 So, we've spent a little time figuring out how to maximize the probabilities, maximize the entropy of a system when it's subject to energy constraints. And what we discovered was this set of equations, the probabilities look like this exponential expression and we're not quite done solving the problem because we didn't determine the value of this constant beta here. So the purpose now is to talk about the meaning of this constant beta to remind us of some terminology. Any time we see this exponential factor, e to the minus beta times an energy, that thing is called a Boltzmann factor. So e to the minus beta times energy is a Boltzmann factor. That Boltzmann factor also shows up in this thing we've called q. If I add up those Boltzmann factors for every possible state the system can be in, that is the thing we're calling q and if I take 1 over q multiplied by the Boltzmann factor for a particular state, the jth state, then that tells me the probability of being in the jth state after we've maximized the entropy. So only step left is to determine what beta is equal to. So one approach would be to solve that problem numerically. We can use the energy constraint to solve for the value of beta. Unfortunately that turns out to be difficult most of the time and it also is different for every individual problem so we can't do it generally for all cases all at once. So what we'll do instead first at least is to talk about the physical meaning of this property beta. Tell us something about what beta means even if we don't have a numerical value for it just yet. So as an example, we can consider something that we'll call a two state system. So in order to calculate these probabilities we need to know what the energies are. So let's say for simplicity there's a system with only two states. The system can either be in state A that has an energy E sub A or state B that has an energy E sub B. So those are the only two states for our two state system. For example this could be something like cyclohexane. So cyclohexane has two main confirmations it can exist in. It can exist in the chair confirmation which has a lower energy than the boat confirmation. So the boat confirmation will call state B and the chair confirmation we can call state A. Boat confirmation has a higher energy so that's this upper state with the energy E sub B. So we could be talking about cyclohexane if we want a concrete specific example but in general let's just say we have a system with two possible states spin up and spin down, boat and chair for cyclohexane or whatever. So we have an expression that tells us what the probability is of finding ourselves in state A or state B if we know the energies of state A and state B and if we know what this constant beta is. But if we think about this problem physically we can control how many of the molecules in a sample of cyclohexane are in the boat state or the chair state. And let's say at a given temperature we have a sample that might be let's say 60% chair and 40% boat. If we want to change that number, if we want to get more of the molecules up into the boat confirmation then in order to do that we have to take many of the molecules that are in the chair confirmation and lift them up into the boat confirmation. So somehow we've got to get more energy into the system in order to get more of the molecules into the B state. So in order to do that we supply energy to the system in some form. We might heat it up, we might increase its temperature. So as we increase the temperature imagine what happens for let's say a hot system. As we increase the temperature or even for let's say an extremely hot system as we increase the temperature without bound temperature becomes very, very large. What happens is the molecules have more than enough energy to make it up into that state B and they don't care anymore. The probability will be irrelevant whether they're in the A state or the B state. The difference in energy between those two states is negligible compared to the amount of the energy we supply to the molecules. So we would expect we ought to be able to get to the point where an equal number of molecules are in the boat or the chair state because at high temperature their confirmation is irrelevant. That energy difference isn't important to them anymore. So one way of controlling the probability is to adjust the temperature. Clearly from this equation another way to control the probability is to control beta. If I change the value of beta that's also going to change the probability. So it turns out there's going to be some connection between the temperature and beta. Under these hot conditions where the probabilities are equal for the two confirmations we can ask what that says about the value of beta. So let's go back to this expression for the probability. The probability that something is in the, let's put B in the numerator. So I'll say probability that something is in the state B is 1 over q e to the minus beta energy of the state B. So I can't solve for that without knowing the value of beta. But if I take that piece of B and if I divide it by the probability that I'm in the state A, probability that I'm in state A is 1 over q e to the minus beta e sub A. And the reason I've done that is because now the 1 over q on the top with the 1 over q on the bottom will cancel each other. So even though I haven't yet calculated the value of q because I don't know beta, those values are going to cancel. And what I have left is e to the minus beta eB from the numerator divided by another exponential. So I can subtract in the exponent. So it's e to the minus beta eB minus e to the minus negative beta eA. And the minus minus shows up like this. So if I prefer, I can write this as e to the minus beta times the difference in energy where e sub B minus e sub A is the difference in energy, the difference between the upper state and the lower state in our two state system. So what we've just discovered is the ratio of these two probabilities is also related to e to the minus beta times the difference in energy. What we're really interested in is solving for beta. So if I rearrange this equation a little bit, let's say take the log of both sides. So the log of PB over PA will be negative beta times epsilon delta E. So if I divide through by delta E with a negative sign, rearranging this equation gives this one. So beta is equal to, if I know the probability of being in the two states and if I know the difference in energy, I can use that to obtain beta. So that's exactly what we know about these conditions under very hot temperature conditions where we've supplied a lot of energy and there's equal number of molecules in each of these two states. Probabilities are equal to each other. So let's say under our very hot temperature conditions, when the two probabilities are equal to each other, this ratio is 1. The natural log of 1 is 0. So we find that beta equals 0 when temperature is very hot. So that's one data point we have in the connection I promised you between knowing something about beta and knowing something about the temperature. When we've made the system very, very hot near infinite temperature, we've essentially made beta very small, very near 0. As a different example, just to take the opposite extreme, imagine what happens if the system gets very cold. So the temperature is approaching 0, and absolute temperature 0 Kelvin. Imagine what you would expect for how many cyclohexane molecules are in the chair state or the boat state as I cool the system down towards 0 Kelvin. If I've cooled it down to 0, there's no energy in the system. So I've removed all the excess energy, any of the molecules that used to be up in the boat state have fallen down to the chair state, the lower energy state. So I expect that it should be true that when the temperature reaches 0, the probability of a molecule being in the lower state approaches 100%. Probability of their molecules in the upper state is approaching 0, and certainly a number much smaller than 100%. So the value of beta, so minus 1 over delta E log of P over Pb. Sorry, Pb over Pa. When Pb is a small number divided by 1, this number is small. The natural log of this small number, whatever it is, is very negative and large. So a large negative number times a negative sign, that's approaching a large positive and infinite number. So when the system is getting very cold, beta is getting very large. When the system is getting very hot with infinite temperature, beta is getting very small. So what we're noticing is physically the meaning of beta is inversely related to the temperature. It turns out that, like we've seen, when the temperature gets small, beta gets big and vice versa. So in fact, we can say beta is inversely proportional to the temperature with some proportionality constant. So if I rearrange that equation, switch the places of the k and the beta, I'm sorry, the temperature and the beta, the temperature is equal to 1 over beta multiplied by or divided in this case by some proportionality constant. And that will help explain what we mean by temperature as it relates to this property beta. So again, if the point of this exercise is just to illustrate physically what's going on when beta changes value, when beta has a large value, that's when the system has a low temperature and almost all of the molecules are in the ground state. When beta has a small value near zero, that's under conditions where the temperature is large and I have nearly equal amounts of the molecules in each of the two states. So physically, that's the meaning for beta. One other thing we can observe from this that we also could have noticed from the original equation is the units on beta. If I use the square brackets to indicate the units of this quantity beta, the units of beta are 1 over energy. That we could have observed just by looking at this expression. The exponent here has to be unitless because it's the exponent of an exponential. So e has units of energy, beta has to have units of 1 over energy in order to make this unitless. So we know beta has units of 1 over energy. We know t, of course, the temperature, has units of temperature. So what that tells us is something about this constant k. The units of k, if I need to multiply k by a temperature in order to, when I turn it upside down, get units of 1 over energy. That means the units of k are going to have to be energy over temperature. For example, if the units of k are energy over temperature, I multiply by temperature, that leaves me with just energy. 1 over energy matches what we expect for the units of beta. So for example, that could be joules per Kelvin. If we're talking about Si units, energy over temperature would be joules per Kelvin. It could be k cal per mole per degree Fahrenheit. It could be any units of energy divided by some units of temperature. But notice that these units of joules per Kelvin match what we know the units of Boltzmann's constant are. The numerical value for Boltzmann's constant we know is 1.38 times 10 to the minus 23 in units of joules per Kelvin. So this is explaining why the units of this Boltzmann's constant have to be in units of joules per Kelvin or energy per temperature. Essentially, what Boltzmann's constant is, is it's a conversion factor that lets us convert back and forth between the value of beta and the value of temperature using either one of these two equations. So it helps us convert back and forth between energy and temperature, which both means related things, but have different units. So we now understand at this point why Boltzmann's constant, this proportionality constant, has the units that it does. We don't yet understand why it has the exact value that it does. We'll see later why that exact value is what it is. So with this physical understanding now of what beta means as the inverse of temperature, a one over temperature, hopefully that helps this equation make a little bit more sense and we're ready now to move on and do some practical example calculations using this Boltzmann energy distribution.