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Chapter 2: Q. 2.6 (page 55)

Calculate the multiplicity of an Einstein solid with 30oscillators and 30units of energy. (Do not attempt to list all the microstates.)

Short Answer

Expert verified

An Einstein solid with 30oscillators and 30energy units has a multiplicity ofΩ(30,30)=591322907824307

Step by step solution

01

Step1:The number of oscillators contained in an Einstein solid.

Using the general Einstein solid multiplicity formula withN=3and q=4:

Ω(N,q)=q+N−1q=(q+N−1)!q!(N−1)!Ω(3,4)=4+3−14=(4+3−1)!4!(3−1)!=15

02

Step2:The total number of microstates

So, if we have30oscillators and energy units, the number of30microstates is:

Ω(30,30)=30+30−130=(30+30−1)!30!(30−1)!=591322907824307

Ω(30,30)=591322907824307

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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! .

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?

Problem 2.20. Suppose you were to shrink Figure2.7until the entire horizontal scale fits on the page. How wide would the peak be?

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(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)?

Rather than insisting that all the molecules be in the left half of a container, suppose we only require that they be in the leftmost 99%(leaving the remaining 1%completely empty). What is the probability of finding such an arrangement if there are 100molecules in the container? What if there are 10,000molecules? What if there are 1023?
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