In this lecture, we now want to take the somewhat abstract principles we developed in the previous lecture. About the relationship of probability, entropy, and spontaneity, into a more chemical context. We want to actually look at what the entropies are for individual molecules. We're going to measure those experimentally and then compare them against our equation. Which relates the entropy to the probability. In other words, I should be able to use this equation to understand why certain materials have higher entropies and others do not. We're going to take it as a given that it is possible to measure the entropy of a compound or a substance just directly experimentally. And not go through the details of how that measurement is made. It takes some extra thermodynamics that we have not gone through to develop those details. It's straight forward, but it takes more time then we want to do at this point. If we make a collection of measurements of entropies, we wind up with a table that looks something like this. This is the entropy measured for each of these substances and with different given phases. So, notice, for example, the first two entries compare the entropy of gaseous water per mole at zero, at 25 degrees centigrade. To the entropy of liquid water at 25 degrees centigrade, and in turn we can also compare the entropy of liquid water to solid water at zero degrees centigrade. It also allows us to make comparisons at different temperatures. Here for example, is the entropy of oxygen at 50 degrees. And here is the entropy of oxygen as 100 degrees. We want to begin to look for particular generalities in this table that might help us understand why things appear in the way that they do. So, for example, one of the observations that we would make immediately is the one we described at the outset. If we compare the entropy of gas to liquid, at the same temperature, or the entropy of liquid to solid at the same temperature. Here's another case with with Amonia. or right. Or with hydrazine down below. And what are the things that we discover in each circumstance? Is if we compare the entropy of the gas to the entropy of a liquid. That the entropy of the gas is always larger and typically actually quite a bit larger than the entropy of the liquid. And in turn the entropy of the liquid is greater than the entropy of the solid at the same temperature. Provided that we hold the temperature constant. That might make sense to us because if we think about the entropy as being the number of ways in which we can arrange the particles. The number of ways in which we can arrange gaseous particles is very, very much larger than the number of ways in which we can arrange those same particles in a liquid phase. Because the gaseous particles have a much greater volume to move around and are more free to move in that greater space. The liquid molecules are much more confined by a very large factor. So, the numbers of ways of arranging those molecules is quite a bit less. But, in turn of course, the molecules in a solid are fairly locked into place. They can't move around, except to wiggle slightly in their locations. And so, the number of ways of arranging the molecules in a solid is considerably smaller than the number of ways of arranging them in the gas. So right off the bat, we can make the measurement and understand that as a generality, the gaseous entropy is greater than the liquid entropy is greater than the solid entropy. And it seems to reflect our equation here that k logarithm w is a measure of the entropy. We can also look at this table again and make conparisons of substances at different temperatures. There is in fact a comparison here of the oxygen molecules at 25 degrees centigrade, 50 degrees centigrade and 100 degrees centigrade. And in doing that comparison we notice that as the temperature increases the entropy increases for the same substance. In this case, oxygen. That must mean that somehow or another, w is increasing. Well, even if we are confining that material to the same volume, which we perhaps are. There are still more ways to arrange the molecules in terms of their energetics. That is at higher temperatures, there's more energy available to be distributed amongst all of the molecules. Hence the number of kinetic energy states available to each of the molecules is greater. So w becomes larger as we increase the temperature. And that will be then our second conclusion. That the entropy increases with the temperature. It turns out also though it's harder to see in our data, that entropy also for a gas increases with the volume, that makes sense. The larger the volume we contain a mole of gas in, the more ways or places there are we can put those gas molecules. So the more ways there are to arrange them so the entropy should be larger. Go back and look at the data yet again. One of the things we observed is that molecules which have greater complexity in general have higher entropy than molecules that have less complexity. So for example, if we compare let's say carbon monoxi, carbon dioxide to carbon monoxide at the same temperature. And in the same phase the carbon dioxide has a greater entropy than does the carbon monoxide. That's an interesting result. And it arises from individual particles in the ways in which those particles themselves can arrange themselves. For example, if I compare them, the carbon monoxide to the carbon dioxide. The number of ways in which the carbon and oxygen can arrange themselves relative to each other is fairly limited because there's only two of them. The number of ways in which the carbons and the oxygens can arranges themselves becomes much greater because of the complexity of the molecule. Each of these bonds can stretch and in addition, they can bend about that central bond as well. That means that the entropy of a molecule that has more atoms is in general going to be greater than the entropy of a molecule which has fewer atoms. So that's our next generalization S increases with molecular complexity because then increases W. The last thing that we're going to look at actually has to do with observing that the largest entropy of any of the materials here is in fact C8H18. Which may have something to do with the complexity of the molecule, but may also be related to the mass of the molecule. To see that, let's actually compare calcium fluoride in the solid to calcium chloride in the solid, to calcium bromide in the solid. Notice, as we replace the fluorines with the chorines the entropy goes up. And as we replace the bromines with the chlorines with bromines, the entropy also goes up. We could similarly make a comparison of, for example, the entropy of iodine gas to bromine gas, as an example. And we would observe that the entropy again goes up. Our observation then, is that, with a larger mass of the molecules, the W must be larger because the S is larger. This turns out to be a somewhat more complicated thing to understand. At the same temperature, these molecules have the same kinetic energy, even as they have different masses. But with larger masses, there are molecules that have more momentum states available to them at the higher mass. And that means there are more ways of arranging their momentum vectors at the larger mass. And so in general, we can conclude that S increases with molecular mass. So, we can interpret this table of absolute entropies by understanding the complexity of the molecules or the freedom of those molecules to move around. Remember though, at the outset though, we began with the assumption that entropy was always going to increase. So, is delta-S viewed from this perspective of these data always going to be greater than zero. This turns out to be challenging given the generalities we have just discovered. Remember, the entropy of the gas is always greater than the entropy of the liquid. But we do know, under certain conditions, liquid will spontaneously condense, I'm sorry, gas will spontaneously condense into liquid. That means spontaneously, we're going from a higher entropy state to a lower entropy state, gas to liquid. So delta S is not always positive. Likewise we know under certain conditions that a liquid will freeze. That means that we're going from a higher entropy liquid state to a lower entropy solid state which means that the entropy is decreasing. So delta S is not always positive. Hence our first stab at the second law of thermodynamics that says that delta S always increases in the spontaneous process is now clearly not true once we have analyzed the entropies the individual materials. We've left something out in our analysis of the second law of thermodynamics. We've understood entropy but we haven't yet fully understood delta S and its relationship to spontaneous processes. We'll pick that up in the next lecture.
