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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) 惟overall=6.892脳1010microstates are available

(c) the probability that all the energy is in solid Aat thermal equilibrium,P=1.453脳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:

惟overallNoverall,qoverall=qoverall+Noverall-1qoverall=qoverall+Noverall-1!qoverall!Noverall-1!

where,

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

so,

width="385">惟overall=20+20-120=(20+20-1)!20!(20-1)!=6.892脳1010

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Most popular questions from this chapter

Use a computer to reproduce the table and graph in Figure2.4: 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 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?

Consider a system of two Einstein solids, Aand B, each containing 10 oscillators, sharing a total of 20units 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) 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?

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 a royal flush (on the first deal)?

Use a computer to plot formula 2.22directly, as follows. Define z=q_{A}/q, so that (1-z)=q_{B}/q. Then, aside from an overall constant that we'll ignore, the multiplicity function is [4z(1-z)]N, where zranges from 0to1and the factor of 4ensures that the height of the peak is equal to 1for any N. Plot this function forN=1,10,100,1000, and 10,000. Observe how the width of the peak decreases asNincreases.

Fill in the algebraic steps to derive the Sackur-Tetrode equation(2.49).
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