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A piston is resting halfway into a cylinder containing gas in thermal equilibrium. The layer of molecules next to the closed end of the cylinder is suddenly flash-heated to a very high temperature. Which best describes what happens next?

  1. The high temperature molecules push out the piston until their energy is reduced enough that the system is in equilibrium.
  2. The molecules with the highest temperature bounce off their neighbors, losing energy to them, and so on until the system is at a new equilibrium with the piston moved out.
  3. The molecules with the highest temperature bounce off their neighbors, losing energy to them, and so on until the system is at a new equilibrium with the piston where it started.
  4. The high temperature molecules push out the piston until their energy is reduced enough that the system is in equilibrium, and then the piston gets sucked back in.

(b)

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Design a macroscopic simulation using reasonably common materials to represent one very high energy particle gradually transferring energy to a bunch of lower energy particles, and determine if you end up with some sort of equilibrium.

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Section summary

  • Disorder is far more likely than order, which can be seen statistically.
  • The entropy of a system in a given state (a macrostate) can be written as
    S = k ln W ,
    where k = 1.38 × 10 –23 J/K is Boltzmann's constant, and ln W is the natural logarithm of the number of microstates W corresponding to the given macrostate.

Conceptual questions

Explain why a building made of bricks has smaller entropy than the same bricks in a disorganized pile. Do this by considering the number of ways that each could be formed (the number of microstates in each macrostate).

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Problem exercises

Using [link] , verify the contention that if you toss 100 coins each second, you can expect to get 100 heads or 100 tails once in 2 × 10 22 size 12{2´"10" rSup { size 8{"22"} } } {} years; calculate the time to two-digit accuracy.

It should happen twice in every 1.27 × 10 30 s or once in every 6.35 × 10 29 s ( 6.35 × 10 29 s ) ( 1 h 3600 s ) ( 1 d 24 h ) ( 1 y 365.25 d ) = 2.0 × 10 22 y

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What percent of the time will you get something in the range from 60 heads and 40 tails through 40 heads and 60 tails when tossing 100 coins? The total number of microstates in that range is 1 . 22 × 10 30 size 12{1 "." "22"´"10" rSup { size 8{"30"} } } {} . (Consult [link] .)

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(a) If tossing 100 coins, how many ways (microstates) are there to get the three most likely macrostates of 49 heads and 51 tails, 50 heads and 50 tails, and 51 heads and 49 tails? (b) What percent of the total possibilities is this? (Consult [link] .)

(a) 3 . 0 × 10 29 size 12{3 "." 0 times "10" rSup { size 8{"29"} } } {}

(b) 24%

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(a) What is the change in entropy if you start with 100 coins in the 45 heads and 55 tails macrostate, toss them, and get 51 heads and 49 tails? (b) What if you get 75 heads and 25 tails? (c) How much more likely is 51 heads and 49 tails than 75 heads and 25 tails? (d) Does either outcome violate the second law of thermodynamics?

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(a) What is the change in entropy if you start with 10 coins in the 5 heads and 5 tails macrostate, toss them, and get 2 heads and 8 tails? (b) How much more likely is 5 heads and 5 tails than 2 heads and 8 tails? (Take the ratio of the number of microstates to find out.) (c) If you were betting on 2 heads and 8 tails would you accept odds of 252 to 45? Explain why or why not.

(a) - 2 . 38 × 10 23 J/K size 12{ +- 2 "." "38"´"10" rSup { size 8{ +- "23"} } " J/K"} {}

(b) 5.6 times more likely

(c) If you were betting on two heads and 8 tails, the odds of breaking even are 252 to 45, so on average you would break even. So, no, you wouldn't bet on odds of 252 to 45.

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10-coin toss
Macrostate Number of Microstates ( W )
Heads Tails
10 0 1
9 1 10
8 2 45
7 3 120
6 4 210
5 5 252
4 6 210
3 7 120
2 8 45
1 9 10
0 10 1
Total: 1024

(a) If you toss 10 coins, what percent of the time will you get the three most likely macrostates (6 heads and 4 tails, 5 heads and 5 tails, 4 heads and 6 tails)? (b) You can realistically toss 10 coins and count the number of heads and tails about twice a minute. At that rate, how long will it take on average to get either 10 heads and 0 tails or 0 heads and 10 tails?

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(a) Construct a table showing the macrostates and all of the individual microstates for tossing 6 coins. (Use [link] as a guide.) (b) How many macrostates are there? (c) What is the total number of microstates? (d) What percent chance is there of tossing 5 heads and 1 tail? (e) How much more likely are you to toss 3 heads and 3 tails than 5 heads and 1 tail? (Take the ratio of the number of microstates to find out.)

(b) 7

(c) 64

(d) 9.38%

(e) 3.33 times more likely (20 to 6)

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In an air conditioner, 12.65 MJ of heat transfer occurs from a cold environment in 1.00 h. (a) What mass of ice melting would involve the same heat transfer? (b) How many hours of operation would be equivalent to melting 900 kg of ice? (c) If ice costs 20 cents per kg, do you think the air conditioner could be operated more cheaply than by simply using ice? Describe in detail how you evaluate the relative costs.

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Questions & Answers

how can I read physics...am finding it difficult to understand...pls help
rerry Reply
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
Ju
hope that helps
Ju
I have a exam on 12 february
David Reply
what is velocity
Jiti
the speed of something in a given direction.
Ju
what is a magnitude in physics
Jiti Reply
Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
and he to multiply
Owofemi
stuff happens
Ju
God plays dice
Ju
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
Practice Key Terms 3

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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