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Use Stirling’s approximation to show that the multiplicity

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 18P Chapter 2

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 18P

Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately

The square root in the denominator is merely large, and can often be neglected. However, it is needed in below Problem (Hint: First show that Do not neglect the  in Stirling’s approximation.)

Problem:

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 N oscillators, 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 above Problem to 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.

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 above Problem to find an approximate expression for the multiplicity of this macrostate.

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 yon 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.

Step-by-Step Solution:

Step 1:

        To show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately

         =

Step 2:

        We know that

         = -----(1)

The basic formula related to n! = n(n-1)!

This gives (n-1)! =         -----(2)

Applying the (2) in the appropriate terms in (1)

 = -----(3)

(N-1)! = -----(4)

Step 3:

        Substituting (3) and (4) in (1)

        We get

         = -----(4)

Step 4:

        We know that from Stirling’s approximation

        n! = -----(5)

        Therefore  

         =  ----(6)

         =  -----(7)

         =  -----(8)

 

Step 5 of 8

Chapter 2, Problem 18P is Solved
Step 6 of 8

Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

This full solution covers the following key subjects: system, large, Multiplicity, Einstein, combined. This expansive textbook survival guide covers 10 chapters, and 454 solutions. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1. The full step-by-step solution to problem: 18P from chapter: 2 was answered by , our top Physics solution expert on 07/05/17, 04:29AM. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. The answer to “Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately The square root in the denominator is merely large, and can often be neglected. However, it is needed in below (Hint: First show that Do not neglect the in Stirling’s approximation.)Problem: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 N oscillators, 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 above 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.________________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 above find an approximate expression for the multiplicity of this macrostate.________________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 yon 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.” is broken down into a number of easy to follow steps, and 301 words. Since the solution to 18P from 2 chapter was answered, more than 408 students have viewed the full step-by-step answer.

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