15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

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 tempera…

Using the table, verify the contention that if you toss 100 coins each second, you can expect to get 100 heads or 100 tails once in $2 \times \text{10}^{\text{22}}$ years; calculate the time to two-digit accuracy.

(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 the table.)

(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) Construct a table showing the macrostates and all of the individual microstates for tossing 6 coins. (Use the table 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 mic…