Kay, welcome back. So, hopefully you realized if you're in the superheat region, things are pretty easy. So temperature and pressure are in fact independent in the superheat region. So, sure, if we have T and P, we can define any other piece of information we need for the thermodynamic properties. Now, within the saturation region, this might have given you a little bit of scratching your head to think about it. If you have pressure and internal energy, can you fully define the state of the system? The answer is yes. Pressure and internal energy are independent in the saturation region. More specifically, what you would do is you would go to the tables or the online calculator or you know, whatever the reference is. But here I'll show you graphically what you would be doing. Is you know the pressure who that's going to define our pressure, whatever let's call this P, state P1 here. You would go look up for that pressure. Remember there is a unique internal energy for the saturated liquid and there's a unique internal energy for the saturated gas. So, what you now have is the internal energy, remember is a function of the quality of the mixture. So one minus the quality times uf plus x times here and now you have this information. That's what was given to you, you won. And you have this information from having looked that up at the state, from the tables or from the graph or wherever you got that information. And that would allow you to extract or calculate what the quality is for the mixture. Once you have the quality you can calculate anything else that you need. So, the key here is that you take pressure and internal energy that gives you quality. And from the quality plus pressure or whatever else you're going to use, you can find anything else, whether it's enthalpy, specific volume density, et cetera et cetera. Okay, so that's the process. So hopefully, we're getting a feel for how these state relations work, how we can determine information from tables. So the, now what I want to do is introduce two very common simplifications of state relations. The first is the incompressible substance model. This is only appropriate for solids and liquids and the name says it all. If it's an incompressible substance I can't compress it anymore. What I'm saying is that the specific volume is a constant or if you prefer the density is a constant, the same thing. In that case the internal energy is a function of, oh sorry, that shouldn't be a little t that should be a capital T. Temperature only, okay? Not time, temperature. Okay. If I have that information, I can now go back and say, hey. The definition of the specific heat. Remember the specific heat at constant volume was defined as the partial derivative. The specific, internal energy with respect to temperature for a constant volume. But because we now know that the internal energy's only a function of temperature. It's not a function of, of volume for this approximation. It tells us mathematically that the specific volu, the specific heat at constant volume is now the simple derivative of the internal energy with respect to temperature. Okay, so what does that mean. Well lets go back and look at CP. CP this specific key that constant pressure, remember, was by definition the partial derivative of the entropy with respect to temperature at constant pressure. But we also know that entropy is by definition the internal energy plus pressure times volume. So we look at this expression, we say hey, but this, we know from our model it's only a function of temperature. The internal energy is only a function of temperature. The definition of this partial derivative is with respect to constant pressure. So, this mathematically is treated as a constant. And this density or specific volume is prescribed as constant in this model. So, these are both constant. So, hopefully, we realize that when we have two constants and you're trying to take a partial derivative of those constants is that in fact they drop out. And what we have is that the specific key to constant pressure, under these circumstances is identical to the definition for the specific key, oh sorry, no, let me erase one of those. This is, oop, took out too many. the specific heat at constant pressure for the incompressible substance model is equal to the definition of the specific unit constant value. Okay? So, they're identically equal for this model. So, what it's saying to us is that the CP is equal to CV and so they can just drop the subscripts. There is only one specific heat. That is necessary for us to describe an incompressible substance or if you prefer we only need one specific heat to describe liquids and solids. Alright, now having said that we can go back and look at these definitions and descriptions for the internal energy. And we can say hey, if I need to know what the change is between the initial state and the final state for the specific, for the specific internal energy. That's simply the integral from the temperature at state one to the temperature at state two of the product of the specific heat times, the differential step in temperature. If we assume that the specific heat is constant, that the specific heat is not a function of temperature, so if we assume C is not a function of temperature. Then we can make the even further simplification that the change in the internal energy is simply given by the product of the change in the temperature of the states and the specific, times the specific heat of the mixture. And you can see we went from these kind of abstract concepts of specific heats, and partial derivatives, to something that looks really easy to use. Okay? And you look at this and you say, hey, because of the assumptions of the incompressible substance model, we can take this information, go back to the definition of the entropy and we can say hey the change in entropy. So the cost you know again by definition the entropy is u plus pv so the change in entropy from the initial state to the final state is simply the change in the, internal energy from the initial to the final state. And then we have plus p2 v2 minus p1, specific volume one. Except again, one of the fundamental assumptions of the incompressible substance model is that there's only one specific volume. And so I can simplify this into an expression that says this. Okay. So what we can see is that the change in the enthalpy is simply given by the change in the internal energy. Which is simply given by the change in the temperature and then we subtract off this piece of the the contribution from the pressure and the density. And sometimes this portion will be really small and it won't matter. So all we care about is this portion. And what that's really telling us is that the change in the enthalpy and change in the internal energy are the same. Okay, so, these are pretty powerful approximations. But we need to remember this caveat, it is for liquids and solids only. One of the most common mistakes I will see is the inappropriate application of a model. So you say ohp, I'm going to apply this in the superheat region. no, you can't use that approximation in the superheat region, only for liquids and solids. Another common mistake that I see, is people will not you'll assume an incompressible substance. And then, at some point, you'll go back and say, oh, I'm going to treat the system as having a variable density. So you, you mix the the fundamental assumptions of the model with the actual output. So in other words you, again, you try not to make these mistakes but don't, don't forget that if you're going to consider the system incompressible. You have to assume the density is constant, the specific volume is constant and the internal energy is a function of temperature only. Okay. So that gives you an introduction to a very common approximation for having the left side of our phase diagrams. Now let's look at the super heat region or the right side of our phase diagrams. And what we have now is an approximation that's called the Ideal Gas Model. And I'm sure you have seen the ideal gas equation before. It's usually introduced in typically physics and chemistry classes. So hopefully, this is a little bit of a review for you. If it's not, no worries and we'll go through this from the very basic steps. So just like the incompressible substance model had two assumptions, that the system had a constant density. And the internal energy was a function of temperature only, so does a gas model, the ideal gas model. The ideal gas model says this expresion has to be true, which is the pressure times a specific volume has to be equal to this product of the gas constant, that's what r is times temperature. So we're going to introduce the gas constant and more more specifically this is the specific gas constant. So you, likely have been introduced to the universal gas constant. So the universal gas constant is denoted in thermodynamics in having the over bar above the R. And it's the specific gas constant is the, universal gas constant divided by the mole cur weight of the fluid. Okay, so this, our bar, is the universal gas constant. And it has a variety of different numbers based on what unit system that you're using, and so we'll, we'll get a chance to see those in later examples. But the universal gas constant it's name means everything. It's universal it's the same number everywhere. The specific gas constant, it's name means, mean everything. It's very specific to the fluid you're using. So if you're using air, it's a different number than the gas constant for water, than the gas constant for nitrogen, than the gas constant for ammonia. So, again one of the most common mistakes that people make with the ideal gas model is there's probably a hundred different ways for you to write this expression. I'll give you a few examples here in a moment. Make sure you're consistent in each one. So here we have pressure times specific volume. Well, I could also write that in terms of [SOUND] density. Okay. I can write this expression in terms of molar units. So this over bar tells me that it's the universal gas constant. It also means that that unit is on a per mole basis. So the over bar means it's, that that unit or that variable is on a per mole basis. Just like the lowercase means per mass basis in thermodynamics, the over bar means at a per mole basis. So I can do this. I can do this. So just make sure you're using a consistent set of units on all of your expressions. Okay? You've also probably seen where you'd separate the total volume instead of using specific volumes and specific densities, you use the total volume and the mass. And one last representation, let me do this, write the expression this way. because we are going to sneak in another definition where we have 'n' is equal to the total number of moles in the system. Okay. So again all these permutations and combinations of how to write the ideas, ideal gas model. Make sure that you are using the right one and you have consistence set of units. Okay. So just like we did before, we say we can look at these expressions and say, well, if the internal energy is a function of temperature only. Then that's going to give a simplification that I can invoke for specific heats and all that sort of stuff. But before we go there, what we want to understand is the fact that the ideal gas model has we defined the compressibility factor that allows us to understand how close we are. How good of an approximation it is that we are treating the system as an ideal gas. So it's a check. We have a check in the system and that's the compressibility factor. So that's this Z. So this is compressibility. Okay. So the compressibility factor, you can see again is essentially pick any of those forms of the ideal gas model and normalize it by itself. So what we know is that as Z of the compressibility factor approaches 1, then that means the ideal gas model is valid. as Z decreases and goes, becomes smaller and smaller as Z approaches 0, we know that we're having what they would say officially is that we'd have significant molecular interactions. And so the molecules actually know each other are there as a consequence, the ideal gas approximations are no longer valid. So, we know as Z approaches 0, we have interaction. And our ideal gas model. That interaction causes that approximation of the ideal gas relation. And in terms of the functional dependence of the internal energy and things like that to break down, so it's no longer appropriate. There are some advanced pneumatic methods that you can use the compressibility factor to correct your approximations and we're not going to cover those in this class. But what we should recognize is there are times when the ideal gas model is not valid and the compressibility factor tells us when those times occur. Now having said that I want you to think about by intuition we know that the ideal gas model is typically valid in the super heat region. So, I want you to visualize the pressure volume diagram and I want to you or try to provide whichever one works for you. And I want you to tell me what pressure in temperature conditions or temperature conditions would you expect the ideal gas law to fail? So I'm talking like high, low, middle things like that. So when would you expect the ideal gas model to not be appropriate? And we'll start with that next time.
