(II) (a) Suppose you have four coins, all with tails up. You...

(II) $(a)$ Suppose you have four coins, all with tails up. You now rearrange them so two heads and two tails are up. What was the change in entropy of the coins? (b) Suppose your system is the 100 coins of Table $1 ;$ what is the change in entropy of the coins if they are mixed randomly initially, 50 heads and 50 tails, and you arrange them so all 100 are heads? (c) Compare these entropy changes to ordinary thermodynamic entropy changes, such as Examples $6,7,$ and 8 from "Second Law of Thermodynamics"

(II) $(a)$ Suppose you have four coins, all with tails up. You now rearrange them so two heads and two tails are up. What was the change in entropy of the coins? (b) Suppose your system is the
100 coins of Table $1 ;$ what is the change in entropy of the coins if they are mixed randomly initially, 50 heads and 50 tails, and you arrange them so all 100 are heads? (c) Compare these entropy changes to ordinary thermodynamic entropy changes, such as Examples $6,7,$ and 8 from "Second Law of Thermodynamics"

Key Concepts

Entropy

Entropy is a measure of the number of possible microscopic configurations (microstates) that a system can have while maintaining its macroscopic properties. It quantifies the degree of randomness or disorder in a system, and changes in entropy reflect changes in the number of ways a system's state can be arranged.

Microstates and Multiplicity

Microstates refer to the individual configurations that a system can adopt. Multiplicity is the total number of these configurations available to the system. When coins are rearranged from one state (e.g., all tails) to another (mixed heads and tails or all heads), the change in multiplicity determines the change in entropy.

Boltzmann’s Entropy Formula

Boltzmann’s entropy formula, S = k log(W), where k is Boltzmann’s constant and W is the number of microstates, provides a quantitative method to calculate entropy. Changes in entropy can be determined by comparing the logarithms of the multiplicities of the initial and final states.

Second Law of Thermodynamics

The second law of thermodynamics states that in an isolated system, the total entropy tends to increase, reflecting the natural progression towards states with greater disorder (i.e., a higher number of microstates). This law underpins why spontaneous processes, including both simple rearrangements (as in coin problems) and more complex thermodynamic processes, tend to occur in the direction of increasing entropy.

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