Problem

(a) What is the change in entropy if you start wi…

Problem 60 Hard Difficulty

(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?

Answer

a) $\Delta S_{1}=6.635 \cdot 10^{-24} \mathrm{J} / \mathrm{K}$
b) $\Delta S_{2}=-1.2879 \cdot 10^{-24} \mathrm{J} / \mathrm{K}$
c) $\frac{P_{2}}{P_{3}}=4.12$

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Physics 101 Mechanics

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Chapter 15

Thermodynamics

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Video Transcript

his problem 60 Chapter 15 on were asked, What is the change in entropy? If you start with 100 coins in the 45 heads and 55 taels micro state awesome and get 51 hoods and 49 tails and then part B asks, What if you get 75 heads and 25 taels and then C s How much more likely is 51 heads and 49 tails in 75 heads and 25 taels? And then for finally for party Does either outcome violate the second law of thermodynamics on DSO For part A. We can go ahead and use this formula to calculate the change in entropy. And so if you reference the table 15.4, we can substitute 9.9 times 10 to the 20th power 28 uh, for W f and 6.1 times 10 to the 28 for W I. And then finally 1.38 times 10 to the negative. 23rd jewels for Calvin for K, which is development constant. Uh, so now we can go ahead and use this formula here and we will go ahead and do that so there's gonna be Delta s is equal to 1.38 times 10 to the negative, 23rd times the natural log of 9.9 times 10 to the 28 all over. 6.1 times 10 to the 20. Um, and this is gonna be equal to 6.7 times 10 to the negative. 24th jewels for Kelly. Uh, and so this is our answer for part A. This is the change in entropy for that. And so now we can go ahead and use the same formula for part B, so Delta s will be equal to K L N and be you f over w all right. And so, no, we can go ahead and substitute our other values. Using the, uh, table and 15.4 Delta s is going to be equal to Bozeman's Constant, which is 1.38 times 10 to the negative. 23rd multiplied by the natural log of 24 or 2.4 times 10 to the 23rd over six point Warne times 10 to the 28 on. This is gonna come out to your negative 1.7 times tend to the negative 20 two jewels for come and then in part, see, uh, were asked how much more likely is 51 heads and 49 tails and then 75 heads and 25 test. So to do that, we need to actually look at our ratio of micro states. And so since the number of micro states and 51 heads and 49 tails is 9.9 times 10 to the 28th and the number of micro states in 75 heads and 25 taels is 2.4 times 10 to the 23rd there's a ratio, and so are our is gonna be equal to 9.9 times 10 to the 28 over 2.4 times 10 to the 23rd. Oh, and so the first date is actually gonna be equal are gonna be 4.1 times 10 to the fifth times more likely than the second state. And then finally for party uh, we were asked, does either outcome violate the second law of thermal dynamics and says, um, and this is actually gonna be no. And tribute values decrease are possible, but unlikely also in the same case. Delta s is a small negative number which indicates that a spontaneous fluctuation in our system will decrease. All right, decreases the entropy, but it's not highly likely.

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