Hey, guys. So up until now, whenever we've seen isobaric or isovolumetric processes, the only thing we could calculate was the work done. And in some rare cases, if you were given the heat transfer, you could also calculate the change in internal energy. But a lot of problems won't give you that heat transfer. For example, the problem we're going to work out down below here, all we know about this problem is that we have some kind of a process on a PV diagram. We have the moles, and we want to calculate the change in internal energy, but we don't have that heat transfer. So in these kinds of situations, you're going to have to calculate it, and that's what I want to show you how to do in this video. I want to give you the 2 heat equations that you need to know for isobaric and isovolumetric processes, and we're going to see that they're very similar to an equation that we've already seen when we studied calorimetry. So let's go ahead. We're going to keep on filling out our table here with some equations and we'll do an example.

So basically, what I'm referring to is an equation that we talked about when we talked about calorimetry, which is the \( q = mc\Delta T \) equation. So remember that this worked really well for solids and liquids. We did lots of problems where you have water that's warming up from 0 to 20, or ice that's melting, or something like that. But it actually doesn't work very well for gases, so we're going to need a different equation, and it's usually because we're not given the mass of the gas. Remember that this little \( c \) in this \( mc\Delta T \) equation refers to the specific heat per kilogram, but we're usually not given the mass of our gases in kilograms. So instead for gases, we're going to use this big \( c \) here, which is the specific heat per mole. So this is also known as the molar specific heat, this big \( c \) over here. So basically, what happens is that our 2 heat equations are not going to be \( mc\Delta T \). They're going to be \( nC\Delta T \), so \( n\cdot C\cdot \Delta T \).

Now, these two processes are different, and so these \( c \) values, these big \( c \)'s, are also going to be different. We give them special names. So, we do the one for isobaric. This is going to be \( C_p \). This is the molar specific heat at constant pressure. Right? Isobaric means constant pressure, so we use \( C_p \). Isovolumetric means constant volume, and so we're going to use \( C_v \). So these are just always the values that we're going to use for these two equations.

Alright, so that's all there is to it. Instead of \( mc\Delta T \), we use \( nC\Delta T \). Now, what are the values for these big \( c \)'s? Well, if you remember from calorimetry, this little \( c \), the specific heat per kilogram, was always different depending on whether you had water or steam or aluminum or something like that. Right? It was always different. Where these big \( c \)'s are actually much simpler. They only actually depend on what type of gas you have. So I've come up with a little table here that shows you all of the different values that you need to know. It really just comes down to whether you're dealing with a monoatomic or a diatomic gas, and here are the values for them. And so you just basically look up on this table here which value you need and then just plug and chug. So it's \( \frac{3}{2} R \) and \( \frac{5}{2} R \) for monoatomic and then it's going to be \( \frac{5}{2} R \) and \( \frac{7}{2} R \) for diatomic.