 Okay, so we've seen this expression that the internal energy, a change, a differential change in the internal energy can be represented as some portion that we refer to as heat and some portion that we refer to as work. So this is an important enough relationship that we give it a name, important enough in fact that we call it the first law of thermodynamics. So we can write that first law of thermodynamics both in this form as a differential expression. We can also use it as an integrated expression. So if I write, if I integrate both sides of that expression, integral of du is integral of dq and the integral of dw, those all look like fairly easy integrals to do. Integral of du is just u, evaluated from some initial to some final conditions. That's going to give us the difference in the internal energy between the initial and final conditions. So integrate du, I get delta u. I have to pay a little more attention when I integrate these inexact differentials. Remember an inexact differential is something that is not directly the differential of some original quantity. So we'll talk about that a little bit more, but the integral of dq I'll write as q and the integral of dw I'll write as w. So these are just the finite sized amount of heat and finite sized amount of work that combine to give us the change in the internal energy of the system. So in some sense the first law of thermodynamics, both of these, both the differential form. So both of these expressions can both be described as the first law of thermodynamics in a differential or an integrated form. And essentially what the first law of thermodynamics is is an energy, a statement of energy conservation. Any change in energy has to come from a form that we call heat or a form that we call work. There's no other types of energy other than heat and work. And there's no energy can be destroyed or created in the sense all the energy is reshuffled from one form to another. So let's talk a little bit about the fact that I didn't write a delta q or a delta w in front of heat and work because they're inexact differentials. And the reason for that is both heat and work rather than being inherent properties of a system like the energy is. I can say the system, this box of gas or this box of molecules has an energy. It has an energy of 100 joules. Its energy changes by 50 joules when I do something to it. I can't do the same about heat and work. Heat and work are very different types of quantities than the energy is. Heat and work are not inherent properties. I can't say the box of gas has some amount of heat or has some amount of work. In fact, those statements should sound a little bit ridiculous. Instead, both heat and work represent transfers of energy. So when I'm talking about heat, if I say I applied 100 joules of heat to the system or I transferred 100 joules of heat into the system, I'm talking about a transfer of energy from outside the system to inside the system or vice versa. Likewise, I can say I did some work on the system. It cost me 100 joules of energy to compress the system. So I did 100 joules worth of work on the system. But the system itself doesn't have any work. I don't have the work. I do work on the system. So that's an important thing to keep in mind. It'll save you definitely a few headaches if you're able to keep in mind as you learn thermodynamics that heat and work always represent transfers. I can't ever talk about the amount of heat or work the system has, only describes how much has been transferred into or out of a system. So the other thing we should talk about for this first law of thermodynamics is the conventions we've assumed for the signs of these quantities. So notice that the q and w both have positive signs in front of them. What that means is if q is a positive number, that's contributing to increasing the internal energy of the system. Likewise, when w is a positive number, it's contributing to increasing the energy of the system. So what that means is we'll talk specifically about the differences when q is a positive number or when q is a negative number. Let's take a system like, so here's our box of gas. Gas molecule is inside a box. If I want to talk about increasing the energy of the system using heat, then of course what I want to do in my bad artistic rendition, here's a flame that I'll use to apply some heat to the system. So in this sense, I am transferring heat into the system. That will increase the energy of the system. So when q is a positive number, that represents a transfer of heat into the system. And a negative q would be exactly the opposite. It's a transfer of heat out of the system. So negative heat would be the case if I didn't heat the system up with a Bunsen burner or a candle. If I took that system and placed it into the refrigerator so that heat was withdrawn from the system and the temperature dropped, that would be a negative value of the q. So q is positive or negative. Positive means transfer in. Negative means transfer out. The way to remember that is positive q results in an increase in the temperature, not the temperature. The internal energy of the system, similarly for work. So in my cartoon, in this case, if I want to do some work on the system, if I take the system and decrease its volume, which has an effect on the energy levels of the system, and therefore increases the energy of the system, if I want to increase the energy of the system, I want to compress it, make it smaller. That costs me work. I'm putting that work, that energy, into the system. So let's say, in this case, if I compress the box down to a smaller size, then I'm doing some work in that case. If I do the work on the system, that's a positive value for w, because I've used that to increase the energy of the system. On the other hand, if the work is a negative number, then the system is doing work on me. If I have a box of gas that's compressed at a high pressure and I release the pressure by pushing less hard, then the system is doing work in pushing back my hand, or the system is pushing back the lid of the container, or whatever weight I put on the lid of the container. So whenever the system does work on the surroundings, when the work is done by the system, that represents a negative value of w. And that's because in order to do that work, it costs the system some energy. The energy of the system is reduced. The energy that it takes to do that work is subtracted from the energy of the system. So these sign conventions for what we call a positive heat or a positive work are important. And the reason it's worth spending some time on them is different disciplines have different sign conventions. This is the convention that most physical chemists, most chemists, most physicists prefer. Engineers, and I won't dwell on this too much or write any equations to avoid confusion, but perhaps to prevent some confusion if you run across equations in textbooks or websites that seem to have the signs wrong, engineers intentionally use the opposite sign convention for work. And that's mainly because to an engineer, someone who's in the business of constructing a system or an engine or a machine, usually you're focused in that sort of system, in that sort of situation is to construct a system that's good at doing work. Engineers like to make machines that do work on their surroundings and therefore do useful things. So in that sense, it's positive, it's a good thing, and work is a positive number whenever the system is doing work. So engineers are more focused on work done by the system. Chemists and physicists and physical chemists are more focused on the properties of the system themselves, so when the energy change is positive, that's because either Q is positive or W is positive or both. And we like positive work and positive heat when we've put energy into the system and use them to increase the energy of the system. So that's a brief summary of what we mean by the first law and the sign conventions to keep in mind. The next step will be to focus a little more closely on work and talk about different types of work that we can do on systems.