 Now, please write down isochoric. How does it look in PV graph parallel to y axis parallel to y axis. So this is P, this is V and the graph will look something like this. Okay, we are going from point one. So let us say point two. What is happening? Is it compression or expansion? None of them. None of them. Volume is constant. There is no compression or expansion. Okay, let's say at point one pressure is P1 and at point two pressure is P2. So this is P1 and P2. So let's say T1, T2 is also given to you and Cp and Cv are given to you. Then delta Q you have to write down and W you have to write down. W will be zero. Delta Q will be and Cv delta T constant volume specific you have to use. Okay, simple. Now, this is isochoric process. Next we are going to talk about isothermal process. By the way, every process has some process equation as well. What I mean to say is that there is a path. So there is a description of the path also. For example, for isobaric, you can say the process equation is pressure is constant, or you can say that volume by temperature is constant. Okay, so this is the process equation. P constant and V by T is constant at every location. Okay, similarly for isochoric process we have volume constant or I can say that at every location pressure by temperature is constant. Alright, so this you should know and from where P by T constant comes because Pv is equal to nrt, P into V is nrt. Basically V is equal to nrt by P. So if V is constant, T by P should be constant. And if T by P is constant, P by T is also constant. Okay, write down isothermal process. So we will try to plot that. We are all the time drawing PV graph only, but it need not be PV graph. It could be P and T, it could be volume and temperature, it could be between any two macro variables. So don't worry if the diagram is something else other than PV diagram. So you can expect other diagrams as well at times. So isothermal process has a process in which temperature is constant. What else I can say? T is constant and P into V is constant. Okay. Because PV is equal to nrt. So if T is constant, P into V is also constant. Now if pressure into volume is a constant, the PV graph will look like this. It will be a curve. It is basically a formula of rectangular hyperbola. So I don't know whether you have learned that in mathematics. So if this is x axis, this is y axis. So the equation of x into y is constant. This graph will be a rectangular hyperbola like this. Same way isothermal process looks like this. Okay. Now over here, I am trying to find out the different values of delta Q and W. Now tell me what is delta Q? Anyone? It is a thermal process. Temperature doesn't change. Delta Q is zero. All of you agree? Temperature of 0.1 is equal to temperature of 0.2. Is the heat absorbed zero? Is delta Q zero in this process? In isothermal process is delta Q zero. This take this pole is delta Q zero. All right. The answer is delta Q is not zero. What is done? How you written midterm exam guys? Anyways, see, why I am telling Nariman, hold your horses. So in the isochoric process, the delta Q is NCV delta T. In isobaric process, it is NCV delta T. But we don't know the specific heat capacity for isothermal process. But subconsciously we will be like, okay, delta T should be there. If delta T is there, then only heat will be supplied. Okay. Okay. What if whatever heat you supply, entire heat got converted into work. If entire heat got converted into work, internal energy doesn't change. Right. If you convert somehow entire energy into work, then internal energy of the gas will not change. Because of that, the kinetic energy of the molecule doesn't change and hence the temperature doesn't change. Or the internal energy doesn't change. Right. One and the same thing. Okay. So delta Q, we don't know how much it is. But whatever it is, if we use the first law of thermodynamics in isothermal process, the value of delta U will be zero or not. Yes. Right. Just give me one second. I'll open the door. Okay. Sorry for that. So what I was asking, will delta U be zero in isothermal process? Yes. Right. Delta U will always be NCV delta T only. Because of that, since delta T is zero, delta U always will be equal to zero. So whatever is a delta Q has to be equal to work done only according to first law of thermodynamics. Okay. Now, I may not have a direct formula of delta Q. So what I'll do is I will find out the work that instead work done is integral of PDV does not matter what is the process. So I'm going to use this formula itself to find out the work done between point one and point two. And then whatever comes, I'll say that is equal to the heat supplied also. Okay. So let's say the volume at point number one is V1 and volume at point number two is V2. Can you attempt and get the answer for the work done over here? The temperature of this process is let's say T. Isothermal process. So entire process temperature is T only. Try deriving it. I know it is written on the book. Derive yourself, all of you. See, right now, when you write PDV integral, the problem right now is that you have two variables, pressure as well as volume. Okay. So you cannot integrate it as such. You need to reduce it to a single variable. Okay. How can you reduce it to a single variable? We know that PV is equal to nRT. This is a state equation which is valid between the state variable or among the state variable at a particular this thing. So pressure can be written as nRT divided by V. A good thing about writing like this is that temperature is constant. So I'm basically writing pressure in terms of volume. So I'm reducing number of variables in this expression. So work done will be equal to integral of nRT by V dV. Okay. And when I'm going from point one to point two, my volume will be from V1 to V2 like this. So work done will be equal to what nRT comes out of the integral because it is a constant from V1 to V2 integral dV by V. And integral dV by V is natural log lnV, V1 to V2. So this can be written as nRT lnV2 minus lnV1. Okay. So work done will be equal to nRT lnV2 by V1. Okay. So this is a derivation of the work done in isothermal process. Okay. If you want to convert it to the natural log, work done will be equal to 2.303 nRT. If you want to convert it to log the base 10, sorry. So 2.303 nRT log to the base 10 V2 by V1. Same thing. This is natural log to the base E. Okay. Now tell me if you want to write down the same formula in terms of pressure. Let's say pressure at 0.1 is P1 and at 0.2 it is P2. Can I write the same expression in terms of pressure? It will be P1 by P2 instead of V2 by V1. Correct. So I know that P1 V1 is equal to P2 V2 as it is an isothermal process. So I will get P1 by P2 equal to V2 by V1. So same work done I can write down in terms of the pressure also nRT ln now P1 by P2. Okay. Now clearly the work done will be positive. If V2 is more than V1, then the log will be more log of the thing which is more than one will be positive. And work done will be negative if V2 is less than V1. Okay. So when final volume is more, the work is done by the system if final volume is less work is done on the system. So that is something which you can visualize also. Mathematically you don't need to find that out. Anyways, is it clear to all of you? So this is the work done and this is also the heat supplied. But if work done is negative, what can I say about the heat supplied? If work done is negative, what can I say about heat supplied? Sir, is it heat released? Correct. Then it is heat released. If work done is positive, then it is heat absorbed. Okay. So we have got delta Q is equal to W. So they are equal not only in terms of magnitudes but also in signs as well. So if W is negative, delta Q is also negative. If W is positive, delta Q is also positive. So like that.