 So, if quantum mechanics can tell us how to calculate the energies that a system is allowed to have, then the Boltzmann distribution can tell us the probability that each one of those states is occupied. The probability of state j with energy e sub j is given by the Boltzmann distribution where q is this term, the sum of all these e to the minus energies over kT's. And remember these terms that look like e to the minus energy over kT, those are called Boltzmann factors. The probability that a state is occupied is proportional to its Boltzmann factor and that proportionality constant is 1 over q, where q itself is the sum of all of the Boltzmann factors for all the energies the system can occupy. So, this quantity q actually turns out to be very important, not just as being the denominator in this expression, but also for other reasons as well. So, we'll spend a little bit of time talking about this quantity and we'll give it a name first of all. The name for this quantity q, the sum of all the Boltzmann factors, is a partition function. We call q the partition function, so that definition tells us the partition function is the sum of all the Boltzmann factors for a system. So, first we'll talk about what we can understand this partition function to mean, what does it mean if a partition function is large or small or has a particular value. So there's two ways to think about what the partition function is. One is given by this expression. It's the denominator in this expression for calculating the probability. The reason q has to be the sum of all the Boltzmann factors is if I take this Boltzmann factor divided by the sum of all of them to calculate the probability, that ensures that when I add up all the probabilities, if I add up the probability for state one and two and three and four, that sum in the numerator will cancel the sum in the denominator and the probabilities will sum to one. So in that sense, the partition function is just a normalization constant. And that's not terribly exciting or terribly interesting. It has the value it has because if it had any other value then the probabilities wouldn't sum up to one and we want the probabilities to sum to one. If you remember where this expression first came up, it was when we did Lagrange multipliers to constrain the probabilities to add to one. So that's a somewhat boring interpretation of the partition function. It's a normalization constant that guarantees the probabilities will sum to one. But there's a much more interesting way to interpret the partition function as well. And to see that, we'll go back to an example like the butane molecule that we've considered in the past. So remember, butane can have an anti or a gauche conformation and the gauche conformation, there's two of them, gauche plus and gauche minus that have the same energy as each other. So here are the different energies of the three different conformations of the butane molecule, two gauche conformations, one anti. When we treated this problem numerically, if we know the value of this difference in energy, if I tell you that difference in energy is 3.6 kilojoules per mole, and if I tell you we're interested in the temperature of 298 Kelvin, that's enough energy to plug into this expression to tell us. And also, if I say the zero of energy is down here, the anti state has zero energy, that's enough to calculate the partition function. We won't repeat that calculation, but what we found was the partition function. e to the minus e anti over kT, e to the minus e gauche over kT, e to the minus e gauche over kT. The sum of the Boltzmann factors for each of these three different states, numerically, since we want to understand what the value of the partition function means. Numerically, the partition function came out to be one when we add e to the zero and then some smaller number for e to the minus 3.6 kilojoules per mole over k times this temperature. So overall, the partition function came out to be 1.44 in that example. So what does that mean if the partition function is 1.44? Well, think about this way, if I ask you how many states are there that the butane molecule can occupy, you'll say three, an anti and two gauche states. And if I say, well, how many of the states can actually be occupied at 298 Kelvin? You say, well, certainly the ground state can be occupied. There's plenty of molecules in the ground state. The gauche states are partially occupied. These Boltzmann factors tell us exactly, if I were to ask, what's the probability of occupying the gauche plus state relative to the anti state? That ratio of this Boltzmann factor to that Boltzmann factor tells us these states are only 22% as occupied as the ground state. There's only 22% as many butane molecules in the gauche plus state as in the ground state. There's only 22% as many in the gauche minus state as in the ground state. So my question, how many of these states are occupied? You might say, well, the ground state is fully occupied. These two gauche states are only 22% as occupied as the ground state. So the total number of occupied states is 1 plus 22% plus 22%. So in that sense, what this 1.44 is telling us is the number of accessible states. The number of states that can be reached at 298 Kelvin is about one and a half, about 1.44. So not all three states are equally accessible. The upper states are less accessible than the ground state. And the number of accessible states is something like 1.44. So in that sense, we can often interpret the partition function as not the total number of states in the system, but the number of states that can be reached at this temperature that we're interested in, the number of accessible states in this sense. So that's a much more valuable way of understanding what the partition function means. We can think of it as the number of accessible states. There's a couple of caveats to be aware of with that interpretation, however. Number one, that interpretation is really only a qualitative way of understanding the system. If I say 1.4 of the three states are accessible, then that statement doesn't actually make a tremendous amount of quantitative sense. What does it mean if only 1.4 states are accessible? We can understand it in the way I've just described. But when I'm counting number of states, it doesn't really make much sense to talk about a fractional number of states, kind of like the average number of children in a family is 2.5. Then either you have two children or you have three children talking about an average number of children that's a decimal or an average number of states that the decimal is not entirely sensible. So it does make a lot more sense, however, to think of this as a qualitative description. For example, if I were to repeat this example, recalculate these probabilities as the temperature drops to zero Kelvin, where the ground state is fully occupied, the population is in the upper states near zero, then as I go to zero Kelvin, the value of the partition function, as this temperature goes to zero, this term is going to be 1, these two terms are going to drop to near zero, so the partition function is going to drop in towards 1. So that means as the temperature gets colder, fewer and fewer states get accessible, only the ground state becomes accessible. That's a meaningful thing to think about. On the other hand, if I increase the temperature, as the temperature increases towards infinity, so if I plug in large values for T in this expression, each of these terms is going to be close to 1 because it's either the very small negative number, so the partition function is going to approach 3. So at high temperatures, I'm verging towards all three of these states being fully or equally accessible. If I don't just think about butane, but if I think about other systems like we'll consider in the future, here I've got a partition function that ranges somewhere between 1 and 3 depending on the temperature, but we'll see systems pretty soon where the partition function is as big as 10 to the 10th or 10 to the 30th. So what that means qualitatively is there's an enormous number of states accessible to the system, whatever the energy levels are, it's not just the first couple that are accessible, but there's 10 to the 10th states on that energy ladder that are accessible to the system at the temperature we're interested in. So at least in a qualitative sense, if we're able to say, well, only one state is accessible, a few states are accessible, or an enormous number of states are accessible to the system, that's what I mean when I say interpret the partition function qualitatively. The other important caveat to remember is that this explanation of what the partition function means relied on the fact that the ground state, when I plug in an energy of zero in here, I get one for the ground state. That's only going to be true if the ground state has an energy of zero. So if the lowest of all of these ground states, the lowest of all the energy levels, has an energy of zero, if that minimum energy is zero, then everything I've said is true. If not, if I say that the ground state has an energy of 500 kJ per mole, and so the excited state has an energy of 503.6 kJ per mole, then the partition function wouldn't be 1.44, it would be very different. So the good news, however, is we can recover this description, the partition function, which is the sum of all these Boltzmann factors. If I have a ground state which is not zero, if the ground state of the system has an energy of 500 in whatever units, I can choose to redefine the energy however I want. And one way to do that would be to say, let's recalculate, so that's the ordinary partition function. If I instead calculate e to the minus energy where I've subtract from each one of these energies the energy of the lowest state, so subtract 500 from each of my energies, then that will make the new difference be zero for the ground state and some higher number for the states above. So I've rewritten the sum with a different zero of energy, and if I factor out that e to the minus e min, so that's going to look like e to the e min, e to the minus ei over kT, and I can rewrite that as 1 over e to the minus e min outside, I should write a little higher. So if I rewrite this expression as 1 over e to the minus e min, multiplying by e to the minus energy over kT, those e sub i's have to be inside the sum, then this piece here is exactly like my original partition function. So rewriting this bit as q tells me that in order to recover this interpretation that the sum counts up the number of accessible states, I may or may not need to throw in an extra factor of 1 over e to the minus energy of the ground state, multiplying the original q, so it's really this quantity, 1 over e to the minus, sorry that's math error, so I've taken 1 over twice, e to the plus e min is 1 over e to the plus e min, so I can do it this way, 1 over e to the positive e min, or I could just get rid of the 1 over and say e to the minus e min times q corresponds to the number of accessible states. For the particular case like we've seen so far where the ground state has an energy of 0, then this leading term e to the 0 is just 1 and we don't have to worry about it. So if the ground state is 0, then q, the partition function, gives us the number of accessible states. If the ground state is something other than 0, then if I just prefix q with this correction factor, then that's still a way of counting the number of accessible states in the system. So this tells us perhaps a little bit about how to interpret the value of q. The next step will be to see what we can do with the partition function itself.