (1 x)g 0 A +xg0 B. We now consider so-called nonideal mixtures in which the mixing increases the energy, denoted by U mixing. There is also an entropy term, S mixing = R[xlnx+(1 x)ln(1 x)]. a) Sketch typical curves of G(x) a different temperatures from T = 0 (2p) to higher T. b) Under certain (rather common) conditions the homogenous mix- (1p) ture is not the stable state but the system splits up into an A-rich phase and a B-rich phase, since this gives a lower free energy. Make a sketch that illustrates this phenomenon and identify x a of the A-rich composition and x b of the B-rich composition in the figure. c) S mixing has infinite slopes both at x = 0 and at x = 1. Why is (1p) that significant? 9 Osmotic pressure and desaliation Seawater has a salinity of 3.5%, which means that if you boil away a kilogram of seawater, you will have 35 g of solids (mostly NaCl) when you are finished. When dissolved, NaCl dissociates into separate Na + and Cl ions. a) Start from the expression for the chemical potential of the solvent (2p) (with N A molecules) in the presence of N B solute molecules, µ A = ( G N A ) T,P,N B = µ 0 (T,P) N BkT N A, (1) where µ 0 isthechemical potential of thepure substance, andshow that the osmotic pressure, i.e. the difference in pressure between

5 Examination in Thermodynamics B, 6hp, the solution and the pure solvent that is needed to prevent osmosis from happening, is P 2 P 1 = N BkT. (2) V Hint: Express the chemical potential of the pure solvent in terms of µ 0 / P and recall that the chemical potential of a pure substance is just the Gibbs free energy per particle. Note that it is possible to solve the following problem without having solved part (a). b) Calculate the osmotic pressure difference between seawater and (1p) fresh water from Eq. (2). Assume for simplicity that all dissolved salts in seawater are NaCl. c) If you apply a pressure difference greater than the osmotic pres- (1p) sure to a solution separated from pure solvent by a semipermeable membrane, you get reverse osmosis: a flow of solvent out of the solution. This process can be used to desalinate (remove the salt from) seawater. Calculate the minimum work required to desalinate one liter of seawater. Discuss some reasons why the actual work required would be greater than the minimum.

2012 Thermodynamics Division C

2012 Thermodynamics Division C Team: Team Number: Team Member Names: 1. 2. Instructions: Answer all questions on the test paper. If you need more room, you may attach extra paper. The test is worth a total of 50 points. Show all work