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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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Use a computer to reproduce the table and graph in Figure: 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?
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 is the total number of molecules and is the number of molecules of type . Use Stirling's approximation to show that this expression is the same as the result of the previous problem when both and are large.
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?
Suppose you flip fair coins.
(a) How many possible outcomes (microstates) are there?
(b) How many ways are there of getting exactlyheads andtails?
(c) What is the probability of getting exactly heads and tails?
(d) What is the probability of getting exactly heads and tails?
(e) What is the probability of getting exactly heads and 10 tails?
(f) What is the probability of getting heads and no tails?
(g) Plot a graph of the probability of getting n heads, as a function of n.
Suppose you flip 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 heads and tails (in any order)?
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