 What we're going to do now is we're going to take a look at the definition of specific heats and we will then use them to compute both enthalpy and internal energy for ideal gases. So specific heats, specific heat itself is energy required to raise a unit mass, so for example 1 kilogram of material by 1 degree Celsius. So it's the energy required to increase the temperature of 1 kilogram of some material, and that's the material that we're determining the specific heat for, and we're increasing it by a unit of temperature rise, in this case 1 degree Celsius. Now there are two different types of specific heats that you will encounter. You'll encounter specific heat at constant pressure and specific heat at constant volume, and the reason why they are different is, and I'll show you in a second, in one case we need to account for the fact that if it is at constant pressure the boundary could be moving, we have to account for the boundary work. So if you look at specific heats at constant volume, and this is assuming that we're dealing with a gas or a liquid, in the case of a solid the specific heat at constant pressure and the specific heat at constant volume would be the same, there'd be no difference, but if you're dealing with either a gas or a liquid you will have expansion when there is heating and consequently you need to take that into account. So looking at specific heat at constant volume, we use capital C with a little subscript V, specific heat at constant volume is defined as being the partial derivative of the internal energy with respect to the temperature, and we'll put a little bracket here with the V denoting the fact that that is at constant volume. So what would our system look like? It may be a box that we have our fluid within and we're adding heat and then we're monitoring the temperature of that and waiting until it goes up by one degree Celsius. So you can see given that we have a fixed box there can be no change in the volume and that's why we call a constant volume. If you look at constant pressure or the specific heat at constant pressure, again we use capital C with a little P to denote the fact that we're looking at a specific heat at constant pressure. The definition for this is the partial of enthalpy with respect to temperature at constant pressure and we need to account for boundary work here. So just to show you a little schematic, so let's say this was the vessel with which we want to test our specific heat of some material in. I'll show the boundary here. So we have a gas or a liquid inside of this and what we're doing is we're injecting or adding heat and we're monitoring the temperature just like we were doing before so we have a thermocouple in there but what is happening here given that we want to have it at constant pressure so let's say we put some mass up here in order to create a load which maintains the pressure. What will happen as we add heat is our piston will lift off of the mechanical stops and when that happens we have boundary work so all of a sudden our system has increased in size and we need to account for the fact that we're doing a little bit of work in the process so that will have an impact upon the specific heat that you're measuring and that's why we have the two different forms of specific heat. Now with the specific heat at either constant pressure or constant volume we can use these to calculate change in enthalpy and change in internal energy and we can do this for ideal gases so that means that we're quite a ways away from the two phase region and the way that we can do this is the fact that both u internal energy and h enthalpy are functions of temperature alone and so what we can write we can change the partial derivative oops sorry that we had earlier into a total derivative so we can use this to integrate these equations on the right in order to calculate the change in internal energy or change in enthalpy and we can make a couple of approximations it depends upon the process that we're looking at and and notice that the specific heat at constant volume as well as specific heat at constant pressure are functions of temperature if the temperature change that we're dealing with is less than about 200 degrees C we can make an approximation by using the average specific heats if it goes beyond that then you need to use the tables that are in the back of your textbook and enabling you then to calculate the change in internal energy or enthalpy of an ideal gas so if the temperature changes more than 200 degrees for the process that we're looking at you need to use the tables at the back of your textbook if the problem that we're looking at has a smaller temperature differential so if delta T is less than 200 degrees Celsius for example then what we can write is the following approximation and that would be an approximation for the change in either internal energy or enthalpy and values of specific heat again there should be tables at the back of your book where you can estimate the values of specific heat for whatever gas you might be looking at a final comment that I want to make about the specific heats are a relationship with the ideal gas constant so we can write Cp is equal to Cv plus R your ideal gas constant and another thing is the ratio of specific heats these are things that we will use throughout this course especially the ratio of specific heats when we're looking at different processes and the ratio of specific heats will depend upon the gases that we're looking at but it is 1.667 for monatomic gases and 1.4 for diatomic gases and air and again you have to be careful with the fact that it changes they change with temperature because that can have an impact upon the ratio of specific heats one final thing if you're studying gas dynamics or fluid mechanics sometimes you'll see this expressed as gamma however in thermodynamics we use k we do not use gamma and so just be aware of that so please don't get confused as you go through working the problems and studying mechanical engineering thermodynamics