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Chapter 2: Q. 2.7 (page 57)

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 equation 2.9.

Short Answer

Expert verified

Possible microstates of dots and vertical lines are Ω(4,2)=10

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01

 Step :1 Possible microstates

For four oscillators N=4and two unit of energyq=2 , the possible microstates are:

02

 Step :2 Using the general formula

Using the general formula for the multiplicity of Einstein solid with N=4and q=2:

localid="1650337110602" Ω(N,q)=q+N-1q=(q+N-1)!q!(N-1)!

localid="1650337124840" Ω(4,2)=4+2-12=(4+2-1)!2!(4-1)!=10

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Most popular questions from this chapter

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?

How many possible arrangements are there for a deck of 52playing cards? (For simplicity, consider only the order of the cards, not whether they are turned upside-down, etc.) Suppose you start w e 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?

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, is80; 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?

Suppose you flip 50fair coins.

(a) How many possible outcomes (microstates) are there?

(b) How many ways are there of getting exactly25heads and25tails?

(c) What is the probability of getting exactly 25heads and 25tails?

(d) What is the probability of getting exactly 30heads and 20tails?

(e) What is the probability of getting exactly 40heads and 10 tails?

(f) What is the probability of getting 50heads and no tails?

(g) Plot a graph of the probability of getting n heads, as a function of n.

Consider again the system of two large, identical Einstein solids treated in Problem 2.22.

(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 macro state. (This is the system's entropy over short time scales, except when there is a large and unlikely fluctuation away from the most likely macro state.)

(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 macro state, 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?
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