Use the Sackur-Tetrode equation to calculate the entropy

Problem 1P Suppose you flip four fair coins. (a) Make a list of all the possible outcomes, as in below Table. ________________ (b) Make a list of all the different “macrostates” and their probabilities. ________________ (c) Compute the multiplicity of each macrostate using the combinatorial formula 2.6, and check that these results agree with what you got by brute-force counting. TABLE: A list of all possible “microstates” of a set of three coins (where H is for heads and T is for tails). Penny Nickel Dime H H H H H T H T H T H H H T T T H T T T H T T T

Problem 4P Calculate the number of possible five-card poker hands, dealt, from a deck of 52 cards. (The order of cards in a hand does not matter.) A royal flush consists of the five highest-ranking cards (ace, king, queen, jack, 10) of any one of the four suits. What is the probability of being dealt a royal flush (on the first deal)?

Problem 3P Suppose you flip 50 fair coins. (a) How many possible outcomes (microstates) are there? ________________ (b) How many ways are there of getting exactly 25 heads and 25 tails? ________________ (c) What is the probability of getting exactly 25 heads and 25 tails? ________________ (d) What is the probability of getting exactly 30 heads and 20 tails? ________________ (e) What is the probability of getting exactly 40 heads and 10 tails? ________________ (f) What is the probability of getting 50 heads and no tails? ________________ (g) Plot a graph of the probability of getting n heads, as a function of n.

Problem 2P Suppose you flip 20 fair coins. (a) How many possible outcomes (microstates) are there? ________________ (b) What is the probability of getting the sequence HTHHTTTHTHHHTHHHHTHT (in exactly that order)? ________________ (c) What is the probability of getting 12 heads and 8 tails (in any order)?

Problem 5P For an Einstein solid with each of the following values of N and q, list all of the possible microstates, count them, and verify below formula. (a) N = 3, q = 4 (b) N = 3, q = 5 (c) N = 3, q = 6 (d) N = 4, q = 2 (e) N = 4, q = 3 (f) N = 1, q = anything (g) N = anything, q = 1 Formula:

Problem 42P 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, 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: 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 6P Calculate the multiplicity of an Einstein solid with 30 oscillators and 30 units of energy. (Do not attempt to list all the microstates.)

Problem 7P For an Einstein solid with four oscillators and two units of energy, represent each possible microstate as a series of dots and vertical lines, as used in the text to prove below equation. Equation:

Problem 9P Use a computer to reproduce the table and graph in Figure 2.4: two Einstein solids, each containing three harmonic oscillators, with a total of six units of energy. Then modify the table and graph to show the case where one Einstein solid contains six harmonic oscillators and the other contains four harmonic oscillators (with the total number of energy units still equal to six). Assuming that all microstates are equally likely, what is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?

Problem 8P Consider a system of two Einstein solids, A and B, each containing 10 oscillators, sharing a total of 20 units of energy. Assume that the solids are weakly coupled, and that the total energy is fixed. (a) How many different macro states are available to this system? ________________ (b) How many different macro states are available to this system? ________________ (c) Assuming that this system is in thermal equilibrium, what is the probability of finding all the energy in solid A? ________________ (d) What is the probability of finding exactly half of the energy in solid A? ________________ (e) Under what circumstances would this system exhibit irreversible behaviour?

Problem 11P Use a computer to produce a table and graph, like those in this section, for two interacting two-state paramagnets, each containing 100 elementary magnetic dipoles. Take a “unit” of energy to be the amount needed to flip a single dipole from the “up” state (parallel to the external field) to the “down” state (antiparallel). Suppose that the total number of units of energy, relative to the state with all dipoles pointing up, is 80; this energy can be shared in any way between the two paramagnets. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?

Problem 10P Use a computer to produce a table and graph, like those in this section, for the case where one Einstein solid contains 200 oscillators, the other contains 100 oscillators, and there are 100 units of energy in total. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?

Problem 15P Use a pocket calculator to check the accuracy of Stirling’s approximation for N = 50. Also check the accuracy of below equation for ln N!. Equation: ln N! ? N ln N – N.

Problem 16P Suppose you flip 1000 coins. a) What is the probability of getting exactly 500 heads and 500 tails? (Hint: First write down a formula for the total number of possible outcomes. Then, to determine the “multiplicity” of the 500-500 “macrostate,” use Stirling’s approximation. If you have a fancy calculator that makes Stirling’s approximation unnecessary, multiply all the numbers in this problem by 10, or 100, or 1000, until Stirling’s approximation becomes necessary.) ________________ b) What is the probability of getting exactly 600 heads and 400 tails?

Problem 13P Fun with logarithms. (a) Simplify the expression ea ln b. (That is, write it in a way that doesn’t involve logarithms.) ________________ (b) Assuming that b ? a. prove that ln(a + b) ? (ln a) + (b/a). (Hint: Factor out the a from the argument of the logarithm, so that you can apply the approximation of part (d) of the previous problem.) Problem: The natural logarithm function, ln, is defined so that eln x = x for any positive number x: (a) Sketch a graph of the natural logarithm function. ________________ (b) Prove the identities ln ab = ln a + ln b and ln ab = b ln a. ________________ (c) Prove that ________________ (d) Derive the useful approximation ln(1 + x) ? x, which is valid when |x| ? 1. Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.

Problem 18P 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 (Hint: First show that Do not neglect the in Stirling’s approximation.) Problem: 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:Treat 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.

Problem 17P 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:

Write \(e^{10^{23}}\) in the form \(10^x\), for some x.

The natural logarithm function, ln, is defined so that \(e^{\ln x}=x\) for any positive number x. (a) Sketch a graph of the natural logarithm function. (b) Prove the identities \(\ln a b=\ln a+\ln b \quad \text { and } \quad \ln a^{b}=b \ln a\) (c) Prove that \(\frac{d}{d x} \ln x=\frac{1}{x}\) (d) Derive the useful approximation \(\ln (1+x) \approx x\), which is valid when \(|x| \ll 1\). Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.

Problem 19P Use Stirling’s approximation to find an approximate formula for the multiplicity of a two-state paramagnet. Simplify this formula in the limit to obtain . This result should look very similar to your answer to below Problem; explain why these two systems, in the limits considered, are essentially the same. 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.

Problem 20P Suppose you were to shrink below Figure until the entire horizontal scale fits on the page. How wide would the peak be? Figure: Multiplicity of a system of two large Einstein solids with manyenergy units per oscillator (high-temperature limit). Only a tiny fraction of thefull horizontal scale is shown.

Problem 23P Consider a two-state paramagnet with 1023 elementary dipoles, with the total energy fixed at zero so that exactly half the dipoles point up and half point down. a) How many microstates are “accessible” to this system? ________________ b) Suppose that the microstate of this system changes a billion times per second. How many microstates will it explore in ten billion years (the age of the universe)? ________________ c) Is it correct to say that, if you wait long enough, a system will eventually be found in every “accessible” microstate? Explain your answer, and discuss the meaning of the word “accessible.”

Problem 21P Use a computer to plot formula directly, as follows. Define z = qA/q, so that (1 – z) = qB/b Then, aside from an overall constant that we’ll ignore, the multiplicity function is [4z(1 – z)]N, where z ranges from 0 to 1 and the factor of 4 ensures that the height of the peak is equal to 1 for any N. Plot this function for N = 1, 10, 100, 1000, and 10,000. Observe how the width of the peak decreases as N increases. Formula:

Problem 24P For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N? = N/2. a) Use Stirling’s approximation to estimate the height of the peak in the multiplicity function. ________________ b) Use the methods of this section to derive a formula for the multiplicity function in the vicinity of the peak, in terms of x = N? – (N/2). Check that your formula agrees with your answer to part (a) when x = 0. ________________ c) How wide is the peak in the multiplicity function? ________________ d) Suppose you flip 1,000,000 coins. Would you be surprised to obtain 501,000 heads and 499,000 tails? Would you be surprised to obtain 510,000 heads and 490,000 tails? Explain.

Problem 25P The mathematics of the previous problem can also be applied to a one-dimensional random walk: a journey consisting of N steps, all the same size, each chosen randomly to be either forward or backward. (The usual mental image is that of a drunk stumbling along an alley.) a) Where are you most likely to find yourself, after the end of a long random walk? ________________ b) Suppose you take a random walk of 10,000 steps (say each a yard long). About how far from your starting point would you expect to be at the end? ________________ c) A good example of a random walk in nature is the diffusion of a molecule through a gas; the average step length is then the mean free path, as computed in Section 1.7. Using this model, and neglecting any small numerical factors that might arise from the varying step size and the multidimensional nature of the path, estimate the expected net displacement of an air molecule (or perhaps a carbon monoxide molecule travelling through air) in one second, at room temperature and atmospheric pressure. Discuss how your estimate would differ if the elapsed time or the temperature were different. Check that your estimate is consistent with the treatment of diffusion in Section 1.7.

Problem 26P 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 22P 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 below Problem to find an approximate expression, for the total number of microstates for the combined system. (Hint: Treat 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 below 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. Problem: 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 above Problem. (Hint: First show that Do not neglect the in Stirling’s approximation.)

Problem 27P Rather than insisting that all the molecules be in the left half of a container, suppose we only require that they be in the leftmost 99% (leaving the remaining 1% completely empty). What is the probability of finding such an arrangement if there are 100 molecules in the container? What if there are 10,000 molecules? What if there are 1023?

Problem 28P How many possible arrangements are there for a deck of 52 playing cards? (For simplicity, consider only the order of the cards, not whether they are turned upside-down, etc.) Suppose you start with a sorted deck and shuffle it repeatedly, so that all arrangements become “accessible.” How much entropy do you create in the process? Express your answer both as a pure number (neglecting the factor of k) and in SI units. Is this entropy significant compared to the entropy associated with arranging thermal energy among the molecules in the cards?

Problem 29P Consider a system of two Einstein solids, with NA = 300, NB = 200, and (qtotal = 100 (as discussed in Section 2.3). Compute the entropy of the most likely macrostate and of the least likely macrostate. Also compute the entropy over long time scales, assuming that all microstates are accessible. (Neglect the factor of Boltzmann’s constant in the definition of entropy; for systems this small it is best to think of entropy as a pure number.)

Problem 30P Consider again the system of two large, identical Einstein solids treated in below Problem 1. a) For the case N = 1023, compute the entropy of this system (in terms of Boltzmann’s constant), assuming that all of the microstates are allowed. (This is the system’s entropy over long time scales.) ________________ b) Compute the entropy again, assuming that the system is in its most likely macrostate. (This is the system’s entropy over short time scales, except when there is a large and unlikely fluctuation away from the most likely macrostate.) ________________ c) Is the issue of time scales really relevant, to the entropy of this system? ________________ d) Suppose that, at a moment when the system is near its most likely macrostate, you suddenly insert a partition between the solids so that they can no longer exchange energy. Now, even over long time scales, the entropy is given by your answer to part (b). Since this number is less than your answer to part (a), you have, in a sense, caused a violation of the second law of thermodynamics. Is this violation significant? Should we lose any sleep over it? Problem 1: 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 below Problem 2 to find an approximate expression, for the total number of microstates for the combined system. (Hint:Treat 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 below Problem 2 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. Problem 2: 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 above Problem 1. (Hint: First show that Do not neglect the in Stirling’s approximation.)

Problem 31P Fill in the algebraic steps to derive the Sackur-Tetrode below equation. Equation:

Problem 33P Use the Sackur-Tetrode equation to calculate the entropy of a mole of argon gas at room temperature and atmospheric pressure. Why is the entropy greater than that of a mole of helium under the same conditions?

Problem 34P Show that during the quasistatic isothermal expansion of a monatomic ideal gas, the change in entropy is related to the heat input Q by the simple formula In the following chapter I’ll prove that this formula is valid for any quasistatic process. Show, however, that it is not valid for the free expansion process described above.

Problem 32P Find an expression for the entropy of the two-dimensional ideal gas considered in below Problem. Express your result in terms of U, A, and N. Problem: 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 36P 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, the 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 35P According to the Sackur-Tetrode equation, the entropy of a monatomic ideal gas can become negative when its temperature (and hence its energy) is sufficiently low. Of course this is absurd, so the Sackur-Tetrode equation must be invalid at very low temperatures. Suppose you start with a sample of helium at room temperature and atmospheric pressure, then lower the temperature holding the density fixed. Pretend that the helium remains a gas and does not liquefy. Below what temperature would the Sackur-Tetrode equation predict that S is negative? (The behavior of gases at very low temperatures is the main subject of Chapter 7.)

Problem 38P The mixing entropy formula derived in the previous problem actually applies to any ideal gas, and to some dense gases, liquids, and solids as well, For the denser systems, we have to assume that the two types of molecules are the same size and that molecules of different types interact with each other in the same way as molecules of the same type (same forces, etc.). Such a system is called an ideal mixture. Explain why, for an ideal mixture, the mixing entropy is given by where N is the total number of molecules and NA is the number of molecules of type A. Use Stirling’s approximation to show that this expression is the same as the result of the previous problem when both N and NA are large.

Problem 37P Using the same method as in the text, calculate the entropy of mixing for a system of two monatomic ideal gases, A and B, whose relative proportion is arbitrary. Let N be the total number of molecules: and let x be the fraction of these that are of species B. You should find ?Smixing = –Nk[x ln x + (l – x) ln (l – x)]. Check that this expression, reduces to the one given in the text when x = 1/2.

Problem 39P Compute the entropy of a mole of helium at room temperature and atmospheric pressure, pretending that all the atoms are distinguishable. Compare to the actual entropy, for indistinguishable atoms, computed in the text.

Problem 40P For each of the following irreversible processes, explain how you can tell that the total entropy of the universe has increased. a) Stirring salt into a pot of soup. ________________ b) Scrambling an egg. ________________ c) Humpty Dumpty having a great fall. ________________ d) A wave hitting a sand castle. ________________ e) Cutting down a tree. ________________ f) Burning gasoline in an automobile.

Problem 41P Describe a few of your favourite, and least favourite, irreversible processes. In each case, explain how you can tell that the entropy of the universe increases.