 In Chapter 12, our end goal is going to be to analyze heating and cooling systems of actual air. Unfortunately, we can't just model actual air as a dry ideal gas. We have to analyze it with the water vapor that's in the air in the form of humidity. The analysis of water vapor as humidity in atmospheric air is called psychrometry. And before we can talk about the mixture of water vapor and air, we need to talk about ideal gas mixtures in general. So we're going to start by describing mixtures in a general case and their properties, and then we'll go into what's specific about the way that we describe humidity in air. Generally speaking, when we talk about mixtures of gases, we can describe the proportions of the mixtures in two ways. The first is on a mass basis, also called a gravimetric analysis. When we describe a mixture on a mass basis, we are describing proportions by mass, which is also by weight. We can also describe mixtures on a molar basis, and note that because the molar mass of the individual substances could be different, the molar proportions might not be the same as the mass proportions. Also note, for ideal gases, the molar proportion is equivalent to the volume proportion. So if we described a proportion of a substance on a molar basis, we are also describing that proportion by volume. And to get a little practice describing proportions in both a mass and molar basis, let's try an example. Say you have a sealed tank containing 25 kg of hydrogen, oxygen, nitrogen, and carbon monoxide. I want to use that information to describe the total mass of the mixture, the mass and molar fractions of all the components, the average mixture molar mass, and the gas constant of the mixture itself. So to begin with, I will draw a table, as I think that's the most convenient way to represent all this information. So I have diatomic hydrogen, diatomic oxygen, diatomic nitrogen, and carbon monoxide. I was told the mass of each of those individual components. I'm going to use the subscript I to represent of one of these substances or species in the mixture. The mass of hydrogen was 25 kg, likewise with oxygen, nitrogen, and carbon monoxide. The next thing I want to do is represent the amount of mass as a proportion of the mass of the whole. That abbreviation is going to be what we call mass fraction, which is abbreviated with an MF for mass fraction, it makes sense. And the mass fraction is going to be the mass of each individual substance or species, divided by the mass of the mixture. The mass of the mixture is just going to be the sum of all of the individual masses appearing. So 25 plus 25 plus 25 plus 25 yields a mass of the mixture of 100 kg. Then the mass fraction for oxygen is going to be 25 kg divided by 100, which is going to be 0.25. The mass fraction for nitrogen is going to be 25 kg divided by 100, and so on, and so on. The mass fraction of each component is one-quarter because each of them contributes one-quarter of the mass to the mass of the whole. To figure out the molar proportion, we're going to have to figure out how many moles are in 25 kg of each of these substances. To calculate that quantity, we're going to need to look up the molar masses for each of these individual substances. I will abbreviate molar mass with an uppercase M, and that's going to be a measurement of kg per kilomole. And to look up those values, we're going to draw upon table A1. In table A1, I can look up a molar mass for each of these. First up, I have oxygen, which is 32. Excuse me. First of all, I have hydrogen. And then oxygen. So hydrogen was 2.016. And then I have nitrogen, which is 28.01, and then I have carbon monoxide, which is 28.01 as well. So the molar mass represents how many kilograms are in a kilomole of substance. Therefore, to calculate the number of moles of each of these individual species, I'm going to take the mass of the substance divided by the molar mass of the substance. I'm going to abbreviate number of moles with a lowercase N. So we've gone about as far as we can without the calculator. So calculator, if you would help us out. First up, we have 25 divided by 2.016. That's going to give us 12.4 kilomoles of hydrogen. Then we have 25 divided by 32, which is 25.30 seconds. Thank you, calculator. Which is 0.78125. Then we have 25 divided by 28.01, which is 0.89254 for both nitrogen and carbon monoxide. And just like we did with the mass of the mixture, we can figure out the number of moles of the mixture by summing together each of the individual number of moles in each of the species across all of the species. So we take 12.4008 plus 0.78125 plus 0.89254 plus 0.89254 and we get 14.967 kilomoles. So just like we did with mass, we can take the number of moles of each of the individual components and divide it by the number of moles of the mixture and that would represent the molar fraction. The molar fraction is also abbreviated, but we already used MF. So what are we going to use instead for molar fraction? Why? Why it's why? I don't know, but that's the standard. And why is the number of moles of each of the individual species divided by the number of moles in the mixture? So we start off with 12.4008, 12.4008 divided by 14 and we get 0.828. We repeat the process, 40.78 divided by 14.967 and we get 0.0522. And then 0.89254 divided by 14 and change gives us 0.059633. So on a mass basis, this mixture is 25% hydrogen, 25% oxygen, 25% nitrogen and 25% carbon monoxide. On a molar basis or volumetric basis, it is 83% hydrogen, 5.2% oxygen and about 6% of each nitrogen and carbon monoxide. As a general rule of thumb, molar fractions are more useful in chemistry and as a result are usually the ones being described. If you were told a proportion and you weren't told whether or not it's a mass fraction or a molar fraction, you assume it's a molar fraction which again is equivalent to a volumetric fraction for ideal gases. For example, if we consider Earth's atmosphere, it's 78% nitrogen, 21% oxygen, about 1% argon and so on. Those are proportions of volume, not mass. Anyway, we have A, B and C done. Next we need to figure out the average mixture molar mass. And for that, there are a couple of ways that we can approach the problem. We could represent the molar mass of the mixture as the mass of the mixture, divided by the number of moles of the mixture and in that case, we already happened to have both those quantities. We could say 100 kilograms divided by 14.967. In which case, we end up with a molar mass of 6.681 and that worked out well because we happened to have the amount of mass of the mixture and the number of moles of the mixture already for this particular problem. But in some cases, it might be convenient to write it out in terms of the mass or molar proportions. But be aware when you're doing that, you have to keep track of what each of these kilograms represents. This is kilograms of substance per kilomole of substance. So you can't just take, for example, 0.25 times 2 plus 0.25 times 32 plus 0.25 times 28 plus 0.25 times 28. I mean, we can try that and we get a completely different number. And the reason that we do is because this mass fraction is kilograms of substance per kilograms of mixture and we're multiplying by kilograms of substance per kilomole of substance. So you might be tempted to say, well, this kilogram cancels this kilograms, we're left with kilograms per kilomole. But it's not the right kilograms here and here to cancel. The kilograms of substance and kilograms of mixture are different. So they don't cancel. Furthermore, what you would be left with anyway is the kilograms of substance divided by kilomoles of substance. The easier way to think through how to build the proportional averages is to actually start with this equation and substitute quantities if you have them. For example, if we wanted to, we could write the mass of the mixture as the sum of all the masses of the individual components. And if we wanted to, we could write the number of moles of the mixture as the number of moles of each of the individual substances summed together. And then we could figure out, well, the number of moles in total is going to be the number of kilograms divided by the molar mass for each of the individual species all summed together. So in the denominator, we could replace the sum of Ni with the sum of mass divided by molar mass. Okay, well, that's not particularly helpful. What if we try it from another angle? What if we leave the denominator as n-mix but try to come up with alternative ways of writing the mass of the substance? Well, I could write the mass of the substance as the number of moles of each of the individual substances multiplied by the molar mass. So instead of writing sum of Mi, I could write sum of the quantity number of moles times molar mass. Well, in this particular arrangement, I can group together this quantity by bringing n-mix inside of the summation, at which point I would have the summation of y times the molar mass. So when we're talking about molar mass of the mixture, we can represent it as the sum of the molar fraction of each of the individual components multiplied by the molar mass. And the way that we know that is because we have expanded and simplified this particular equation. You can do that for a lot of these relationships. You can algebraically rearrange them to write them in terms of proportions or in terms of total quantities depending on what you have and what's useful. But I'm of the opinion that it is worth it to just treat everything as a total quantity as much as possible. Don't get too caught up in trying to rearrange things algebraically. So just to prove a point here, let's take all of these and we should end up with about 6.7. Hey, we did! We got the same molar mass. I mean, worst case scenario, if you're in a situation where you have mass fractions or molar fractions, instead of a total quantity, you can always just assume an amount and figure out what the other proportion is. I mean, if I told you that you had 25% hydrogen, 25% oxygen, 25% nitrogen, and 25% carbon monoxide on a mass basis, and I said, what is the molar mass of the mixture? You can always just assume one kilogram, figure out the masses, figure out the number of moles, figure out the molar fraction, divide the two masses, excuse me, divide the mass of the mixture by the number of moles of the mixture, or do the molar fraction times the molar mass, and that will work out even though you don't actually only have one kilogram. By assuming a quantity, you can figure out the proportions and figure out the average quantities and it doesn't actually affect anything because the amount of substance doesn't matter. What matters is the relative amounts, if that logic makes sense. Anyway, the last thing I wanted us to calculate was the gas constant of the mixture. So for that, we will step back to how we calculated gas constants in Thermo 1. The gas constant for the mixture on a mass basis is going to be the universal gas constant divided by the molar mass of the mixture. Well, the universal gas constant comes from the inside of our front cover of our textbook. We can take 8.314 kilojoules per kilo mole Kelvin and then divide it by our shiny new molar mass of our mixture which is 6.681 kilo moles, cancels kilo moles, leaving me with kilojoules per kilogram Kelvin. And I get a gas constant of 1.2444 kilograms, excuse me, kilojoules per kilogram Kelvin. So take a minute and reflect on why it might be useful to know the gas constant of this mixture. Using the gas constant allows us to apply the ideal gas law to the mixture as a whole. If I told you that these 100 kilograms were sitting in a 100 cubic meter container at a certain temperature and I asked you to figure out what the pressure is, you can use the ideal gas law to model the relationship between temperature pressure and specific volume and figure out what the pressure is. Doing so would require the specific gas constant for our actual mixture.