Question

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) 0 J/K, 100 J/K, 51 times more likely, No
b) 10 J/K, 100 J/K, 75 times more likely, No
c) 20 J/K, 150 J/K, 100 times more likely, Yes
d) 30 J/K, 200 J/K, 125 times more likely, Yes

Answer 1

In the context of a coin toss, entropy increases when moving towards a more probable macrostate, such as from 45 heads and 55 tails to 51 heads and 49 tails. The outcome of 51 heads to 49 tails is more likely than 75 heads to 25 tails. Neither outcome violates the second law of thermodynamics.

Step-by-step explanation:

Understanding Entropy in a Coin Toss

The concept of entropy is a measure of disorder or randomness in a system. In a coin toss, entropy correlates with the number of ways (microstates) a particular outcome (macrostate) can occur. The initial state of 45 heads and 55 tails or any other specific distribution has lower entropy compared to an equal distribution because there are fewer ways to arrange the coins in that specific pattern.

(a) Change in Entropy for 51 Heads and 49 Tails

To calculate the change in entropy, we would use the Boltzmann's entropy formula, S = k * ln(W), where k is the Boltzmann constant and W is the number of microstates. However, a specific change in entropy in joules per kelvin (J/K) for this scenario is not provided, so we'd need additional information to complete this calculation. For educational purposes, without specific values, we can say that going from a less likely macrostate to a more likely one (like from 45 heads and 55 tails to 51 heads and 49 tails) will result in an increase in entropy.

(b) Change in Entropy for 75 Heads and 25 Tails

This distribution is less likely and more ordered than the initial state; thus, the entropy would decrease if randomly tossing the coins resulted in this outcome.

(c) Relative Likelihood of Outcomes

51 heads and 49 tails is a more probable outcome than 75 heads and 25 tails because it is closer to the peak of the binomial distribution (which would be 50 heads and 50 tails for 100 coins). To find out how much more likely one is over the other, we would use combinations and calculate the ratio of their respective number of microstates.

(d) Second Law of Thermodynamics

Neither outcome violates the second law of thermodynamics, as this law applies to closed systems moving towards equilibrium over time. A single event, like a coin toss, does not represent a trend in the thermodynamic behavior of the system.