 Write down the heading, spontaneous and non-spontaneous process. So what is a spontaneous process? Spontaneous process is the process which happens on its own. We can say on its own or after a proper initiation. So in this, you write down spontaneous process. See, first of all, you understand what is spontaneous process. I'll give you the definition also of it. The spontaneous process are those process which happens on its own or after a proper initiation. For example, I have this pen in my hand, right? When I leave this pen, it will go down, right? It will fall down. It is a spontaneous process, right? Water flows down the hill. Spontaneous process, rusting of iron, spontaneous process, diffusion of gas from high pressure to low pressure, spontaneous process, right? Flow of current from high potential to low potential, spontaneous process, right? So these are the processes which happens on its own, right? Now, when you talk about the burning of fuel, right? It just need an initiation. You have to spark it. Once you spark it, it is done, it will burn on its own. It does not require a continuous support, correct? That is also a spontaneous process. Are you getting it? So spontaneous process are all those process which happens on its own or after a proper initiation. Means continuous support is not required. You just, you know, help the process to start with. Once it starts, it goes on its own. It does not require your or any external continuous support. Okay? Examples I have given you. I will just dictate the definition also I will write down. Now, you see one thing. This is the candle. Suppose we have it's burning, right? Now, if I cover this with a vessel, completely cover. What happens after some time? We have this candle is burning and when we cover this up properly, what happens after some time? Yes. Yes, the flame is extinguished, right? Yes, the candle goes off, correct? Why it is happening? Why it is happening? Because it requires for the burning of candle, it requires the continuous supply of oxygen. Yes. Once you've covered it up properly, then the supply of oxygen is not there. You cut off the supply of oxygen. So whatever oxygen was present inside it, it will get burned once it's over. The flame goes off, right? The fire extinguished, you know, the candle or the flame extinguished. Right? So why it happens? Because the burning of candle or the flame, it requires the continuous support of oxygen. Correct? And that is why, you know, this process is non-spontaneous process. Because it does not happen on its own or after an initiation. It requires continuous support of what? Support of any external support, you can say. Right? One very good example of this also you can consider for the understanding of the two processes, spontaneous and non-spontaneous. You must have heard boiling and evaporation, right? So tell me, I think you have a fair bit of understanding what is spontaneous, what is non-spontaneous, boiling and evaporation. Which one is spontaneous? Which one is non-spontaneous? Or both spontaneous, both non-spontaneous. Tell me. So evaporation is spontaneous. Evaporation is just a second, guys. Yeah. So, yeah. So boiling is non-spontaneous and the boiling happens at a fixed temperature. Like for water it is 100 degrees Celsius. But spontaneous happens at normal temperature, right? At room temperature. Correct? So you need to heat water and its vapor pressure when equals to atmospheric pressure, whatever the temperature it is, that temperature is called the boiling point of that particular liquid. Right? So boiling water requires continuous support of heat but evaporation requires nothing, correct? So evaporation is spontaneous, boiling is non-spontaneous. So you cannot remove sun, Gayatri. Right? It's something like you're talking about if I say this pen, if I leave, right? It will go down, correct? It will go down. And then you will say, sir, if I remove gravitation, then what happens? The process won't be spontaneous then. So there is something which is there, which is and everything is based on these universal facts. Okay. So you cannot remove the universal facts itself. Yes, Gayatri understood. Correct. So there's something that you need to take as it is. So all the theory based upon all these facts that we had already. Evaporation is spontaneous. That's right. Evaporation is spontaneous. No doubt. So I hope you understood the difference between the two processes. Let us write down the definition with you for this. Write down spontaneous process, a process which has, which has a natural tendency, natural tendency occur either by its own, either by its own or after, or after a proper initiation. Initiation, you know, let's be required here, maybe required proper initiation, right? Under a given set of condition, set of condition examples, write down flow of water down the hill, flow of current, high potential to low potential, high potential to low potential, diffusion of gas from high pressure to low pressure. Okay. Diffusion of gas from high pressure to low pressure burning of fuel, burning of fuel, it requires external support. Okay. Rusting of iron. All these are spontaneous process. Yes. Nuclear fusion reaction also you can consider spontaneous. Once it starts, it will go on. Yeah. That's right. Correct. So all these are examples we have. Correct. Of spontaneous process. Non-spontaneous process to write down. Write down. This is the process. This is the process, which neither occurs. This is the process. See, you are applying some condition here, right? Like humidity and all, but how do we define evaporation? Evaporation is just a process by which liquid converts into its vapor form on its own, right? At room temperature. So we are talking about that process. Correct. Now, if you're talking about high humidity, less humidity, or suppose, you know, in the atmosphere, if you keep the glass of water in vacuum, then obviously there it will be difficult. So condition, we are not applying. We're just, you know, trying to understand the term evaporation. What is the definition we have? And that process is spontaneous or not. That is what we are talking about. If you apply some condition, if there is some unexpected condition there, then it then that's a different case. But as far as evaporation is concerned, the definition of evaporation is what? It is a process of conversion of liquid into its vapor form at normal temperature, right? So that process is spontaneous. Okay. So right now, non-spontaneous process, a process which neither occur, on its own, a process which neither occurs on its own, nor by initiation. Neither occurs on its own, nor by initiation is called non-spontaneous process. Example, write down, flow of heat from, flow of heat from cold to hot body, diffusion of gas from low pressure to high pressure, diffusion of gas from low pressure to high pressure. So obviously these things are not possible on its own, right? But it does not mean the process is not possible at all. Correct. The process is possible. We can do some external work in order to get these kinds of processes, like we can push gas from low pressure area to high pressure area by applying pressure, right? That is a process, what we call it as, have you heard reverse osmosis or osmosis? So osmosis and reverse osmosis process are two different types of process, two different way to move, right? Opposite to osmosis. So both are possible. Both are happening. Osmosis is also possible. Reverse osmosis is also possible, right? So it does not like, you know, that non-spontaneous is not possible at all. It is possible, but it requires continuous external support. Candleball example, you see. Candleball example, if you continue, you know, providing oxygen, it will burn, right? It will never extinguish. Once it gets over, it's a different thing, but it will, you know, the flame will continue, right? So that's how the thing is, non-spontaneous process does not mean that process is not possible. If the continuous support is there, then process is possible there, okay? That's what you must keep in mind. Okay, now these are the two processes here. Now, why we are discussing this process here and what is the use of this, correct? So for your board exam and all, sometimes in comparative exam also, they will give you the four different options, like which of this statement is correct or wrong for spontaneous process or non-spontaneous process. So you must have the understanding of a true process. Then you can answer these kinds of questions, correct? In a school exam, they ask sometimes, you know, what is the difference between this spontaneous and non-spontaneous process, right? So all these points can help you. Now, why this process is happening? Okay, whether it is spontaneous and non-spontaneous process, why this process is happening, correct? What is the driving force for this kind of process? So what is the driving force for this kind of process? Right? There must be something which, you know, which is driving this process to take place. What is the actual reason, actual factor for this kind of processes? Like for example, you see, if you talk about some process which is exothermic in nature, right? The process which is exothermic in nature. Like I'll give you some examples you see here. Suppose I'll write down this reaction. Okay, listen to me carefully here. H2 gas plus half of O2 gas. Hydrogen-oxygen combines, forms H2O liquid. This delta H is negative. Delta S is negative. Means the process is exothermic. Is exothermic. The process is exothermic. Is exothermic. Spontaneous also. I'm giving you the example of spontaneous process. Spontaneous. If I take this example, carbon solid and O2 gas converts into CO2 gas. Delta H is again less than zero. Right? This one is also exothermic and the process is spontaneous. So spontaneous process means what? Exothermic process means what? That the energy is decreasing, right? Because amount of energy is releasing. So overall the energy of the system decreases. Right? So what we can say in exothermic process, the entire system is going towards the lesser energy state which we can say it is more stable. Isn't it? Can we say that? This is the process is exothermic. Right? Exothermic process. Hence we can say energy releases in this process and the entire system is going towards the lesser energy state. Isn't it? Right? And since it is going towards the lesser energy state, which is more stable state, hence we can say the process is spontaneous. It is going towards more stability. Right? Okay. So exothermic process to we can understand that the energy is decreasing and hence the stability is increasing. That's why it is a natural process. So till here the thing was fine because logically we can correlate things that stability is increasing. Hence it is spontaneous. After that the problem is there are some endothermic process also which are spontaneous in nature. Like for example you see, this one I will go to the next slide. You need to write down all these examples. Okay? There are some endothermic process, endothermic processes which are spontaneous. Which are spontaneous. For example you see, if you take ice from the freezer, after sometime what happens? It melts and converts into liquid. Is it a spontaneous process? Is it spontaneous? Conversion of ice into liquid? Yes it is spontaneous. It is taking heat from the surroundings and converting into ice. Happens on its own. So it is the endothermic process but what? But spontaneous. Similarly we will take the example of evaporation. We have discussed it. It is also spontaneous. One more reaction you see. CaCO3 calcium carbonate, solid interest. And when you heat this, it converts into CaO solid plus CO2 gas. So all these processes are spontaneous process. So now the problem is when we were talking about exothermic process that is understandable because the energy is overall decreasing and stability is increasing. Here the thing is exactly opposite. It is consuming energy and the energy of the system is increasing. So logically it should be less stable and the process should not be spontaneous but this is spontaneous process. Correct? So freezing requires low temperature, right? Yes, Gayatri? So if that temperature is not that low it won't... the liquid or whatever it is it won't get freeze. So you need to have that condition continuously. So if you talk in terms of spontaneity and non-spontaneity the process won't be spontaneous over there. Because we are considering for a spontaneous process we are considering normal condition basically. Not at extremely low temperature or extremely high temperature because minus below 0 degrees Celsius that temperature is not normally it's not there we need to have that arrangement to achieve those kind of those temperatures, right? Yes, normal condition we have. STP you can consider to some extent NTP you can consider basically under normal conditions. Okay? Yes, so here the point I am trying to make you understand is exothermic is fine clearly we were getting it why it is spontaneous but why endothermic process are spontaneous that we did not understand but since it is a spontaneous process and this information I am giving you so obviously you can understand this that delta H is still less than 0 for isothermic process delta H is not the only criteria for spontaneity of any process can we say that? Yes, tell me guys right? If delta H is the only criteria then this these processes should not be spontaneous process but since it is so obviously delta H is not the only criteria we have for spontaneity also for is spontaneity right? And that criteria is what is randomness have you heard this talk? Apart from enthalpy randomness is the another criteria we have randomness is the another criteria we have this means what the process always goes in that direction where the randomness is increasing randomness or randomness both are same thing actually right so everything is it is moving randomly in all directions there is no fixed pattern nothing is there that we call it as randomness or disorderness right down this point here any process has natural tendency to go towards maximum randomness to go towards maximum randomness okay correct that is the natural tendency we have that process will move towards more randomness now how do we measure randomness of any subsystem to measure randomness we we have given a new thermodynamic term that is entropy okay so listen to me first of all very very carefully here we have two criteria of spontaneity what are those criteria's what are those criteria's of spontaneity yes that is enthalpy delta H and randomness these are the two criteria we have for spontaneity how do we measure randomness to measure randomness a new thermodynamic term is given which we call it as entropy represented by S so entropy is the thermodynamic quantity which measures randomness of any system right down it is a thermodynamic state quantity right down it is a thermodynamic thermodynamic state quantity which measures randomness randomness or disorderness randomness or disorderness randomness or disorderness molecules of the system right next time it is an extensive property and state function the state quantity you have written already right now it is an extensive property since it is state quantity so change in entropy s is delta s equals to s final minus s initial this is a change in entropy unit of entropy is is calorie per Kelvin or joule per Kelvin the unit of entropy entropy also we say that it is a number of possible ways by which the energy is distributed within the system or once again I said it is the number of possible ways by which the energy is distributed in the system so if you talk about the entropy of gas because gaseous particles will have maximize or maximum random motion so entropy of gas is more than to that of liquid and entropy of liquid can move easily is more than to that of gas sorry gas liquid and solid here this is the order of entropy we have for solid liquid and gas okay the mathematical definition of entropy you write down mathematically it is defined as it is defined as as the integral of of all integral of all terms involved in involved in terms involved in reversible process with just a second all terms involved in in heat exchanged in heat exchanged divided by divided by the absolute temperature divided by the absolute temperature in a reversible isothermal process in a reversible isothermal process so here ds is equals to dq reversible by t heat exchange in reversible process so delta s equals to since it is reversible isothermal so temperature outside dq reversible this is the change in entropy we have copied this is the mathematical definition of entropy and you see if heat absorbed by the system by the system so what happens temperature increases entropy also increases temperature increases entropy also increases if heat is released by the system if heat is evolved once again heat is evolved then what happens temperature of the system decreases and hence entropy change also decreases less so keep that in mind we have two terms here for you know to understand the spontaneity of any process one is enthalpy other one is entropy this one entropy we use to measure the randomness or disorderness of the system right now you see for an spontaneous process in isolated system condition we are applying here okay for an spontaneous process spontaneous process in isolated system right since the system is isolated so there's no exchange there is no exchange of energy or matter energy or matter between system and surrounding between system and surroundings like hence delta s would be zero in this case but what happens here only suppose system is isolated and gases are mixing suppose we have a system here isolated system right no nothing is that there's no interaction between system and surroundings but this gases particles are there and these gases particles are mixing with each other right they're interacting and it's mixing with each other so under this process only if you have mixing of gases if we have mixing of gases then delta s total greater than zero because of gases it is mixing so these are some facts we have if the system is not isolated is not isolated then we can write the change in system entropy of the system total is equals to the change in entropy of you know system plus the change in entropy of surrounding I've written delta h here by mistake just let me correct it delta s total is equals to delta s of the system plus delta s of surroundings this is the relation if the system is not isolated next point write down during spontaneous process write down all of you first of all you copy this huh all of you copy this down during a spontaneous process the entropy of the system goes on increasing the entropy of the system goes on increasing till the system attains the equilibrium state till the system attains equilibrium state attains equilibrium state so at equilibrium next line so at equilibrium the entropy of the system is maximum so at equilibrium the entropy of the system is maximum and there is no further increase in entropy right I'm repeating so at equilibrium the entropy of the system is maximum and there is no further increase in entropy should I repeat this is again let me just check This is second try. Let me just finish this. I'll repeat the entire point again. So at equilibrium, the entropy of the system is maximum and there is no further increase in entropy. Okay, so I'm repeating from the beginning here. During a spontaneous process, the entropy of the system goes on increasing till the system attains the equilibrium state. Till the system attains the equilibrium state. So at equilibrium, the entropy of the system is maximum and there is no further increase in entropy. So what happens when there is exchange in energy between system and surroundings? So what we can say, if the entropy of system is increasing, then the entropy of surrounding will decrease. So for one, the entropy is increasing, for the one, the entropy is decreasing. So you will have an equilibrium state in between somewhere. So if you talk about the system whose entropy is increasing, its entropy will be maximum when the equilibrium is achieved. And at equilibrium, what we can write? The change in entropy is what? At equilibrium, delta S, at equilibrium, the change in entropy delta S equals to zero because this is the maximum value. More than this, entropy cannot increase. One more factual statement we have here, like for non-spontaneous process, the entropy change is negative. This also you must keep in mind. Now you see, we have different different conditions and in different different conditions, we'll find out the entropy change. So first one we have entropy change in isothermal reversible process, isothermal reversible process. What I am assuming here, I'm assuming suppose a system absorbs q amount of heat from surroundings at temperature T. So since the system is absorbing heat, so its change in entropy is positive plus q by T. This is for system, delta S for system is this. Equal amount of heat is being lose by surroundings, so it will be minus q by T. Right? System loses, surrounding loses heat and system gains heat. So it is minus, it is plus. What is the total entropy change? Delta S total is equals to system plus surroundings plus delta S of surrounding. This would be equals to zero. So delta S total is zero here. Total entropy change is zero in isothermal reversible process. So this is what we need to keep in mind. If it is reversible isothermal process, then entropy change is total entropy change is zero. Then, okay. Now, if you have reversible adiabatic process, then what happens? Write down heading, reversible adiabatic, reversible adiabatic process. Okay? For adiabatic process, we can say q is equals to zero. There is no heat exchange. When q is equals to zero, delta S of system is what? That is also equals to zero because this is q reversible by T. Right? So this is equals to also zero. Delta S of system is zero. Delta S of surroundings is also zero. Delta S total that is system plus surroundings equals to zero. This also you can keep in mind for a reversible adiabatic process. Again, delta S system zero, delta S surrounding zero, total delta S equals to zero. Copy it. Yes, done, guys. Are you getting it? All of you, respond? Yes, anyone there? What happened? Are you sleeping? Now you see the next third condition what we have. The entropy change in reversible process, in irreversible, I'm sorry, in irreversible process, irreversible process. I'm assuming system is at temperature T1 and surroundings is at temperature T2. See, guys, one thing you must understand here, we are just trying to understand the entropy change in different, different conditions. And one of these conditions you will get in the question in the test. Right? So what are the conditions we have? All those conditions we need to understand. And based on conditions, how we can apply in this concept of entropy, that is what we are doing here. So I'm assuming irreversible process, both system and surroundings at different, different temperature T1 and T2. What I'm assuming next, I am assuming T1 is greater than T2. You can also assume T2 greater than T1, anyone you can assume. You will get the same answer. Whatever you'll assume, you'll get the same answer. This is also our assumption we have. So I'm assuming system is at more temperature than surroundings. So obviously what happens? Which one will lose heat here? Which one will lose heat? System loses heat to the surroundings, right? Minus Q to the surrounding because system is at higher temperature, correct? Q amount of heat is released by the system. So can we write delta S of system equals to minus Q by the temperature of system that is T1, is it correct? Similarly, delta S of surroundings would be what? Surroundings receives heat Q amount by T2. And what is the total entropy change? Delta S total, that would be equals to this plus this plus Q by T2 minus Q by T1 and then we'll write Q T1 minus T2 by T1 T2, okay? Now you see, what is, we have this condition, right? T1 greater than T2. So what we can write? T1 minus T2 is greater than 0 from this condition. This means this term is greater than 0. So delta S total is also greater than 0. Means if you have an irreversible process, a total entropy change always increases. That is what the conclusion we have, clear? Done? Okay, done. Can you move on? Tell me? Yeah. Now, heading right down, calculation of change in entropy. Basically, we are going to have some formula of entropy and that is what you have to memorize and you will get questions based on this formula only, right? On calculation of, calculation of change in entropy, that is delta S. How do we do this? You see, we know the definition of entropy change. It is delta S is equals to DQ reversible by T, okay? If you calculate DQ from this, DQ reversible is equals to TDS, okay? DQ is equals to TDS. Now, from FLOT, from FLOT, what we can write? DQ is equals to DU plus DW. DQ is TDS is equals to DU is NCV DT and DW is PDV, okay? DS is equals to NCV DT by T plus, we can write P is NR T by V into DV into T, right? Any doubt in this, can I tell me? So, DS is equals to NCV DT by T plus NR DV by V. Tell me any doubt here? Till here in doubt, clear? Now, we can integrate it, very simple. We can integrate it and we will get the change in entropy for this, right? So, what formula we get here, you see? The formula we get here is delta S is equals to NCV ln T2 by T1 plus NR ln V2 by V1. This is the formula we have for entropy change if temperature and volume are variable. Any doubt in this? If temperature and volume are variable, then you can use this formula. I'll write down here. Yes, yes, sign you will get on your own. We always use sign convention in the data which is given. For the term which we need to find out, we won't put sign from our own side, right? Whatever answer comes, it will come with the sign which is required, positive negative, okay? So, this formula of entropy change will use for when temperature and pressure are variables, okay? Now, you see one more thing here. You can directly V2 by V1, you can substitute in terms of pressure and temperature. So, another formula that you get here is delta S is equals to NCP ln T2 by T1 plus NR ln P1 by V2. This is another formula we have which we can easily derive from the given one, right? And this formula we use when temperature and pressure is the variable. This two formula you need to memorize, then rest all conditions you can apply in this formula to find out the entropy change formula over here. Tell me how many of you understood this two formula? How did we get this? First one to be derived. How did we get the second one? Yes, right. So, what we need to do here, we can write down, I'll just write down short over here. We can write down P1 V1 by T1 is equals to P2 V2 by T2. So, V2 by V1 from this, you substitute here in terms of pressure and temperature. Then you use log formula, you'll get Cv plus R over here which becomes Cp over here. It should be a volume, yeah. So, mistake, it should be volume here. When temperature and volume are variable, correct. Yes, all of you understood this? Now, if you know this formula, you can derive other formula easily. Now, suppose we have the condition for isothermal process, for isothermal process. So, isothermal process, what we can write T1 is equals to T2, there's no change in temperature. So, this becomes ln1, this becomes ln1 which is 0. This term and this term you have to ignore. You have to eliminate, remove. So, the formula of delta S would be, in this formula, if you remove this, it becomes nR ln V2 by V1 in terms of volume is equals to nR ln, what we can write? P1 by P2. Tell me any confusion in this? You'll get direct question on this, all the data will be given just need to substitute and get the answer. Any doubt here? Now, for isomeric process, what we can do? Look at the first expression, correct. Look at the first expression, delta S is equals to nCv ln T2 by T1 right and pressure is constant here. So, we can write V1 by T1 is equals to V2 by T2 for isomeric process. So, what is T2 by T1? T2 by T1 is nothing but V2 by V1. So, another expression is equals to nCv ln V2 by V1. This is the formula. Is it clear? Any doubt in this? In the first expression, we remove the volume term, we get this and temperature we convert into volume since the pressure is constant by this expression. So, how many formula we have done? We have done here 1, 2 isothermal isomeric. If it is isochoric, then what happens? Volume constant. So, first term, just a second, we have isomeric. Isomeric is constant, so this is 0, nCp ln T2 by T1. I think I have done a mistake over nCp ln T2 by T1. Isomeric pressure is constant. So, this term will be 0 by mistake. I have taken the first term. This won't be there. This expression will be there. So, nCp ln T2 by T1 delta S equals to nCp ln T2 by T1. And T2 by T1, we can write this as V2 by V1, nCp ln V2 by V1. This is the expression we have. If the process is isochoric, then this volume term would not be there. This volume term won't be there. So, it is nCv ln T2 by T1, nCv ln T2 by T1. So, delta S equals to nCv ln T2 by T1. And this T2 by T1, we can apply in terms of pressure. Volume is constant. So, pressure and temperature directly proportional. So, nCv ln T2 by T1. This is the formula. Just we are substituting nothing much with the help of gas law. Tell me it's fine. Now, let's see the next condition we have. Mixing of ideal gases. Suppose A and B, two gases are mixing. You can have more than two also. I am assuming two over here. So, delta S of the mixture is equals to minus 2.303R na log of xA plus nB log of xB. Where na is the number of moles of A, xA is the mole fraction of A. So, n is simply, I'll write down number of moles. x is the mole fraction. Correct? If you have C, another guess also. We write down nC plus log xC and so on will go. The last three formula we have here, the important one on this, they have asked question also in when there is a phase change. So, entropy change during phase change. So, delta S of fusion. What is fusion? Fusion is the process in which solid converts to liquid. Correct? Solid to liquid. Then whatever energy involved in this process, delta H of fusion divided by the temperature at which the fusion is taking place. Right? That is a fusion temperature. Right? Similarly, delta S of vaporization, liquid to vapor. Again, same formula, delta H involved in the process of vaporization divided by the temperature which is the boiling point of the liquid. Because vaporization takes place at boiling point in general. Delta S of sublimation. What is sublimation? It is solid to vapor. So, whatever, again the enthalpy exchange in this process divided by the temperature at which the sublimation is taking place. Right? So, this is the three formula we have during phase change. The first one on this, the question has been asked in JEE. All the data will be given. You just need to substitute and get the answer. Copy this down. Done? Yeah. So, on these basics, changing and property formula, you will get numericals. Okay, that you need to solve. We will see some questions here. Try this. Yeah. Did you get the answer? Tell me. 19.4 you are getting. That is close. It is right. Okay? Approximately 20. Yeah. So, basically this question, this kind of question is to write down the data given and use the formula of entropy. Right? So, you see in this question, entropy change is given, temperature change is given, volume change is given. It means delta S formula, you need to write down in terms of temperature and volume. That is Ncv ln of T2 by T1 plus Nr ln V2 by V1. If you get confused, you just, you have to do this. Suppose formula, you forgot in the exam, what you need to do, you see. You know, first law of thermodynamics, dq is equals to du plus dw, right? From here, you can get anything. dq is Tds, Ncv dt is du, it is pdv is dw. Just you divide this by T, Ncv dt by T, p is Nrt by V. So, it is Nr dv by V. Then you have to integrate this. You will understand that what would be here. Either it is T1 by T2 or T2 by T1, you will understand. Okay? There are two steps you need to do. So, let me just remove this, not required. Basically, you understand from where, which expression we derived this particular formula. There are two steps you need to write, and it is done. So, Ncv, it is 3 mole. Cv is 12.5. Take care of units in this kind of questions. For temperature, you do not have to bother because it is the ratio. So, it is 320 by 300. 320 by 300 plus 3 into r is 8.314. Don't take it in 0.021 into ln. V2 by V1 is 10 by 5. Just you need to solve this approximately. The answer you would get 19.71 joule per Kelvin. Without calculator, it is a bit difficult to solve ln value. See, you can do two, three things. If there is complex expression, the values of antilog will be given in the question. So, that value you have to use. Antilog value will be given. If not, then you should know the value of these three things. That is a log 2 value you should know, 0.30. Log 3 is 0.48 approximately, and log 5 is 0.70 approximately. So, with these three values, if you know, you can do some, you can use some mathematical tools, some mathematical formula, and you can find out the answer, the expression you will get in such a way, the data will be given in the question in such a way, so that you can calculate with the help of these three values. If not, then the value would be given in the question itself. Yeah, understood. Sometimes what happens, here you see, what is, you need to take care over here because you will get this kind of question. I don't want to, you know, just give you the formula, substitute, and calculate, and tell me the answer. You can do it on your own after the class. What is the key point that you need to understand over here, that sometimes what happens here, they will give kilojoule. Okay. So, R value you are taking in joule here, you see, it is joule per mol kelvin. So, either you convert this joule into kilojoule here, or this kilojoule into joule you need to convert. So, this kind of, you know, alertness you must have while solving these kind of questions, because there only you can make some mistakes, if you do not take care of units. Correct, understood. See, one more thing you must understand over here. Sometimes they won't ask you to find out the absolute value like this. Okay. But in the options, expression will be given in and dropping. Like, suppose this obviously you can substitute in terms of CP, right? CP minus CV is equals to R. So, CV you can substitute in terms of CP. And then you can multiply and take this NR common, because what happens? CV is equals to what we can write. We have CP minus CV is equals to R. So, CV is equals to our CP minus R. So, if the expression is given in terms of CP, then this CV here, you can substitute as CP minus R. So, this would be CP minus R over here. Then you multiply from N ln T2 by T1 to this term. And then you see this NR, this NR you can take common and there you will get a different kind of expression probably in terms of pressure you may get. Okay. So, that also you can do, or they can give you the expression that way in the option. So, you can do that once you know the relation. Got it? Clear? Understood? Okay. Fine. So, we have done with this. Okay. Next, we need to start with second law of thermodynamics. Okay. So, this part probably will finish today. Second law of thermodynamics we need to start. That we'll do after the break. Okay. So, we'll take a break now. We'll resume at 6.30. Take a break, guys.