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Q. 2.2

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An Introduction to Thermal Physics
Found in: Page 51
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 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)?

(a)The possible outcomes microstate are 1048576 microstates

(b) The probability of getting the sequence isP=9.536×10−7

(c)The probability of getting 12 heads and 8 tails are P=0.12

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step1:possible outcome of macrostate(part a)

(a)We suppose we flip 20 coins, n=20. The total number of microstates will be:

Calculation method

2n=220

=1048576microstate

So, the answer is 220 or 1048576 microstates.

Step2:The probability of sequence(part b)

(b)The likelihood of obtaining any specific sequence of heads and tails (for example HTHHTTTHTHHHTHHHHTHT) is

P= Favorable microstate Total number of microstates

=12n=1220

=11048576

=9.536×10−7

As a result, the answer is for the sequence of heads and tails. (for example HTHHTTTHTHHHTHHHHTHT) is 1220 or 9.536×10-7.

Step3:Probability of 12 heads and 8 tails(part c)

(c)Regardless of order, the probability of getting the macrostate of 12 heads and 8 tails is given by:

P=Nn Total number of microstates

where n denotes the number of heads and N denotes the number of coins, so:

P=20121048576

After arranging we get,

2012=20!12!(20−12)!=125970

Finally solution is

P=1259701048576=0.12

Most popular questions for Physics Textbooks

The natural logarithm function, ln, is defined so that eln⁡ x=xfor any positive number x.aSketch a graph of the natural logarithm function.b Prove the identitieslocalid="1650331641178" ln⁡ab=ln⁡a+ln⁡b and localid="1650331643409" ln⁡ab=bln⁡a(c) Prove that localid="1650331645612" ddxln⁡x=1x. (d) Derive the useful approximation

localid="1650331649052" ln⁡(1+x)≈x

which is valid when localid="1650331651790" |x|≪1. Use a calculator to check the accuracy of this approximation for localid="1650331654235" x=0.1 and localid="1650331656447" x=0.01.

For an Einstein solid with four oscillators and two units of energy, represent each possible microstate as a series of dots and vertical lines, as used in the text to prove equation 2.9.

Fill in the algebraic steps to derive the Sackur-Tetrode equation (2.49).

Consider an ideal monatomic gas that lives in a two-dimensional universe ("flatland"), occupying an area A instead 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.

For an Einstein solid with each of the following values of N and q , list all of the possible microstates, count them, and verify formula Ω(N,q)=q+N−1q=(q+N−1)!q!(N−1)!

(a) N=3, q=4

(b) N=3, q=5

(c) N=3, q=6

(d) N=4, q=2

(e) N=4, q=3

(f) N=1, q= anything

(g) N= anything, q=1
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