Okay. So, when do you think the ideal gas model is going to fail? Well, again, I asked you to visualize in your head the state diagram, so let's take a look at that. Here's my state diagram and, again, we know the super-heat region is out here in the high temperature, low density region. What we see here are the high specific volume engine if you prefer. So, if this is where the ideal gas law is valid, the closer I get to phase change is when I'm going to see breakdowns in the appropriateness of using the ideal gas model. So, you would say, "Hey, any time I get close to the dome or any time I look like I'm going to have a chance for those molecules to interact with each other, because gases, the molecules are far apart and they don't sense each other." So, that's the fundamental basis for the ideal gas model. Anything that causes those molecules to come close together is going to cause a breakdown on the ideal gas approximation. So, that's going to be essentially really high pressures can do that, or alternatively, really low temperatures, anything that would lead to phase change. So, that's going to tell you when you should really check your compressability factors. Okay. So, we're getting armed and dangerous, we know we have the ideal gas model which told us a relationship between the pressure, the specific volume, the specific gas constant, and the temperature, and we know that an approximation of the model is that the internal energy is a function of temperature only. So, just like with the incompressible substance model, we can now go back and look at the definition for the internal energy, and we'll skip a few steps since we went through this in the last segment. We know that that function of temperature only allows me to write this expression where we have a simple derivative, giving us the relationship between the specific heat and constant volume, and that's going to allow us to say "Hey look, anytime I have a change in internal energy between two states, all I have to do is evaluate," and I'm going to be very explicit here, "how the specific heat varies as a function of temperature over that interval." Now, for solids and liquids, the specific heat doesn't vary very much under those conditions. Generally, it's a pretty good approximation to say the specific heat is constant for most of the states that you'd be considering. You always what to check that. Ideal gases, not so much. The ideal gases, we know that gases have a very strong relationship between the specific heats and temperature, so I am writing that the specific heat at constant volume is a strong function of temperature, and generally, we write them as a polynomial expression. So we would say, let me go ahead and write the internal energy because for reasons that will become obvious in later segments, or excuse me, the enthalpy. We like to use enthalpy more often than internal energy. So, this is our crutch to get us, the internal energy is our crutch to get us here. So again, the enthalpy we now again is described by this expression, again, since it's a function of temperature only, here's kind of an elegant thing. Remember that since the internal energy's a function of temperature only and we have this model, we take these two pieces of information together and what it tells us is that "Hey," again, by definition, enthalpy is given by this expression, but look at that, we know that the product of the pressure times a specific volume is defined by, in the ideal gas model, the product of the specific gas constant and the temperature. If this is a function of temperature only, that's a constant. So, this portion of the expression is a function of temperature only, and that means that the enthalpy for an ideal gas is a function of temperature only, too. That allows me to make this integral statement here, okay. So, we have these two integral expressions, what we typically would do is we would look up the specific heat at constant pressure is actually listed as a polynomial expression as a function of temperature. It's a pretty complex expression and that there are many components to it, and these are just fit coefficients. So, in other words, people took the data from those phase diagrams and from those tables, and they did a polynomial fit between the specific heat and temperatures, and you can take those expressions and you can plug them into this integral and you can evaluate them and they're pretty straightforward integral to evaluate. So, again, that's something that I think is pretty easy to do, nine times out ten, we're going to just look up the information either in a table or using an online calculator. Okay, but the summary of this, the moral of this story is that the ideal gas approximation, the specific heats are still unique and they are a function of temperature. By unique, I mean the CP is not equal to CV like we had for the incompressible substance model. I also want you to remember the ideal gas model is only valid for ideal gases. So these expressions here are only valid for ideal gases. You never want to apply them anywhere near the saturation region or in the sub-cooled liquid region. Okay. So, we saw this little, again, I shouldn't have used that by definition, [inaudible] here, so I'm going to take that top bar off, because this is the simplification. This expression is the simplification of the definition. Okay, so I took that off. So let's take a look at that, just want to show you one more relationship state relation. So again, we have for the ideal gas, this is true. Okay, remember, ideal gases only, okay, don't go applying that willy nilly. So, if we took the derivative of this expression with respect to temperature, we would have this. We would apply the chain rule, remember that from your math days. We'd say "Hey, these specific constant is a constant so that part of the chain rule drops out." So, all we're left with is this, and dT, dT is simply one. You look at these expressions and say "Hey, that's the simplified definition of a specific heat at constant pressure, as a function of temperature. That is a specific heat at constant volume, also as a function of temperature, plus R. So, this relationship between the specific heats and the specific gas constant, again, is unique to the ideal gases. So, I can tabulate the specific heats of constant pressure and from that, I can get the information that I need for the specific heat at constant volume. Okay. So again, engineers are all about efficiency and generally being lazy. So what we, not really, I don't mean it. But I mean to say is that, we like to tabulate information, we don't want to be redundant, we don't want to have a lot of redundant tables so we tabulate the saturated vapor and the saturated liquid data but we don't give you all the information as a function of quality, you can calculate that. We're not going to tabulate the specific heats of constant volume because you can get that information from the tables of specific heat of constant pressure. But remember, only valid for ideal gases. So now let's come back and we snuck in that introduction to polytropic processes in a few segments ago. Now we want to go back and look at that again. Because now, we're empowered with a lot more information. So let's take a look at those polytropic processes. We said by definition, a polytropic process is one where the pressure times the volume raised to a certain power is equal to a constant. Well, now we can say we have a whole slew of different processes that we can define using this expression. So if n will be equal to zero, then we know volume raised to the zero power is telling us, well that's just pressure is a constant. That's the definition of an isobar. If we say that n is approaching positive or negative infinity, we know that the pressure term becomes negligible and what you have instead is that the volume is approximately constant. And this defines a line of constant specific volume or constant volume. It's an asymmetric line. For the ideal gases, we took that Pv=RT expression, and we can say, hey look, if we have a system that's an ideal gas approximation is valid, then we know from state one to state two, this expression is true. So that tells us that P2, V2, T2 is equal to P1. And that's going to ultimately, if we take our information here and we use it with this, what we find is that, again pushing a lot of algebra along and we're not going to go through this or a number of different places you can see this derivation. So these are ideal gas expressions. And ultimately, what we're going to find is that that end is going to be equal to the ratio of specific heats. So again, polytropic processes has a number of those that are important in thermodynamics and there's just few of those are shown for you here. And we'll keep coming back to these so you'll get a chance to see them again and again. All right. We spent a lot of time looking at state relations and understanding states. So you can see in thermodynamics, being able to define a state and being able to move fluently with flexibility between state information like pressure and specific volume to temperature and enthalpy is critical. That's our fundamental toolbox for understanding thermodynamics. Our next most critical tool is the conservation property, the conservation principles. So, we're going to come back to the conservation of energy. We want to consider now control volumes. So previously, we had considered kind of a control mass. We introduced a very baby steps. So I'll be more rigorous as we go through the next few segments about the development and the conservation of energy. I want to start with seeding this information, planting a seed here where control volumes and control masses are related to each other. So if a control volume allows mass to cross the system boundaries, that's the way we defined it in our first unit, all you have to do to get the control mass equations is just stop mass transfer from occurring across the system boundaries. Okay? So the control mass is the limit of the control volume where there's no mass transfer across the boundaries. So having said that, we know that, again, we'll define some arbitrary volume. There's my arbitrary volume. Kind of looks like a stomach. So this is my control volume. And here's my system boundaries, I'm going to use that as a dash line. Okay, like that. And we have mass that can enter the system and mass that can leave the system and we're going to denote mass flow rate using again the over-dot symbol. So we have mass in and we have mass out, and we can say that in general, the mass with the subscript i is the instantaneous mass flow rate, m.in at the inlet and then m.out is the, get lazy here, instantaneous mass flow rate at the exit or outlet if you prefer. And I've shown only one entrance and one outlet. Of course you could have as many as you wanted. You could have a T-junction coming in or you could have a showerhead coming out. It doesn't matter. We just modify our system diagram and we add those outlets and you can index them one, two, three, four, five, however you want to do. So the conservation of mass, C, O, M, again that's conservation of mass for a control volume says that the mass within the control volume, so the mass within the control volume as a function of time has to be related to the mass flow rate of the stuff coming into the system for all possible inlets, so i equals one to all the inlets minus the mass flow rate of the stuff leaving the system, so that's i equals one to all the system coming out. In minus out equals change. That pattern will hold true for the conservation of energy as well. In the limit, that nothing enters the system or leaves the system, you have simply ncvdt=zero which tells you the mass is a constant. So again, in the limit that no mass passes the system boundaries, we recover the control mass system. So that's the conservation of mass. This is in its most general and differential form. All right, now having that in your head, what I want you to do is to write the simplified conservation of mass equation just using variable notation for a host that's being used to water a garden. So draw the system, draw the system boundaries, label the mass transfers, and then use all that information to write a simplified conservation of mass equation. Okay, we'll talk about that next time.
