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Chapter 2: Q. 2.15 (page 63)
Use a pocket calculator to check the accuracy of Stirling's approximation for . Also check the accuracy of equation for .
By using pocket calculator as,
and
By using Stirling's approximation as,
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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 macrostates are available to this system?
(b) How many different microstates 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 behavior?
Using the same method as in the text, calculate the entropy of mixing for a system of two monatomic ideal gases, and , whose relative proportion is arbitrary. Let be the total number of molecules and let be the fraction of these that are of species . You should find
Check that this expression reduces to the one given in the text when .
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?
The mathematics of the previous problem can also be applied to a one-dimensional random walk: a journey consisting of steps, all the same sic, cache chosen randomly to be cither 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 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 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 traveling through air) in one second, at room temperature and atmospheric pressure. Discuss how your estimate would differ if the clasped time or the temperature were different. Check that your estimate is consistent with the treatment of diffusion in Section
Consider a two-state paramagnet with 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."
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