Q. 2.10 Use a computer to produce a tabl... [FREE SOLUTION]

魅影直播

An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 2: Q. 2.10 (page 60)

Use a computer to produce a table and graph, like those in this section, for the case where one Einstein solid contains $200$ oscillators, the other contains $100$ oscillators, and there are $100$ units of energy in total. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?

Short Answer

Expert verified

The most likely macrostate is when $67$ energy units out of the total energy units are in solid $A, q_{A} = 67$ and $q_{B} = 33$ , with a probability of $0.07315$ ; the least likely macrostate is when all energy units are in partition ${Bq}_{A} = 0$ or when $q_{B} = 100$ , with a probability of $2.6926 脳 10^{- 37}$ .

Step by step solution

Calculation for overall multiplicity

Probability is,

P (q_{A}) = \frac{惟_{A} 惟_{B}}{惟_{overall}}

Over all multiplicity is,

$惟_{overall} (N_{overall}, q_{overall}) = (\begin{matrix} q_{overall} + N_{overall} - 1 \\ q_{overall} \end{matrix})$

$= \frac{(q_{overall} + N_{overall} - 1)!}{q_{overall}! (N_{overall} - 1)!}$

Where,

$q_{overall} = q_{A} + q_{B} = 100$

$N_{overall} = N_{A} + N_{B} = 300$

Then,

$惟_{overall} = (\begin{matrix} 100 + 300 - 1 \\ 100 \end{matrix})$

$= \frac{(100 + 300 - 1)!}{100! (300 - 1)!}$

$= 1.681 脳 10^{96}$

Calculation for total multiplicity and probability

Total multiplicity is,

$惟_{total} = 惟_{A} 惟_{B}$

$N_{A} = 200$

$惟_{A} = (\begin{matrix} q_{A} + 199 \\ q_{A} \end{matrix})$

$= \frac{(q_{A} + 199)!}{q_{A}! (199!)}$

$惟_{B} = (\begin{matrix} q_{B} + N_{B} - 1 \\ q_{B} \end{matrix})$

For $N_{B} = 100$ , $q_{B} = 100 - q_{A}$

$惟_{B} = (\begin{matrix} 100 - q_{A} + 100 - 1 \\ 100 - q_{A} \end{matrix})$

$= \frac{(199 - q_{A})!}{(100 - q_{A})! (99)!}$

probability is,

$P (q_{A}) = \frac{惟_{A} 惟_{B}}{惟_{overall}}$

$= \frac{1}{1.681 脳 10^{96}} \frac{(q_{A} + 199)!}{q_{A}! (199!)} \frac{(199 - q_{A})!}{(100 - q_{A})! (99)!}$

$P (q_{A}) = \frac{1}{1.681 脳 10^{96}} \frac{(q_{A} + 199)!}{q_{A}! (199!)} \frac{(199 - q_{A})!}{(100 - q_{A})! (99)!}$

Python program for creation of graph

Graph for probability versus energy

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 魅影直播!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

魅影直播

Short Answer

Step by step solution

Calculation for overall multiplicity

Calculation for total multiplicity and probability

Python program for creation of graph

Graph for probability versus energy

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Mechanics and Materials

Oscillations

Physics of Motion

Electricity

Radiation

Study anywhere. Anytime. Across all devices.