Chapter 21, Problem 19SP

When 100 coins are tossed, there is one way in which all can come up heads. There are 100 ways in which only one tail comes up. There are about $1\times {10}^{29}$ ways that 50 heads can come up. One hundred coins are placed in a box with only one head up. They are shaken and then there are 50 heads up. What was the change in entropy of the coins caused by the shaking?

To determine

The change in entropy of coins when 100 coins placed in the box with only one head up is shaken and then there are $50$ coins with heads up.

Answer to Problem 19SP

Solution:

$8.6\times {10}^{-22}\text{ J/K}$

Explanation of Solution

Given data:

There is one way in which all heads come up if 100 coins are tossed.

There are100 ways in which only one tail comes up if 100 coins are tossed.

There are about $1\times {10}^{29}$ ways in which 50 heads come up if 100 coins are tossed.

A total of 100 coins are placed in a box with one head up.

The coins are shaken until there are $50$ heads up.

Formula used:

Entropy is calculated by the formula:

$S=k_B\ln \Omega$

Here, $\Omega$ is the number of ways in which the system can achieve the state, $S$ represents the entropy, and $k_B$ is the Boltzmann constant.

The number of ways in which a number of things can be arranged is calculated by the formula:

$P{}_r=\frac{n!}{\left(n-r\right)!}$

Here, $n$ is the total number of objects, $r$ is the number of places in which $n$ objects have to be arranged, and $P{}_r$ is the number of ways in which $n$ objects can be arranged at $r$ distinct places.

Explanation:

Recall the expression for calculating the number of ways in which the 100coins can be arranged with one head up:

$P{}_r=\frac{n!}{\left(n-r\right)!}$

Understand that thenumber of ways is to be calculated for the case in which one coin with its head facing up can be arranged at100 different places.

Substitute $100$ for $n$ and $1$ for $r$

$\begin{array}{c}P{}_1=\frac{100!}{\left(100-1\right)!}\\ =\frac{100!}{99!}\\ =100\end{array}$

Understand that the number of ways in which 100 coins can be arranged with one coin withits head facing up is equal to the number of ways in which one coin with its head facing up can be arranged at100 different places. Therefore,

${\Omega }_1=P{}_1$

Substitute $100$ for $P{}_1$

${\Omega }_1=100$

Here, ${\Omega }_1$ is the number of ways in which 100 coins can be arranged with one coin facing headup.

Recall the expression for entropy when 100 coins are arranged with one head up:

$S_1=k_B\ln {\Omega }_1$

Here, $S_1$ is the entropy when 100 coins are arranged with one head up.

Consider the standard value of Boltzmann constant is $1.38\times {10}^{-23}\text{ J/K}$.

Substitute $1.38\times {10}^{-23}\text{ J/K}$ for $k_B$ and $100$ for ${\Omega }_1$

$\begin{array}{c}{S}_1=\left(1.38\times {10}^{-23}\text{ J/K}\right)\ln 100\\ =6.36\times {10}^{-23}\text{ J/K}\end{array}$

Recall the expression for entropy when 100 coins are arranged with $50$ heads up:

$S_2=k_B\ln {\Omega }_2$

Here, $S_2$ is the entropy when 100 coins are arranged with $50$ heads up and ${\Omega }_2$ is the number of ways in which 100 coins can be arranged with $50$ coins facing headup.

According to the problem, 100 coins can be arranged with $50$ coins having their heads up in $1\times {10}^{29}$ ways.

Substitute $1.38\times {10}^{-23}\text{ J/K}$ for $k_B$ and $1\times {10}^{29}$ for ${\Omega }_1$

$\begin{array}{c}{S}_2=\left(1.38\times {10}^{-23}\text{ J/K}\right)\left[\ln \left(1\times {10}^{29}\right)\right]\\ =92.1\times {10}^{-23}\text{ J/K }\end{array}$

Calculate the change in entropy when the box is shaken till there are $50$ coins with heads up:

$\Delta S=S_2-S_1$

Here, $\Delta S$ is the change in entropy during the process.

Substitute $92.1\times {10}^{-23}\text{ J/K }$ for $S_2$ and $6.36\times {10}^{-23}\text{ J/K}$ for $S_1$

$\begin{array}{c}\Delta S=\left(92.1\times {10}^{-23}\text{ J/K }\right)-\left(6.36\times {10}^{-23}\text{ J/K}\right)\\ \simeq 8.6\times {10}^{-22}\text{ J/K}\end{array}$

Conclusion:

The change in entropy is $8.6\times {10}^{-22}\text{ J/K}$.