I want to look in this lecture at differentials and state functions. So, let me define a state function. A state function is a property that depends only upon the state of the system. It does not depend on how the system got to that state. That is, it is independent of the path. And so, what defines a state, well we've seen in some instances for instance, the specification of particular variables. So we've worked with partition functions up 'til now for example that have specified number of particles, temperature and volume. And so a state function within that [UNKNOWN] would be something that depends indeed only on number of particles, temperature and volume. Not how I got there, not how I might have changed the temperature until I got to the current temperature. A key property of a state function is that its differential can be integrated as a mathematical quantity in a normal path independent way. And so, in particular, energy is a state function, internal energy. That is, I can think of integrating the differential of the energy from state one to state two, that will be equal to the difference between the energy of state two and state one, and I can write that as just delta u. So, the energy difference, and it's important to emphasize that in thermodynamics, we're almost always interested in differences in quantities. It's quite rare we calculate something absolutely, indeed, we often have to take conventions to define where zero is, and that makes quite clear that we're usually interested in differences. But, in any case we can integrate a differential for a state function to get a change in a simple and path independent way. Work we've already seen and we'll also see heat, these are not state functions. Instead they depend on the path that's taken. So work minus the integral from one to two. State one to state two of the external pressure dV. You know it depends on how we got from state one to state two and we exercised various examples of that in the last lecture. Looking in particular at the difference between reversible or taking off a single weight or taking off two weights as we change the external pressure. So since work depends upon how a process is carried out. Work is not a state function we call it a path function. And we use slightly different notations. So we will write an integral from state one to state two, not of d, little d the differential symbol in mathematics. But little delta, the small Greek character delta to emphasize that this is an inexact differential. And we don't say w2 minus w1 because there is no concept of a work at state two or a work at state one. We just say w. Alright, and it's going to have to have some specification of path in order to get a value for w. Energy is a state function. So just to illustrate the differences, then, on one slide. So here is the friendly differential of internal energy, and a normal integration. And here is what we would do with path functions. They're defined by q for heat, w for work, but we would not say delta q or delta w unless it's some special situation where they are not behaving generally when they're path functions. But maybe under special circumstances they could be state functions, we'll see that happen occasionally. And so here, then Is the first law of thermodynamics, which we expressed in words early on in this week, we said energy is conserved, it was a very simple way to express the first law. But here's the mathematical way, the manipulation way, that one would express the first law. In differential form, we would say that dU state function, so d is equal to delta q plus delta w. The sum of the heat and the work. If we were to write it as a, as a change equation as opposed to a differential equation, having integrated over differentials. Delta u is equal to q plus w. So that's a, a remarkable result in some sense. It says that I take the sum of two path-dependent quantities, two inexact differentials, and yet their sum, the internal energy, is an exact differential. It's really a, a very deep result. And so, in order to understand the relationship between heat and work and internal energy, we're actually going to make a comparison of some different paths, all leading to the same place. So, in each case, I want to go from an ideal gas at an initial pressure, I'll use one for my initial state. An initial pressure, volume, and temperature to a ideal gas at a different pressure and a different volume but the same temperature. So I actually will do this well the final temperature will be the same temperature. The path I take may be a little different. And so, path A which is shown here in sort of violet is going to be a reversible isothermal expansion. That is the temperature is staying the same at every point along this curve. This is the ideal gas PV curve for a constant temperature. I'll also consider a different path shown here in blue. Path B will be a reversible adiabatic expansion. I'll define what adiabatic means momentarily. And then, once I get to my final volume v2, I'll be at a lower pressure, I'm going to heat it until it gets to the desired higher temperature. And so that is a heating, sorry a higher pressure, a heating at constant volume. And the third path I want to consider is the path de, the red path. So, that is a constant pressure expansion from the initial volume to the final volume. Followed by cooling to get down to the original temperature. So that's a cooling at constant volume. Delta u must be the same for all paths. Why? Because u is a state function. All it cares about is where it is. It is defined here by a p, a v, a t. And, that's that. However, will q and w be the same along each of these possible sets of paths. Well, we'll take a look at that. Now before we leave I'd like to spend a moment here, I'll ask you to do a quick self assessment, a little bit of a review of some earlier material. But earlier material that's going to help us, as we explore these different paths. Alright, armed with everything we've learned up to this point, let's actually take a look at what goes on along these different paths. And so, we will look at the characteristic ideal gas expansion paths.