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Chapter 2: Q.2.8 (page 59)

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?

Short Answer

Expert verified

(a) 21 macrostates are available

(b) Ωoverall=6.892×1010microstates are available

(c) the probability that all the energy is in solid Aat thermal equilibrium,P=1.453×10-4

(d) the probability in this case isP=0.1238

Step by step solution

01

find macrostates and microstates 

Suppose we have two systems of Einstein solids, Aand Bwith NA=NB=10and qA+qB=20

(a) the number of macrostates is therefore:

q+1=20+1=21

because we start counting 0 to 20.

(b) the number of microstates formula is given by:

ΩoverallNoverall,qoverall=qoverall+Noverall-1qoverall=qoverall+Noverall-1!qoverall!Noverall-1!

where,

qoverall=qA+qB=20,Noverall=NA+NB=20

so,

width="385">Ωoverall=20+20-120=(20+20-1)!20!(20-1)!=6.892×1010

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Most popular questions from this chapter

Use a pocket calculator to check the accuracy of Stirling's approximation forN=50 . Also check the accuracy of equation 2.16forlnN! .

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?

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.

Show that during the quasistatic isothermal expansion of a monatomic ideal gas, the change in entropy is related to the heat input Qby the simple formula

∆s=QT

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.

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 Noscillators, 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 Problem2.18to 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. Answer: 24N/8πN.)

(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 Problem 2.18to find an approximate expression for the multiplicity of this macrostate. (Answer:24N/(4πN) .)

(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 you 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.
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