 We have connections now between heat and two quantities, the energy and the enthalpy and the connections are very simple and the connection to the energy is very simple when we're doing things at constant volume and the connection to the enthalpy is particularly simple when we're doing things at constant pressure. So we've done this work partly to get away from this path dependent quantity of the heat and be able to talk about these state functions, the energy and the enthalpy instead. But it's worth pointing out that heat is, despite the fact that it's a path dependent quantity, a very natural, intuitive and useful quantity experimentally. Think about when you do a chemical reaction in the lab. Sometimes you can just do that chemical reaction at room temperature but fairly frequently you want to do that reaction either at an elevated temperature or a lowered temperature. So what you do is perhaps you stick a Bunsen burner under your flask or perhaps you put your flask in a heat bath that's at hot water heat bath or you put it in an ice bath if you want to lower the temperature. So in every one of those cases what you've done is you've applied some heat, either you've put heat into the reaction vessel or you've removed some heat from the reaction vessel. What you really want to do is affect the temperature. You want to elevate or reduce the temperature but the way we control the temperature of something is by the application of heat or the removal of heat from the system. So experimentally we're frequently using heat to control the temperature of a reaction. And it's a very common thing to need to know how much heat do I need to change the temperature by a certain amount. In other words, what's the rate of change of Q with respect to T or sometimes vice versa. And that quantity is something we define as the heat capacity. So the amount of heat per unit change of temperature is, the name's a little bit old fashioned but we still nonetheless call it a heat capacity. The amount of heat's not something that can be held, the objects don't have a capacity for heat in the way that this term implies but the amount of heat transfer that's required to change the temperature by one degree is called the heat capacity. So you notice this derivative looks a little bit weird. I've had to write a path dependent DQ, an inexact differential DQ above this differential DT in the denominator because heat is a path dependent quantity. So instead it's often more useful to think about state functions like energy and enthalpy. So if we happen to be at constant volume conditions then the heat is equal to the internal energy then the heat capacity in that specific circumstance would be DUDT and I'll write, I've written this as a partial derivative and I'll write that as DUDT while holding V constant to remind me that that derivative is taken at constant volume. On the other hand, if I do this at constant pressure, under constant pressure the heat is equal to the enthalpy so DQ DT becomes DH DT. So the energy and the enthalpy are not the same thing because Q is path dependent that derivative either looks like DUDT or looks like DH DT under different conditions. Again to remind us of those conditions I can write the subscript V or the subscript P to remind us that this is the heat capacity at constant volume, this is the heat capacity at constant pressure. So we've got two different definitions of the heat capacity now and mainly what I've done this for is to get rid of the path dependence. When I define the constant volume heat capacity as DUDT at constant V these are all state functions. So this definition is always going to be true. This is just a quantity I've defined. It happens to be equal to the change in heat with respect to temperature when I'm at constant volume likewise this term is a state function that's not going to depend on the path I take. It will happen to equal DQ DT when I'm at constant pressure. So we've got these expressions these are both extensive properties I've written the U and the H without a bar on top so if I prefer to think about intensive properties temperature is already intensive so I can define if I just divide on either side of this equation by the amount of moles of material I have then I've got the change in the molar energy with respect to temperature and that's equal to the molar constant volume heat capacity or the intensive equivalent on this side the constant pressure molar heat capacity Cp bar is equal to the molar enthalpy derivative of molar enthalpy with respect to temperature. So those are definitions of constant volume and constant pressure heat capacity either the extensive or the intensive versions of those I suppose since we already know for the specific case of an ideal gas specifically an ideal gas that a gas that we can describe well with the 3D particle in a box model. The energy is three halves NrT the enthalpy is five halves NrT we can take the derivatives of those expressions derivative of energy with respect to T in fact if I want to do this the molar energy is energy divided by the number of moles so that's just three halves Rt the molar enthalpy would be five halves NrT divided by N so I've got five halves Rt if I want to know what is the molar heat capacity of this gas constant volume heat capacity just take the derivative of this expression with respect to T that's a relatively simple derivative and I find out that the molar heat capacity is three halves R after taking the derivative of the T goes away and likewise the constant pressure heat capacity would be five halves R so again those these equations on the top here are true for anything this is just the definition of the heat capacity these equations three halves R five halves R are the heat capacities not of everything but of an ideal gas specifically an ideal gas that behaves like a 3D particle in a box. So we have these expressions for the constant pressure and constant volume molar heat capacities those allow us to do things relatively easily like calculate the change in the energy or the enthalpy when I change the temperature of something and to show you how that works let's see do that over here let's take this expression du dt at constant v is equal to cv if I rearrange this expression actually let's rearrange this one du dt at constant v is equal to cv I'll rearrange that question and say du is equal to cv dt I've just broken up this derivative left the du on the left move the dt over to the right if I integrate both sides of that expression du becomes delta u what's rewrite cv now the the extensive heat capacity is number of moles times the intensive heat capacity if I want to know how much the energy of an object changes its moles times its constant volume heat capacity molar constant volume heat capacity integrated over a change in temperature from some t1 to some t2 in the specific case so it may be the case that the heat capacity depends on temperature anything that depends on temperature I have to leave inside the integral if something doesn't depend on temperature the number of moles certainly doesn't depend on the temperature so I can pull that out of the integral if the heat capacity does not depend on temperature I can pull it out of the integral as well so I'll note that that is true only if cv is constant if it doesn't depend on the temperature but if I can do that then this integral is relatively simple integral of dt from t1 to t2 is just t evaluated between t1 and t2 so t2 minus t1 that's just the change integral of dt is delta t so we have this expression delta u is n times heat capacity times delta t as long as the heat capacity is is a constant like three halves are or five halves are something that doesn't depend on the temperature then internal energy is just moles times molar heat capacity times the change in temperature if I heat something up this is how much the energy changes there's a very similar expression if I were to do the same thing for enthalpy we have the expression enthalpy change is n times cp delta t so those are worth putting in a box because those are expressions we'll use all the time in both those cases we have to remember the caveat that the heat capacity either constant volume or constant pressure heat capacity has to be constant but if we can assume the heat capacity is constant those are relatively easy expressions to use for example and this will make it clear why they're useful if we repeat the same example that we've done two different ways now using this expression let's say we have one mole of an ideal gas that we want to heat from 298 kelvin to 348 kelvin if we do that let's say we want to know what is the enthalpy change for that process delta h is n cp delta t so all I need to do to calculate the enthalpy change for that process is multiply these three things together one mole molar constant pressure heat capacity for an ideal gas that's five halves are so five halves times the gas constant times the change in temperature from 298 kelvin to 348 kelvin I've increased the temperature by 50 kelvin if I multiply those numbers together 8.3 if 1 4 times 50 times two and a half you will perhaps not be surprised to see the answer we get works out to be a little over a thousand joules exactly the same answer we got as in the previous example where we first asked what were the heat and work required to increase the temperature of a mole of an ideal gas by 50 kelvin and then later once we define the enthalpy what is the enthalpy change for heating a gas the same amount we've now done the the same thing but notice the calculation was much simpler so if all we want to do is calculate the energy or the enthalpy change for an object that's being heated or cooled down then these expressions are often the easiest way to go notice I'll make one comment about this point notice that I didn't have to even tell you what the pressure and the volume of this gas were doesn't matter whether I've done the heated this gas at a pressure of one atmosphere as we used in the prior examples or even if I had a gas compressed to two atmospheres and I heated it from 298 kelvin to 348 kelvin turns out to get the same result so this this calculation is much simpler we don't have to know anything other than the amount of the temperature change and how much of the gas we have also I'll point out that this state function the enthalpy I didn't have to tell you this was a a reversible or a constant pressure or a constant volume process ordinarily you might think if I'm calculating an enthalpy it would only be valid for a constant pressure process but once I've defined this state function the enthalpy change when I heat something from 298 kelvin to 348 kelvin a mole of an ideal gas is going to be 1040 joules regardless of what path I take in getting from state one to state two so because it's a state function it does not depend on path I can calculate the enthalpy change for any path I want and I'll get the same answer so regardless of what path I actually take I can I can use these expressions to calculate the enthalpy change so those are several convenient features of using this type of expression thinking about heat capacities when we measure the energy or the enthalpy change of an object under a change in temperature