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Q. 2.16

An Introduction to Thermal Physics
Found in: Page 63
An Introduction to Thermal Physics

An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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Short Answer

Suppose you flip 1000 coins.
a What is the probability of getting exactly 500heads and 500 tails? (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 , or 1000, until Stirling's approximation becomes necessary.) b What is the probability of getting exactly 600 heads and 400 tails?

Part a

aThe probability of getting exactly P(500)0.02523.

part b

bhe probability of getting exactly P(600)4.635×1011.

See the step by step solution

Step by Step Solution

Step: 1 Finding probability: (part a)

A larger coin tossing experiment is underway. N=1000 coins are flipped. Stirling's approach may be used to calculate the likelihood of receiving exactly 500 head and 500 tails. The total number of possible outcomes is as follows:


Where n=500 heads as


Using Stirling's approximation as


Step:2 Dervative: (part a)

From the above equation,


Probability approximately getting exactly role="math" localid="1650333432268" 500heads as


Substituting,we get


Step: 3 Derivative probability: (part b)

The number getting n=500 heads as


Using Stirling's approximation as

role="math" localid="1650333713655" N!NNeN2πNΩ(1000,600)=1000!600!400!10001000e10002π×1000600600e6002π×600400400e4002π×400Ω(1000,600)10001000×e1000×1000400400×600600×e1000×600×400×2π

Step: 4 Equating part: (part b)

From the above equation,


Probability approximately getting exactly 500 heads as


Substituting,we get


Step: 5 Finding probability value: (part b)

The ratio is not applicalable, so simplify as






Where as Maple is working large exponents directly.

Most popular questions for Physics Textbooks

A black hole is a region of space where gravity is so strong that nothing, not even light, can escape. Throwing something into a black hole is therefore an irreversible process, at least in the everyday sense of the word. In fact, it is irreversible in the thermodynamic sense as well: Adding mass to a black hole increases the black hole's entropy. It turns out that there's no way to tell (at least from outside) what kind of matter has gone into making a black hole. Therefore, the entropy of a black hole must be greater than the entropy of any conceivable type of matter that could have been used to create it. Knowing this, it's not hard to estimate the entropy of a black hole.
a Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c2, where G is Newton's gravitational constant and c is the speed of light. Calculate the approximate radius of a one-solar-mass black holeM=2 ×1030 kg .b In the spirit of Problem 2.36, explain why the entropy of a black hole, in fundamental units, should be of the order of the maximum number of particles that could have been used to make it.

cTo make a black hole out of the maximum possible number of particles, you should use particles with the lowest possible energy: long-wavelength photons (or other massless particles). But the wavelength can't be any longer than the size of the black hole. By setting the total energy of the photons equal to Mc2 , estimate the maximum number of photons that could be used to make a black hole of mass M. Aside from a factor of 8π2, your result should agree with the exact formula for the entropy of a black hole, obtained* through a much more difficult calculation:

Sb.h. =8π2GM2hck

d Calculate the entropy of a one-solar-mass black hole, and comment on the result.


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