Reading assignment: 22.8, especially the marble example but not the “Adiabatic Free Expansion: One Last Time” example. Also: “What is entropy?” handout posted to website, through Example 1.
Using ideas from both the reading and from the last lecture, explain why
heat flows from hot to cold when the process of energy exchange
between two objects is "random". (How can you get directed motion of
heat, when energy is being exchanged both ways?!)
The heat flow is random, but there is more energy on one side (the hot side) than the other. So even though the flow is random, there is a higher chance that energy on the hot side (lots of energy) will randomly move to the cold side than there is that energy on the cold side will flow to the hot side. In fact, both things are always happening but flow from the hot side to the cold side is happening more often (simply because there is more energy on that side that can flow).
We learned last lecture that entropy always tends to increase. However, when two systems A and B can exchange energy, the entropy of system A
always decreases when system A gives energy to system B. If that's
so, why would energy ever spontaneously flow from system A to system
B? (It often will. When? Why?)
The key concept is that energy flows to maximize the *combined* entropy, not the entropy of one or the other. If the entropy of B increases more than the entropy of A decreases, then the process will occur spontaneously.
In the "What is entropy?" handout, what was significant about the equation
dS1/dE1 = dS2/dE2?
That equation allows one to identify a quantity which is the same for two different systems in thermal equilibrium, just like temperature is. It makes the connection between "regular" temperature and the far more esoteric definition of entropy in terms of microstates.