 Welcome back, let us continue with our study of the ideal gas. Now we should remember that any ideal gas was characteristic with Obey's-Boil's law all over its state space. Combine this with our definition of the ideal gas Kelvin scale of temperature and you get its equation of state which can either be written down as pv equals mrt which also equals n into universal gas constant into T. We should also remember that because of the relationship between mass and moles, there is a relationship between the gas constant R and the universal gas constant R u and that is R equals R u divided by molecular weight. So if you know R u which is a famous number 8.314 kilojoule per kilo mole Kelvin and divided by the molecular weight of that particular gas, we will get a gas constant for that gas. The third characteristic which we have not seen so far is that any ideal gas obeys Joule's law. Just the way Boil experimented with the pvT characteristic of the equation of state of an ideal gas, Joule experimented with the thermal energy of an ideal gas and he discovered that for an ideal gas, any ideal gas the thermal energy we will write it as specific thermal energy is a function only of temperature. Notice that Joule's law is a restrictive law because any gas is a fluid system, we will require two properties to define any other property. So Boil's law and equation of state tell us that T and V define temperature or T and V define pressure. So even internal energy should be specified in terms of two properties. It could be pressure and volume or temperature and volume or could be pressure and temperature. However it turns out that it can be expressed always as a function only of temperature, you need not have the second variable which could either be pressure or volume. As a consequence of this, let us say the corollary of this is when we define H called enthalpy say derived property it is defined as U plus the product of pressure and volume. Let us work in terms of specific enthalpy, divide this definition throughout by the mass of our system and we get H is U plus pV. Now consider an ideal gas. What happens for an ideal gas? Let us see on the right hand side. U is a function only of temperature because of Joule's law and what is pV? pV by Boil's law and equation of state is only RT. So the first term is a function only of temperature, the second term is a function only of temperature. Hence it turns out that for an ideal gas H also is a function only of temperature. Now let us plot for a typical ideal gas the specific thermal energy as a function of temperature and the specific enthalpy as a function of temperature. Since there is no effect of pressure or volume, this is a curve which is almost a straight line. Similarly H is also a curve which is almost a straight line and because of this the fact that the variation of U with temperature is a reasonably accurate straight line over a reasonably wide range of temperature and the H T behavior is also similar. It is important for us to look at the slope that becomes interesting, slope of the U against T curve and slope of the H against T curve. This slope we can write as dU by dt and this slope we can write as dH by dt. This slope and this slope here give us the definition of specific heats. The general definition of specific heats is the specific heat at constant volume is defined as the variation of U with respect to T at constant volume and the specific heat at constant pressure is defined as the variation of H with respect to T at constant pressure. Why is it constant volume here? Why is it constant pressure here that we will appreciate later when we consider properties of various fluids that will occur quite sometime later after our study of the second law of thermodynamics. One thing to note is that the name here specific heat at constant volume and the name here specific heat at constant pressure are historical terms which we continue to use. Remember the most important thing to note that although they are called specific heats they are just derivatives of one property with respect to another. They have no direct relation with the heat interaction. It is only for historical reasons that we continue to use the term specific heat. Now coming back to our ideal gas characteristic, it turns out that for an ideal gas, any ideal gas since U is a function only of temperature and H is a function only of temperature, Cv turns out to be du by dt. We do not have to say constant volume because U depends only on temperature and Cp turns out to be the ordinary derivative of H with respect to T because H is a function only of temperature. To emphasize this we may even write this is du by dt where U is a function only of temperature and dH by dt where H is a function only of temperature. Finally we have a still simpler approximation. If we notice what we said that the U against T characteristic is almost a straight line. Similarly the H against T characteristic is almost a straight line. So let us further approximate and let us consider an ideal gas where these two behaviors variation of U with T and variation of H with T are exact straight lines and that would mean that our specific heats Cv and Cp would be constant because the slope is the same throughout for U as well as for H and that gives us a simpler approximation an ideal gas with constant specific heats. The advantage of this is for such a gas Cv is some constant Cp is also some other constant and because of this if we consider two states say state 1 and state 2 in terms of say T1, T1, V1, T2, T2, V2. Notice that any two of these say T and P are sufficient to specify the state because if we specify T and P the specific volume will come out of the equation of state. Then it is very easy to determine U2 minus U1 which will be delta U from 1 to 2. This in the general case will be integral Cv dT from state 2 state 1 to state 2 but because now Cv is a constant this can be written down as Cv into T2 minus T1 which will be Cv into delta T from 1 to 2. In a similar fashion delta H the change in enthalpy from state 1 to state 2 which will be H2 minus H1 which will be in the general case integral from 1 to 2 Cp dT this will be simply written down as Cp into T2 minus T1 which will be Cp delta T 1 to 2. We should notice that this part is true for any ideal gas whereas the simplification on the right is true for an ideal gas with constant Cp Cv that is constant specific heats. Finally, let us look at some other relation and definitions for an ideal gas. Let us go back to our definition of enthalpy. Enthalpy is defined as U plus Pv. So, for an ideal gas this turns out to be H of T is U of T plus RT. It does not matter whether the ideal gas here at constant specific heats or not. Now let us differentiate this equation with respect to temperature that is easy to do because each and every term here is a function only of temperature and you will get dH by dT is du by dT plus R and using the relation that for an ideal gas dH by dT will be Cp and du by dT will be Cv. We can write that Cp equals Cv plus R remember that this is only for an ideal gas. You should also notice that Cp Cv R all have the same units. These would be joule per kilogram Kelvin or kilojoule per kilogram Kelvin and finally the definition. Since Cp and Cv have the same units, we quite often find it useful to define its ratio. This ratio is known as the ratio of specific heats. Common symbol is the Greek letter gamma and it is defined simply as Cp by Cv. Thank you.