So, that's how, that gave you the general formula for how you would evaluate specific volume within the saturation region. So now, we just follow that exact same pattern for internal energy, which we've already talked about before. And now we're going to introduce a new term, which is enthalpy. Okay. so I'm going to write that expression up here again, so we have it available to see how these patterns develop. So here's that expression. We could similarly just you know, do a little bit of algebra, push that stuff around and you get this. And depending on the types of tables that you use, this difference which is the density difference that occurs due to phase change will be listed as Vfg. Okay. it is the increase in this case, it is the increase in specific volume upon vaporization. Okay. So again, sometimes the tables will list they offer the saturated liquid and the saturated vapor data. And sometimes the list just the saturated that saturated, the difference between those two properties and then the saturated liquid data. So they'll do some combination of those three variables, so it just depends on what's tabultaed in the system. Okay. So, we'll recall, remember the total internal energy is the upper case U, so this is total internal energy. And the specific internal energy is the total energy divided by the mass total mass in the system. And we could go through the exact same algebra that we did before, and what we would find is that, the specific internal energy is simply 1 minus when we're in the saturation region, it's 1 minus the quality times the saturated liquid properties plus the quality times the saturated gas. They're the exact same process. So, again, every moment is a teaching moment, so what I'm going to now do is introduce enthalpy. And I'm going to use this very specific notation that says, enthalpy by definition. So the enthalpy is by definition the specific internal energy plus the product of the pressure times the specific volume. Okay. on a total basis, we would say that H is the eth, the enthalpy is equal to the internal energy plus pressure times volume. Okay. Again, by definition, the enthalpy is a variable that combines information about the internal energy and this pressure volume term. And it essentially, it will be really obvious when we start looking at conservation of energy for control volumes, why we define this system this way. It essentially saves us a lot of handwriting. It saves us a lot of time because, this is going to represent an aspect of energy transfer that occurs in all conservation, in all control volumes. So we don't quite have the tools yet to explain why we define enthalpy this way, so we're just going to put it out there, that enthalpy is defined in this manner. If we have it normalized by mass, it's an intensive property; if we have the total amount, it's an extensive property. Enthalpy is a thermodynamic property. And you can see, it's related to the internal energy of the system. Okay. Having written these definitions down, they are the definitions. So, these are true at all times, okay? It's the definition of enthalpy. It doesn't matter if it's a solid, a liquid, a vapor, if the system is undergoing face change, this relationship between enthalpy and internal energy is always true. Okay. So, remember this, we're going to come back to this quite a few time, this definition of and relationship between enthalpy and internal energy. This is reffered to as state relation. It's where we have information about the state that allows us to move between pressure and volume, and internal energy to enthalpy. Okay, so this is a state relation. And this is one of the outcomes of thermodynamics. Okay. Now, Since internal energy within the, so this is, this expression here. Again remember, it's only defined within the saturation region. So this is only valid in saturation region. This expression is valid for all phases, so this is true at all times. If we're within the saturation region, then the enthalpy can be written in terms of the quality of the mixture, just like we can write the internal energy in terms of the quality of the mixture. Just like we can write the density and the specific volume in terms of the quality of the mixture. So, just like we defined the increase in specific volume upon vaporization using this expression here, we can also define Ufg as being the difference between the specific volume of a saturated gas minus the specific volume of the saturated liquid, just like we can define the difference in the enthalpy of the saturated gas minus the saturated liquid. And hopefully you can see even though we call these hfg, the gas phase actually occurs first. And that's primarily because we'd like these numbers to be positive numbers. The enthalpy of the gas is always higher than the enthalpy of the liquid. The internal energy of the gas is always higher than the enthalpy of the liquid, during that phase change for a constant temperature or constant pressure process. Okay. So again, when we go to the tables, you would go there to extract information with. If you're within the saturation region, you would go there to extract information about the saturated gas and the saturated liquid properties. You would then take that information and combine it with the quality in order, in order to define the actual state. So, in other words, if you were within this region here. Again, if this is a quality of one and this is a quality of zero, and your state is somewhere between those two. Let's say a quality of 3, 30%, 30%, well, that looks a little bit more like its 40, 50, close to 40 or 50%. You would take this information plus the saturated liquid information and the saturated gas information, and combine it to get the actual state properties. Okay. So it's a little bit two-step. It's a two-step, so it's not exactly the quickest most efficient way to determine your state information. But is it very efficient in terms of the that we store the information, in that we only have to understand how the saturated fluids, how the saturated liquid and the saturated gases behave. Cool. Notice my little qualifier down here. All thermodynamic property data are actually taken relative to a reference state. And so, I'm just going to introduce this right now and we're going to come back to it. But, we will be taking the difference between state conditions. So the fact that everything is relative to a reference state is going to cancel. So I'm going to plant that seed right now, and we'll come back to it in just a few segments. But all thermodynamic properties have associated with them a reference condition. And the key issue is that the reference conditions aren't always the same, 90% of the time they are. But if you are to take information from two different resources, you need to be very careful in understanding that the reference states are the same for the two different databases. Okay. But again, we'll cover that some more later. Just wanted to plant that seed. Okay, whew. Lots of state information. I snuck in the, I introduced a new variable called enthalpy. Internal energy, I'm sure you might have some feel for it. Kind of sounds like internal energy would by like oh, how much energy is within the system. We know that enthalpy is related to that. So we think okay, how goes internal energy is how goes enthalpy. Now, we're going to keep going and introduce a couple more state relations in thermodynamic properties. We going to introduce the specific heats at constant pressure, and the specific heat at constant volume. Specific heats are also referred to the heat capacities, and that'll be kind of obvious, pretty quickly here why they're called that. And there also intensive properties, intensive thermal dynamic properties of simple compressible substances. So, let's give you those definitions. It gets a little bit complex mathematically, but as you might expect in this class, we're going to very quickly try and simplify that math. Okay. The specific heat at constant volume is by definition the partial derivative of the specific internal energy with respect to temperature, while holding the specific volume constant. So that's what this notation means. It's a partial derivative. This is not a substantial derivative. This is not a differential, it's a partial derivative. So we have the partial derivative of the internal energy with respect to temperature, while holding the volume constant, is the definition of the specific heat at constant volume. Sounds really complicated, all we want to know is that there's a way to relate a specific heat of a system to internal energy, temperature and volume. Okay. It's sibling, took the sister or brother if you prefer of the con, specific heat of constant volume is the specific heat of constant pressure. It is by definition the partial derivative of the enthalpy with respect to demp, temperature while holding the pressure constant. If we take these two variables, and we put them together in a ratio form, we need to find the ratio of the specific heats, that's k. and this is the ratio of specific heats. Okay. And you can look at these expressions and you should be able to determine what the units are for the specific heats. So let's take internal energy. This is a specific internal energy. So, we know it's going to have units of energy divided by mass, because it's energy, this is specific internal energy. And we know that the temperature has units of kelvin, because remember, we talked about this before. We need to work in an absolute temperature scale at all times in thermodynamics. No degrees Celsius, no degrees farenheit. It's kelvin and Rankine every single time. If we parrallel the units to, from SI to British units, then we would say, then that's BTU units are my units of energy. My mass is a unit of pound mass, and then, and sometimes we'll drop the little subscript m here, but you have pound mass and pound force. This is pound mass, so it's a mass unit, and our absolute temperature scale is Rankine. Since internal energy and enthalpy have the same units, because recall by definition, the enthalpy is the internal energy plus PV. So, you know internal energy has units of specific energy specific energy. So they're kJ per kg, and you know this has to have this product of pressure times specific volume also has to have the same units. And so the units of enthalpy are also kJ per kg. Okay. So, hopefully all that logic was something that you thought of intuitively. Okay. so these are the specific heats, we're going to use them quite a bit. And again, they're state relations. And for these fundamental definitions, these are state relations. And these definitions are always true regardless of state. Okay. So, as long as the system is equilibrated. That's a caveat that we have that blankets the entire subject of thermodynamics, right? We always have to have equilibrium. So the state relations are always true, these are their definitions regardless of state or phase, little footnote, always assuming that we have equilibrium. Okay. So, now what I want you to think about, having gone through all these states, we've introduced a number of new variables. Remember what we talked about in the previous unit. We said, in order for us to fully define a system, we need to independent intensive properties. And we discussed where for example temperature and pressure are not independent. So, what I want you to do is think about these two questions here. Can you fully define the state if you're in the super heat region, and I give you a temperature and pressure? And what that's ultimately going to mean is, if I give you temperature and pressure, and you're in the super heat region, can you tell me the enthalpy, the internal energy, the specific the specific heat at constant volume, the specific heat at constant pressure, the density, the specific volume everything? That's what the power of these definitions are. If you have two independent properties, you can define everything else. So the question is, in the superheat, are T and P enough? And then the next question is, if you go to the saturation region, you're in the dome, are pressure and internal energy enough for you to define this, to fully define the state? So I'll leave you with that, and we'll take that up next time.