Hello, in this video, we're going to talk about Mixtures of Ideal Gases. On this first slide, there were a number of relationships. You should have been exposed to these in undergraduate thermo. If you took a full year of thermo, the first equation defines the mole fraction, which is essentially the normalized number of moles. In other words, it's the moles of a given species say species i, divided by the total number of moles or it could be the ratio of the number density of species i divided by the total number densities. It's a molecular based, so it's the relation of the number of molecules to the total number of molecules. And since it's a fraction, the sum has to be equal to one of all the mole fractions. In the next line, we define the mass fraction. The mass fraction is the mass of the individual species divided by the total mass. And like the mole fraction, the sum of the mass fractions have to equal one. We can convert back and forth between mole and mass fractions by taking into account the molecular weight of the individual species. So if you know the mole fraction, you can get the mass fraction by multiplying the mole fraction by the molecular weight of that particular species divided by the molecular weight of the mixture. And likewise you can get the mole fraction from the mass fractions by multiplying the mass fraction times the molecular weight of the mixture divided by the molecular weight of the individual species. The molecular weight of the mixture can be defined either in terms of molar mass fractions as shown in the next equation. So it has either the sum over the product of the mole fractions time the molecular weights. Or it is 1 over the sum of the mass fractions divided by the molecular weights of the individual species. In either case, to know the mole fraction of the mixture you need to know the composition of the mixture. In other words, you need to know either the mole fractions or the mass fractions. We also define the partial pressure. The partial pressure being the product of the mole fractions times the total pressure. And therefore, we can write the total pressure as the sum of the partial pressures. In mixtures an important issue is energy referencing, we referred to this a little bit when we talked about vibrational energy of diatomic molecules. The usual method is to reference the enthalpy using the enthalpy of formation. This is the enthalpy reference to the so-called standard reference state. The usual notation is delta sub f h super not or h in the normalized form h sub f super not and I put a bar over the h, lowercase h in this case because it refers to the molar enthalpy, enthalpy per unit moles. So the enthalpy can be written as the enthalpy of formation plus the change in enthalpy between the given temperature and standard temperature which in the SI system of units is 298.15 Kelvin. So that last term accounts for the change in enthalpy from the standard reference state. So what is the standard reference state? For liquids and solids, the standard state is the thermodynamically stable state at a pressure of 1 atmosphere and a specified temperature. The stable state could be liquid or solid. For gases, the standard state is the gaseous state at a pressure of 1 atmosphere and a specified temperature based on the assumption of ideal gas behavior. Typically, we specify the temperatures 298.15 Kelvin. And then for dissolved species, which will not be dealing with the standard state is a 1 molar solution at a pressure of 1 atmosphere and a specified temperature based on the assumption of ideal solution behavior. So if we have a mixture, what are the properties of the mixture? Well, the mixture internal energy is just the per unit mass is equal to the mass fractions times the internal energy per unit mass of each species. And likewise the enthalpy of the mixture is equal to the sum of the product of the mass fractions times the enthalpy per unit mass of each species. Or if it's on a molar basis, then it would be the internal energy would be equal to the mole fractions times the internal energy per mole of each species and likewise for enthalpy. So I've already mentioned how we reference enthalpy but we can reference internal energy exactly the same way. So we can write say the molar version, the molar internal energy is equal to the molar internal energy of formation plus the change in internal energy between the temperature and the standard reference state and likewise for enthalpy. And as I mentioned in the previous slide, the reference temperature for the standard reference state is to 298.15 Kelvin 25 degrees Celsius, and the pressure is 1 atmosphere. It turns out that the enthalpies of formation are zero for elements in there naturally occurring state at the reference state conditions. Note that you can write the change in internal energy or enthalpy for an ideal gas is the integral over the appropriate specific heat dT between the reference temperature and and the desired temperature. And this just shows a diagram, a plot of what specific heats look like to remind you that only for a monatomic gases are specific heats and not a function of temperature. We have to be careful with entropy because entropy is a function of both temperature and pressure. And therefore, we have to use the partial pressure when evaluating the component entropy before calculating the entropy of the mixture and that shown here for the per unit mass and per unit mole entropies. Enthalpy of formation data is tabulated in quite a few sources, but in particular in NIST database, they were originally tabulated in something called the JANAF tables. JANAF at the time stood for Joint Army Navy Air Force and later included NASA. And this was a effort that started during the second world war to tabulate thermal chemical data for the purpose of doing calculations related to propulsion both air propulsion and rocket propulsion. But the tables look something like this and you can see there's a column for temperature,specific heat, entropy actually, Gibbs free energy, which we'll get to and enthalpy. And in both of these, both the Gibbs and the enthalpy tabulations are for the difference between the standard reference state and some temperature. And then delta h sub f is the next column and delta s sub g and the log k f we'll get to that in a little bit. So, how do we use this information? Well, one of the ways we use it is in talking about chemically reacting systems and in particular combustion reactions. So typical combustion reaction is shown in this slide. Here we define the term Stoichiometry, a reaction balanced equation like the one written here for a general hydrocarbon is really a statement of conservation of mass. That is if I start off with a hydrocarbon with x number of carbons and y number of hydrogens and I add in say air to it and it chemically reacts, then I better end up with the same number of atoms that I started with. This reaction is what's called a global reaction. We'll get into these differences later, but the conservation concept applies both the global and elementary reactions. By global we mean that this is the overall reaction of this hydrocarbon with air assuming that we have so-called complete combustion, which means they're all goes to carbon dioxide, water vapor, and of course the nitrogen at this level that we're talking about is unreactive. An elementary reaction is a reaction that actually occurs and there can be a large difference between the two, methane for example requires hundreds of reactions for methane to oxidate to equilibrium products at the end. In combustion, the mixture of fuel and oxidizer that results in complete combustion is called the stoichiometric ratio or mixture. You can define various kinds of ratios air fuel/ratio is the mass of air over the mass of fuel or you could define it in terms of moles, I mean, moles of air over moles of fuel. But a common definition is that is used is the so-called equivalence ratio, which in air-to-fuel notation is the ratio of the stoichiometric air to fuel ratio divided by the by the local particular air to fuel ratio. Or it could be defined as fuel air ratio over the stoichiometric fuel to air ratio and that is in my field how it is typically defined as a fuel ratio. So that if phi is one you have a stoichiometric mixture, if it's less than one you have excess air, if it's greater than one you have excess fuel. But you gotta make sure whether you're talking mass or moles because it would be the other way around for the mass piece definition. So, how how do we use all this? Well, we might do a first law analysis. For example suppose, we have a control mass. There's a change in time, some heat transfer takes place. The first law is the energy at state 2 minus the energy at state 1 is going to be equal to the net heat in minus the net work out. And the energy of course is the total energy is the mass in the control mass times the specific internal energy plus one half V squared plus gz. We also could be doing a steady state analysis, first law analysis. For example, if we have a control volume a flow in, a flow out perhaps there's some net heat transfer in, some net work out and then the first law is written as shown the mass flow rate times the enthalpy plus v squared over 2 plus gz out minus the same thing in is equal to Q dot in minus w dot out. Another type of problem isn't the adiabatic flame temperature problem where we have a insulated reactor or a furnace and we put reactants in combustion takes place. We get reactants out and they are going to be at a higher temperature, there's no heat transfer or work. And then finally, heat of combustion problem where we put reactants in and we have products out. We bring the products back to the reactant temperature that would generate the most amount of heat and that heat we call it heat of combustion. So there's a wide variety of these types of problems that ultimately involve mixtures, and as a result you have to take into account mixture properties the way we have discussed in this video. Thanks for listening and have a great day, bye.
