 All right, we have this thermodynamic connection formula, which will turn out to be very useful in the future that relates the internal energy to the partition function, if we already know what the partition function is. And just as a quick recap of where this came from, because it's going to be useful again in this video lecture, we made a connection between the thermodynamic energy, which we wrote as related to the microscopic energy, which is the average of all the individual energies multiplied by their Boltzmann factors, and then connected that. We did some work to find out that this summation can be written as a derivative of the partition function, and then that itself we could turn into this later expression. So here's the key result, but we can do this same process for other thermodynamic properties other than the energy. So for example, we can do that for the entropy. So if we remember the microscopic definition of the entropy as sum of probability, log probability for all the different states of the system multiplied by negative Boltzmann's constant. Again if we use Boltzmann to tell us what these probabilities are, this is going to look like the probabilities are 1 over q e to the minus energy over kT. Same thing is true for this probability as well as this probability. So I've got log of 1 over q e to the minus energy over kT. So it's log of that entire thing. So far so good. That's a little bit of a mess. We can clean it up a little bit. In particular the log of this quotient, log of the exponential divided by q, I can write that as leaving the first part alone. I've got negative log of q and positive log of the exponential, but of course log of an exponential is just the exponent. So I've got a minus eI over kT as the log of the exponential. So I've done two things here. I broke up this log of A over B as log of A minus log of B, and I've also written the log of the exponential as just the exponent. So if I now have two terms inside the brackets, so if I multiply each of those terms by this 1 over q e to the minus energy over kT inside the sum, so that gives me two different sums. I can cancel all these negative signs, so this negative sign cancels each of those negative signs, and I've got a positive k and a 1 over q and a sum of an exponential times log q. That's the first term, and I've got the same thing, positive k 1 over q exponential times energy over kT inside the sum, so energy over kT, e to the minus energy over kT. So I've got two separate sums. I've got the sum of these terms, and I've also separately got the sum of these terms. If I simplify each one of those a little bit further, this log q doesn't involve any i, so I can pull that outside the sum. So the first term is I've got 1k times 1 over q times log q times the sum of these Boltzmann factors, e to the minus energy over kT. And the second term, the only thing I can pull out is the 1 over kT. I've got a k outside and a 1 over k inside, so those cancel, and I have 1 over q, 1 over t, sum of energies times e to the minus energies over kT. So each of those terms now we can do something useful with. Notice this sum, this is just a sum of Boltzmann factors. By definition, the sum of the Boltzmann factors is q. This quantity is q. So that q inside the sum will cancel this entire sum, which is equal to q. We'll cancel the q in the denominator right here, so that simplifies that first term quite a bit. The second term, I don't have a sum of Boltzmann factors. I have a sum of energies times Boltzmann factors. But from the work we did in calculating the thermodynamic connection formula for the energy, we found that the sum of energies times Boltzmann factors, that is kT squared times d q dt. So I can rewrite it the same way here. And what I'm left with is after the q's cancel, I've got k log q for the first term. And in the second term, I've still got the 1 over q, 1 over t, and the sum of e energy times e to the minus energy over kT becomes kT squared dq dt. One last bit of cancellation. I have a t in the denominator, t squared in the numerator, so that one of those factors of t cancels. And also, again, just as before, we have a 1 over q d, q dt that we ended up rewriting as derivative of log q dt, derivative of log is 1 over. And then Chain Rule tells us that we also need the derivative of q. So this 1 over q dq dt, I'm going to take that portion of it, I'm going to rewrite it as d log q dt. So the final result is k log q. After the cancellation of t's, I've got a kT. And then 1 over q dq dt becomes derivative of log q with respect to t. So that's all the result that I've gotten for calculating what the entropy is. And now I've converted my microscopic definition of the entropy. Entropy in terms of knowing what each of the probabilities are for each of the individual states that I'm interested in. After all this simplification, I'm down to what we call a thermodynamic connection formula. All I need to know now is an expression for the partition function. If I have a closed form expression for the partition function, then I can just calculate the entropy directly from the partition function. So now we've seen not only a thermodynamic connection formula for the energy, but also one for entropy. And we have one more to calculate that we'll do in the next video lecture.