An Introduction to Thermal Physics 1st Edition solutions

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
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 an

An Introduction to Thermal Physics 1
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)

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
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:

An Introduction to Thermal Physics 1
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.)

An Introduction to Thermal Physics 1
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:

An Introduction to Thermal Physics 1
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)

An Introduction to Thermal Physics 1
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 con

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
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 th

An Introduction to Thermal Physics 1
Problem 13P Fun with logarithms. (a) Simplify the expression ea ln b. (That is, write it in a way that doesnt 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

An Introduction to Thermal Physics 1
Write \(e^{10^{23}}\) in the form \(10^x\), for some x.

An Introduction to Thermal Physics 1
Problem 15P Use a pocket calculator to check the accuracy of Stirlings approximation for N = 50. Also check the accuracy of below equation for ln N!. Equation: ln N! ? N ln N N.

An Introduction to Thermal Physics 1
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 Stirlings approximation. If you have a fancy

An Introduction to Thermal Physics 1
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:

An Introduction to Thermal Physics 1
Problem 18P Use Stirlings 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 negl

An Introduction to Thermal Physics 1
Problem 19P Use Stirlings 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

An Introduction to Thermal Physics 1
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 s

An Introduction to Thermal Physics 1
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 well 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

An Introduction to Thermal Physics 1
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 e

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
Problem 24P For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N? = N/2. a) Use Stirlings 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 multi

An Introduction to Thermal Physics 1
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 m

An Introduction to Thermal Physics 1
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.

An Introduction to Thermal Physics 1
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 i

An Introduction to Thermal Physics 1
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 mu

An Introduction to Thermal Physics 1
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 acces

An Introduction to Thermal Physics 1
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 Boltzmanns constant), assuming that all of the microstates are allowed. (This is the systems entropy over long time scale

An Introduction to Thermal Physics 1
Problem 31P Fill in the algebraic steps to derive the Sackur-Tetrode below equation. Equation:

An Introduction to Thermal Physics 1
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 followi

An Introduction to Thermal Physics 1
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?

An Introduction to Thermal Physics 1
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 Ill prove that this formula is valid for any quasistatic process. Show, however, that it is not valid for

An Introduction to Thermal Physics 1
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 s

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
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

An Introduction to Thermal Physics 1
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 wi

An Introduction to Thermal Physics 1
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.

An Introduction to Thermal Physics 1
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 w

An Introduction to Thermal Physics 1
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.

An Introduction to Thermal Physics 1
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: Add