Thermodynamics: Entropy

From Warmup I've heard people say that the Entropy Statement of the Second Law of Thermodynamics disproves God. Why is this? What are your thoughts? The second law of thermodynamics is a statement about information. That is, information about the microscopic state of the universe is always being lost and can never be recovered. Reconciling this law with the idea of an omniscient God requires some work. For my thoughts are not your thoughts, neither are your ways my ways, saith the LORD. For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts. (Isaiah 55:8-9) Do you think that Outer Darkness is in this state and that approaching heat death is the extinguishing of their spirits? You could ask someone who knows more about it than me Ask, and it shall be given you; seek, and ye shall find; knock, and it shall be opened unto you: (Matthew 7:7)

From Warmup In order for heat death to occur, everything needs to be in a state of uniform temperature and density (according to the book). Through gravity, the center of the earth is at a higher density and temperature than the outer layers of the earth. Does that mean that in order for heat death to occur, everything needs to be perfectly balanced so that there is no "center" in terms of gravity? Yes, that is what that means. More generally, gravity has some interesting implications for heat death (beyond the scope of this course.) Density gradients can be used to do useful work with gravity just like temperature gradients can in thermodynamics. How likely and unlikely something else is kind of hard to determine without some kind of control. Likely and unlikely has to be compared to something else. What do we compare it to? We compare the relative likelihood compared to other states, e.g., we could observe these 5 things, which of the five is the most likely? We calculate the absolute probability of how likely something is to occur in a given time and space interval. For example, how many times during the lifetime of the universe will be find all the air molecules in the left half of this room? Answer on Wed!

From Warmup I don't really understand the math involved here. It's been a while since I've done probability. Is there a review of the math needed for the section somewhere? We ll review basic combinatorics today I thought it was interesting how long it took them to say what entropy is when it isn't that hard of a concept. (it was like 3 pages!) What has more entropy, a metal or a crystal? (Most) Metals have a crystal structure so What is the unit of entropy? Same units as Boltzmann s constant Joules/Kelvin in SI units.

Up to now We have introduced many concepts: State Variables Energy, Temperature, Pressure, Volume, Transfer Variables Heat, Work, Processes Reversible Irreversible Forbidden processes These are combined with a hodgepodge of equations. Can we unify them into a single theory? The next three lectures are heavy on math and theory (light on demos)

Entropy Some processes are reversible, others are irreversible, and still others are forbidden. Speculate: there must be another variable, S DS = 0 for reversible processes DS > 0 for irreversible processes DS < 0 for forbidden processes That is, S never gets smaller, it only grows. Is S a state variable or a transfer variable? Can we say anything else about S?

Consider Heat Flow Q T1 T2 S Q T1 T2 Q T1 Q T2 Since heat flows from hot to cold: S increases when T1 > T2 S decreases when T1 < T2 Therefore: S vs Q is flat when T1 = T2.

What is temperature? Q T1 T2 Temperature is something that is equal when two objects are in thermal equilibrium. Not quite.. Actually: Temperature is a measure of how likely something is to give away thermal energy.

What is temperature? Why not define temperature as a measure of microscopic kinetic energy? This requires we know atomic and kinetic theory. The definition would be different for different materials/objects. Historically, kinetic theory was not universally accepted when thermodynamics was developed. Thermodynamics, as a theory, does not require kinetic theory to be true. Thermodynamics is applicable to other systems without knowing the microscopic dynamics, e.g. black holes

What is entropy? Entropy helps us relate the macroscopic world to the microscopic world. Macrostate: a state defined on the macroscopic level (e.g. Pressure, Volume, Energy) Microstate: a state defined on the microscopic level (e.g. position, momentum of all particles in a gas) Entropy quantifies how many microstates are consistent with a macrostate

From Warmup 8 random choices on an 8x8 grid: Rank these 5 outcomes in order of likelihood These are each microstates. They have the same probability. All are equally probable, because no matter where certain spaces are occupied, it will be the same probability to get each space to be a certain way.

Microstates vs Macrostates You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing! --Richard Feynman

From Warmup Consider all of the gas particles in the room where you are sitting right now. Thinking about all the positions, speeds, and directions of the particles in the room, is this a likely microstate? or an unlikely one? I assume it is really likely, because it is currently happening. It is an unlikely microstate because it has only one possibility. Basically every state of gas particles in the room would be unlikely because there are so many possibilities.

Disordered This is an adjective for macrostates, not for microstates! Macrostates have a multiplicity, microstates don t. Macrostates don t fully specify the system. Multiplicity is the number of microstates consistent with the macrostate. Disordered macrostates have large multiplicity

Distinguishing Macrostates and Microstates You have a shoebox containing 35 pennies. The pennies are numbered 1 thru 35. You shake the box. Peering inside you find that the sequence is: HTTTHHHTTTHTHTTHHTHTTTHTTHHTTTTHTTT True (A) or False (B): The specific sequence is a microstate. Can you describe this macrostate?

More on Macrostates You have a shoebox containing 35 pennies. The pennies are numbered 1 thru 35. You shake the box. Peering inside, what are you most likely to see?

Combinations, Permutations, and entropy

Entropy increases during heat flow: Example

Entropy increases during heat flow: Example After lots of random exchanges.

Entropy increases during heat flow: Example Intuitively, which of these two cases is the most disordered?

Entropy increases during heat flow: Example How many arrangement are there having 19 cards of one color and 7 cards of the other in one half of the deck? How many arrangements are there having 13 cards of one color and 13 cards of the other in one half of the deck?

Entropy increases during heat flow: Example What is the relative probability of these two macrostates?