15.7 Statistical interpretation of entropy and the second law of  (Page 5/8)

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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).

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 $2×{\text{10}}^{\text{22}}$ years; calculate the time to two-digit accuracy.

It should happen twice in every $1.27×{\text{10}}^{\text{30}}\phantom{\rule{0.25em}{0ex}}\text{s}$ or once in every $6.35×{\text{10}}^{\text{29}}\phantom{\rule{0.25em}{0ex}}\text{s}$ $\begin{array}{ll}\left(6.35×{\text{10}}^{\text{29}}\phantom{\rule{0.25em}{0ex}}\text{s}\right)\left(\frac{\text{1 h}}{\text{3600 s}}\right)& \left(\frac{\text{1 d}}{\text{24 h}}\right)\left(\frac{\text{1 y}}{\text{365.25 d}}\right)\\ =& 2.0×{\text{10}}^{\text{22}}\phantom{\rule{0.25em}{0ex}}\text{y}\end{array}$

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 $1\text{.}\text{22}×{\text{10}}^{\text{30}}$ . (Consult [link] .)

(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) $3\text{.}0×{\text{10}}^{\text{29}}$

(b) 24%

(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?

(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) $-2\text{.}\text{38}×{\text{10}}^{–\text{23}}\phantom{\rule{0.25em}{0ex}}\text{J/K}$

(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.

10-coin toss
Macrostate Number of Microstates ( W )
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?

(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)

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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