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

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## Test prep for ap courses

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)

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.

## 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
$S=klnW,$
where $k=1.38×{10}^{–23}\phantom{\rule{0.25em}{0ex}}\text{J/K}$ is Boltzmann's constant, and $lnW$ 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).

## 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.

Determine the total force and the absolute pressure on the bottom of a swimming pool 28.0m by 8.5m whose uniform depth is 1 .8m.
for the answer to complete, the units need specified why
That's just how the AP grades. Otherwise, you could be talking about m/s when the answer requires m/s^2. They need to know what you are referring to.
Kyle
Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
how would I work this problem
Alexia
how can you have not an integer number of protons? If, on the other hand it supposed to be 1e12, then 1.6e-19C/proton • 1e12 protons=1.6e-7 C is the charge of the protons in the speck, so the difference between this and 5e-9C is made up by electrons
Igor
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Fidelis
Why does earth exert only a tiny downward pull?
hello
Islam
Why is light bright?
an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
thanks so much. i undersooth well
what is physics
is the study of matter in relation to energy
Kintu
physics can be defined as the natural science that deals with the study of motion through space,time along with its related concepts which are energy and force
Fidelis
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Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
did you solve?
Shii
1.75cm
Ridwan
my name is Abu m.konnek I am a student of a electrical engineer and I want you to help me
Abu
the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
Kings
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
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Godfred
Godfred
If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag