Use the thermodynamic identity to derive the heat capacity

Problem 4P Can a “miserly” system, with a concave-up entropy-energy graph, ever be in stable thermal equilibrium with another system? Explain.

Problem 39P In below Problem 1 you computed the entropy of an ideal monatomic gas that lives in a two-dimensional universe. Take partial derivatives with respect to U, A, and N to determine the temperature, pressure, and chemical potential of this gas. (In two dimensions, pressure is defined as force per unit length.) Simplify your results as much as possible, and explain whether they make sense. Problem 1: Find an expression for the entropy of the two-dimensional ideal gas considered in below Problem 2. Express your result in terms of U, A, and N. Problem 2: Consider an ideal monatomic gas that lives in a two-dimensional universe (“flatland”), occupying an area A instead of a volume V. By following the same logic as above, find a formula for the multiplicity of this gas, analogous to equation 2.40.

Problem 3P Below Figure shows graphs of entropy vs. energy for two objects, A and B. Both graphs are on the same scale. The energies of these two objects initially have the values indicated; the objects are then brought into thermal contact with each other. Explain what happens subsequently and why, without using the word “temperature.” Figure: Graphs of entropy vs. energy for two objects. .

Problem 1P Use Table 3.1 to compute the temperatures of solid A and solid B when qA = 1. Then compute both temperatures when qA = 60. Express your answers in terms of ?/k, and then in kelvins assuming that ? = 0.1 eV.

Problem 2P Use the definition of temperature to prove the zeroth law of thermodynamics, which says that if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is in thermal equilibrium with system C. (If this exercise seems totally pointless to you, you’re in good company: Everyone considered this “law” to be completely obvious until 1931, when Ralph Fowler pointed out that it was an unstated assumption of classical thermodynamics.)

Problem 5P Starting with the result of Problem, find a formula for the temperature of an Einstein solid in the limit q ? N. Solve for the energy as a function of temperature to obtain U = N?e??/kT (where ? is the size of an energy unit). Problem: Use the methods of this section to derive a formula, similar to below equation, for the multiplicity of an Einstein solid in the “low-temperature” limit, q? N. Equation:

Problem 6P In Section 2.5 I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of any such system is proportional to UNf/2, where Nf is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula for ? cannot be valid when the total energy is very small?

Problem 7P Use the result of below Problem 1 to calculate the temperature of a black hole, in terms of its mass M. (The energy is Mc2.) Evaluate the resulting expression for a one-solar-mass black hole. Also sketch the entropy as a function of energy, and discuss the implications of the shape of the graph. Problem 1: A black hole is a region of space where gravity is so strong that nothing, not even light, can escape. Throwing something into a black hole is therefore an irreversible process, at least in the everyday sense of the word. In fact, it is irreversible in the thermodynamic sense as well: Adding mass to a black hole increases the black hole’s entropy. It turns out that there’s no way to tell (at least from outside) what kind of matter has gone into making a black hole. Therefore, the entropy of a black hole must be greater than the entropy of any conceivable type of matter that could have been used to create it. Knowing this, it’s not hard to estimate the entropy of a black hole. a) Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c2, where G is Newton’s gravitational constant and c is the speed of light. Calculate the approximate radius of a one-solar-mass black hole (M = 2 × 1030 kg). ________________ b) In the spirit of below Problem 2, explain why the entropy of a black hole, in fundamental units, should be of the order of the maximum number of particles that could have been used to make it. ________________ c) To make a black hole out of the maximum possible number of particles, you should use particles with the lowest possible energy: long-wavelength photons (or other mass less particles). But the wavelength can’t be any longer than the size of the black hole. By setting the total energy of the photons equal to Mc2, estimate the maximum number of photons that could be used to make a black hole of mass M. Aside from a factor of 8?2, your result should agree with the exact formula for the entropy of a black hole, obtained* through a much more difficult calculation. ________________ d) Calculate the entropy of a one-solar-mass black hole, and comment on the result. Problem 2: For either a monatomic ideal gas or a high-temperature Einstein solid, the entropy is given by Nk times some logarithm. The logarithm is never large, so if all you want is an order-of-magnitude estimate, you can neglect it and just say S ~ Nk. That is, tire entropy in fundamental units is of the order of the number of particles in the system. This conclusion turns out to be true for most systems (with some important exceptions at low temperatures where the particles are behaving in an orderly way). So just for fun, make a very rough estimate of the entropy of each of the following: this book (a kilogram of carbon compounds); a moose (400 kg of water); the sun (2 × 1030 kg of ionized hydrogen).

Problem 8P Starting with the result of Problem 1, calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem. A more accurate model of solids at low temperatures is presented in Section 7.5.) Problem 1: Starting with the result of Problem 2, find a formula for the temperature of an Einstein solid in the limit q ? N. Solve for the energy as a function of temperature to obtain U = N?e??/kT(where ? is the size of an energy unit). Problem 2: Use the methods of this section to derive a formula, similar to below equation, for the multiplicity of an Einstein solid in the “low-temperature” limit, q ? N. Equation:

Problem 9P In solid carbon monoxide, each CO molecule has two possible orientations: CO or OC. Assuming that these orientations are completely random (not quite true but close), calculate the residual entropy of a mole of carbon monoxide.

Problem 10P 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. (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 the 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?

Problem 11P In order to take a nice warm bath, you mix 50 liters of hot water at 55°C with 25 liters of cold water at 10°C. How much new entropy have you created by mixing the water?

Problem 12P Estimate the change in the entropy of the universe due to heat escaping from your home on a cold winter day.

Problem 3.15. In Problem 1.55 you used the virial theorem to estimate the heat capacity of a star. Starting with that result, calculate the entropy of a star, first in terms of its average temperature and then in terms of its total energy. Sketch the entropy as a function of energy, and comment on the shape of the graph.

Problem 14P Experimental measurements of the heat capacity of aluminium at low temperatures (below about 50 K) can be fit to the formula Cv = aT + bT3, where CV is the heat capacity of one mole of aluminium, and the constants a and b are approximately a = 0.00135 J/K2 and b = 2.48 × 10–5 J/K4. From this data, find a formula for the entropy of a mole of aluminium as a function of temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing your answers both in conventional units (J/K) and as unitless numbers (dividing by Boltzmann’s constant). [Comment: In Chapter 7 I’ll explain why the heat capacity of a metal has this form. The linear term comes from energy stored in the conduction electrons, while the cubic term comes from lattice vibrations of the crystal.]

Problem 18P Use a computer to reproduce Table 3.2 and the associated graphs of entropy, temperature, heat capacity, and magnetization. (The graphs in this section are actually drawn from the analytic formulas derived below, so your numerical graphs won’t be quite as smooth.)

Problem 13P When the sun is high in the sky, it delivers approximately 1000 watts of power to each square meter of earth’s surface. The temperature of the surface of the sun is about 6000 K, while that of the earth is about 300 K. (a) Estimate the entropy created in one year by the flow of solar heat onto a square meter of the earth. ________________ (b) Suppose you plant grass on this square meter of earth. Some people might argue that the growth of the grass (or of any other living thing) violates the second law of thermodynamics, because disorderly nutrients are converted into an orderly life form. How would you respond?

Problem 17P Verify every entry in the third line of Table 3.2 (starting with N? = 98).

Problem 16P A bit of computer memory is some physical object that can be in two different states, often interpreted as 0 and 1. A byte is eight bits, a kilobyte is 1024 (= 210) bytes, a megabyte is 1024 kilobytes, and a gigabyte is 1024 megabytes. (a) Suppose that your computer erases or overwrites one gigabyte of memory, keeping no record of the information that was stored. Explain why this process must create a certain minimum amount of entropy, and calculate how much. ________________ (b) If this entropy is dumped into an environment at room temperature, how much heat must come along with it? Is this amount of heat significant?

Problem 19P Fill in the missing algebraic steps to derive equations 3.30, 3.31, and 3.33.

Problem 20P Consider an ideal two-state electronic paramagnet such as DPPH, with ? = ?B. In the experiment described above, the magnetic field strength was 2.06 T and the minimum temperature was 2.2 K. Calculate the energy, magnetization, and entropy of this system, expressing each quantity as a fraction of its maximum possible value. What would the experimenters have had to do to attain 99% of the maximum possible magnetization?

In the experiment of Purcell and Pound, the maximum magnetic field strength was 0.63 T and the initial temperature was 300 K. Pretending that the lithium nuclei have only two possible spin states (in fact they have four), calculate the magnetization per particle, M/N, for this system. Take the constant \(\mu\) to be \(5 \times 10^{-8} \mathrm{eV} / \mathrm{T}\). To detect such a tiny magnetization, the experimenters used resonant absorption and emission of radio waves. Calculate the energy that a radio wave photon should have, in order to flip a single nucleus from one magnetic state to the other. What is the wavelength of such a photon?

Problem 23P Show that the entropy of a two-state paramagnet, expressed as a function of temperature, is S = Nk[ln(2 cosh x) – x tanh x], where x = ?B/kT. Check that this formula has the expected behavior as T ? 0 and T ? ?.

Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy. Use separate columns for the energy, multiplicity, entropy, temperature, and heat capacity. To calculate the temperature, evaluate \(\Delta U / \Delta S\) for two nearby rows in the table. (Recall that \(U=q \epsilon\) for some constant \(\epsilon\).) The heat capacity \((\Delta U / \Delta T)\) can be computed in a similar way. The first few rows of the table should look something like this: (In this table I have computed derivatives using a “centered-difference” approximation. For example, the temperature .28 is computed as 2/(7.15 – 0).) Make a graph of entropy vs. energy and a graph of heat capacity vs. temperature. Then change the number of oscillators to 5000 (to “dilute” the system and look at lower temperatures), and again make a graph of heat capacity vs. temperature. Discuss your prediction for the heat capacity, and compare it to the data for lead, aluminium, and diamond shown in below Figure. Estimate the numerical value of \(\epsilon\), in electron-volts, for each of those real solids.

Sketch (or use a computer to plot) a graph of the entropy of a two-state paramagnet as a function of temperature. Describe how this graph would change if you varied the magnetic field strength.

Problem 25P In below Problem 1 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately (a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why the factors omitted from the formula have no effect on the entropy, when N and q are large. ________________ (b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is U = q?, where ? is a constant.) Be sure to simplify your result as much as possible. ________________ (c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity. ________________ (d) Show that, in the limit T ?? the heat capacity is C = Nk. (Hint: When x is very small, ex ? 1 + x.) Is this the result you would expect? Explain. ________________ (e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot C/Nk vs. the dimensionlcss variable t = kT/?, for t in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminum, and diamond shown in Figure. Estimate the value of ?, in electron-volts, for each of those real solids. ________________ (f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x 3 in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than (?/kT)2 in the final answer. When the smoke clears, you should find . Problem 1: Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately The square root in the denominator is merely large, and can often be neglected. However, it is needed in below Problem 2 (Hint: First show that Do not neglect the in Stirling’s approximation.) Problem 2: This problem gives an alternative approach to estimating the width of the peak of the multiplicity function for a system of two large Einstein solids. a) Consider two identical Einstein solids, each with N oscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly 2N. How many different macrostates (that is, possible values for the total energy in the first, solid) are there for this combined system? ________________ b) Use the result of above Problem to find an approximate expression, for the total number of microstates for the combined system. (Hint: Traet the combined system as a single Einstein solid. Do not throw away factors of “large” numbers, since you will eventually be dividing two “very large” numbers that are nearly equal. ________________ c) The most likely macrostate for this system is (of course) the one in which the energy is shared equally between the two solids. Use the result of above Problem to find an approximate expression for the multiplicity of this macrostate. ________________ d)You can get a rough idea of the “sharpness” of the multiplicity function by comparing your answers to parts (b) and (c). Part (c) tells you the Height of the peak, while part (b) tells yon the total area under the entire graph. As a very crude approximation, pretend that the peak’s shape is rectangular. In this case, how wide would it be? Out of all the macrostates, what fraction have reasonably large probabilities? Evaluate this fraction numerically for the case N = 1023. Figure: Measured heat capacities at constant pressure (data points) forone mole each of three different elemental solids. The solid curves show the heatcapacity at constant volume predicted by the model used in Section 7.5, with thehorizontal scale chosen to best fit the data for each substance. At sufficiently hightemperatures, CV for each material approaches the value 3R predicted by theequipartition theorem. The discrepancies between the data and the solid curvesat high T are mostly due to the differences between CP and CV. At T = 0 alldegrees of freedom are frozen out, so both CP and CV go to zero. Data from Y. S.Touloukian, ed., Thermophysical Properties of Matter (Plenum, New York, 1970).

Problem 26P The results of either of the two preceding problems can also be applied to the vibrational motions of gas molecules. Looking only at the vibrational contribution to the heat capacity graph for H2 shown in below Figure, estimate the value of ? for the vibrational motion of an H2 molecule. Figure: Heat capacity at constant volume of one mole of hydrogen (H2) gas.Note that the temperature scale is logarithmic. Below about 100 K only the threetranslational degrees of freedom are active. Around room temperature the tworotational degrees of freedom are active as well. Above 1000 K the two vibrationaldegrees of freedom also become active. At atmospheric pressure, hydrogen liquefiesat 20 K and begins to dissociate at about 2000 K. Data from Woolley et al. (1948).

A liter of air, initially at room temperature and atmospheric pressure, is heated at constant pressure until it doubles in volume. Calculate the increase in its entropy during this process.

Problem 27P What partial-derivative relation can you derive from the thermodynamic identity by considering a process that takes place at constant entropy? Does the resulting equation agree with what yon already knew? Explain.

Problem 30P As shown in below Figure, the heat capacity of diamond near room temperature is approximately linear in T. Extrapolate this function up to 500 K, and estimate the change in entropy of a mole of diamond as its temperature is raised from 298 K to 500 K. Add on the tabulated value at 298 K (from the backof this book) to obtain S(500 K). Figure: Measured heat capacities at constant pressure (data points) forone mole each of three different elemental solids. The solid curves show the heatcapacity at constant volume predicted by the model used in Section 7.5, with thehorizontal scale chosen to best fit the data for each substance. At sufficiently hightemperatures, CV for each material approaches the value 3R predicted by theequipartition theorem. The discrepancies between the data and the solid curvesat high T are mostly due to the differences between CP and CV. At T = 0 alldegrees of freedom are frozen out, so both CP and CV go to zero. Data from Y. S.Touloukian, ed., Thermophysical Properties of Matter (Plenum, New York, 1970).

Problem 31P Experimental measurements of heat capacities are often represented in reference works as empirical formulas. For graphite, a formula that works well over a fairly wide range of temperatures is (for one mole) where a = 16.86 J/K, b = 4.77 × 10–3 J/K2, and c = 8.54 × 105 J.K. Suppose, then, that a mole of graphite is heated at constant pressure from 298 K to 500 K. Calculate the increase in its entropy during this process. Add on the tabulated value of S(298 K) (from the back of this book) to obtain S(500 K).

Problem 33P Use the thermodynamic identity to derive the heat capacity formula 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.

Problem 32P A cylinder contains one liter of air at room temperature (300 K) and atmospheric pressure (105 N/m2). At one end of the cylinder is a massless piston, whose surface area is 0.01 m2. Suppose that you push the piston in very suddenly, exerting a force of 2000 N. The piston moves only one millimeter, beforeit is stopped by an immovable barrier of some sort. (a) How much work have you done on this system? ________________ (b) How much heat has been added to the gas? ________________ (c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the internal energy of the gas increase? ________________ (d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).

Problem 29P Sketch a qualitatively accurate graph of the entropy of a substance (perhaps H2O) as a function of temperature at fixed pressure. Indicate where the substance is solid, liquid, and gas. Explain each feature of the graph briefly.

Problem 34P Polymers, like rubber, are made of very long molecules, usually tangled up in a configuration that has lots of entropy. As a very crude model of a rubber band, consider a chair, of N links, each of length l (see below Figure). Imagine that each link has only two possible states, pointing either left or right. The total length L of the rubber band is the net displacement from the beginning of the first link to the end of the last link. (a) Find an expression for the entropy of this system in terms of N and NR, the number of links pointing to the right. ________________ (b) Write down a formula for L in terms of N and NR. ________________ (c) For a one-dimensional system such as this, the length L is analogous to the volume V of a three-dimensional system. Similarly, the pressure P is replaced by the tension force F. Taking F to be positive when the rubber band is pulling inward, write down and explain the appropriate thermodynamic identity for this system. ________________ (d) Using the thermodynamic identity, you can now express the tension force F in terms of a partial derivative of the entropy. From this expression, compute the tension in terms of L, T, N, and l. ________________ (e) Show that when L ? Nl, the tension force is directly proportional to L (Hooke’s law). ________________ (f) Discuss the dependence of the tension force on temperature. If you increase the temperature of a rubber band, does it tend to expand or contract? Does this behaviour make sense? ________________ (g) Suppose that you hold a relaxed rubber band in both hands and suddenly stretch it. Would you expect its temperature to increase or decrease? Explain. Test your prediction with a real rubber band (preferably a fairly heavy one with lots of stretch), using your lips or forehead as a thermometer. (Hint: The entropy you computed in part (a) is not the total entropy of the rubber band. There is additional entropy associated with the vibrational energy of the molecules; this entropy depends on U but is approximately independent of L.) Figure A crude model of a rubber band as a chain in which eachlink can only point left or right.

Problem 37P Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgz in addition to its kinetic energy. (a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz: (You can derive this result from either the definition ? = –T(?S/?N)U,V or the formula ? = (?U/?N)S,V.) ________________ (b) Suppose you have two chunks of helium gas, one at sea level and one at height z, each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is N(z) = N(0)e?mgz/kT, in agreement with the result of below Problem 1. Problem 1: The exponential atmosphere. (a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest , the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air. ________________ (b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation called the barometric equation. ________________ (c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e–mgz/kT. Show also that the density obeys a similar equation. ________________ (d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft , 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.) Problem 2: Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%).

Problem 35P In the text I showed that for an Einstein solid with three oscillators and three units of energy, the chemical potential is ? = –? (where ? is the size of an energy unit and we treat each oscillator as a “particle”). Suppose instead that the solid has three oscillators and four units of energy. How does the chemical potential then compare to –?? (Don’t try to get an actual value for the chemical potential; just explain whether it is more or less than –?.)

Problem 38P Suppose you have a mixture of gases (such as air, a mixture of nitrogen and oxygen). The mole fraction xi of any species i is defined as the fraction of all the molecules that belong to that species: xi = Ni/Ntotal. The partial pressure Pi of species i is then defined as the corresponding fraction of the total pressure: Pi = xiP. Assuming that the mixture of gases is ideal, argue that the chemical potential µi of species i in this system is the same as if the other gases were not present, at a fixed partial pressure Pi.

Problem 36P Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate “particle.” (a) Show that the chemical potential is ________________ (b) Discuss this result in the limits N ? q and N ? q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?