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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

Consider an ideal monatomic gas that lives in a two-dimensional universe ("flatland"), occupying an area Ainstead of a volume V. By following the same logic as above, find a formula for the multiplicity of this gas, analogous to equation 2.40.

Suppose you flip 20 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 12 heads and 8 tails (in any order)?

Using the same method as in the text, calculate the entropy of mixing for a system of two monatomic ideal gases, Aand B, whose relative proportion is arbitrary. Let Nbe the total number of molecules and letx be the fraction of these that are of speciesB . You should find

螖Smixing=鈭�Nk[xln鈦�x+(1鈭�x)ln鈦�(1鈭�x)]

Check that this expression reduces to the one given in the text whenx=1/2 .

Suppose you flip1000 coins.
a What is the probability of getting exactly 500heads and 500tails? (Hint: First write down a formula for the total number of possible outcomes. Then, to determine the "multiplicity" of the 500-500"macrostate," use Stirling's approximation. If you have a fancy calculator that makes Stirling's approximation unnecessary, multiply all the numbers in this problem by 10, or 100, or1000, until Stirling's approximation becomes necessary.)
bWhat is the probability of getting exactly 600heads and400 tails?

Use the methods of this section to derive a formula, similar to equation2.21, for the multiplicity of an Einstein solid in the "low-temperature" limit,q鈮�N .
See all solutions

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