Welcome, this lecture is going to cover some supplemental material on entropy and the second law of thermodynamics. We covered some very quantitative analysis using the second law in our previous lectures. And several people have shown interest in trying to understand more about the theory and maybe some of the conceptual interpretations as opposed to the quantitative or analytical interpretations. So in this lecture, we are going to play with a couple of fun proofs that are associated with the second law of thermodynamics and we will discuss how you might relate entropy in the second law of thermodynamics to the practical processes that you see around you every day. We know that the second law of the thermodynamics establishes entropy as a thermodynamic property. We use that in our analysis of processes and devices. We also know the second law is used to determine the best or ideal performance that we can expect from a device or process. We understand that ideal or best performance is not achievable, so this tells us things like the maximum amount of work we can take from a turbine or from a process. Or the minimum amount of work that's required to pump a fluid, we also know the second law to tell us the prediction for the direction of processes. The first law of thermodynamics for the conservation of energy says, whether or not the process is allowed, but it doesn't tell us which direction it will go in. And we're going to talk about that more in just a couple of moments. It's the second law that tells us the intuition, that supports our intuition for how processes, what direction processes occur in. We also use the second law to evaluate and quantify the effects of irreversibility, and that's really what we practiced in the class. We said, how much work can I expect out of a turbine? And if the turbine isn't perfect, what's the effect of those irreversibilities? We also use the second law to establish conditions for equilibrium. And again, we'll discuss that qualitatively in this lecture. And finally, the second law of thermodynamics allows us to define is a process reversible or irreversible? So what's the practical interpretation of the property of entropy, and the process term, the generation of entropy? That's a question that we saw quite a bit in the forum discussions, and that is a really difficult question to answer. Temperature and pressure are very tactile thermodynamic properties. They're good, tangible examples. We know if the temperature is high, we understand what that means. If the pressure is low, we have a very good physical and intuitive understanding of what that means. Entropy doesn't really relate that closely to things that we can visualize, so it makes it very abstract and very difficult to say, what is entropy? Well, this maybe something that will help you get a handle on understanding entropy. It's a measure of disorder in the system. The generation of entropy is the measure of irreversibilities that occur in a process. So we have entropy which is a thermodynamic property and then we have the generation of entropy which is associated with the process. So if entropy measures this order that change in entropy is associated with the irreversibilities that are present in the system. And that's what the second law tells us particularly when we write it down in equation form. So let's think about a reversible process which is defined as a process that can occur and then be reversed, and the system is returned to its exact original condition without changing the surroundings of the system. So again, let's go through that. A reversible process with a reversible system. So essentially I can start and have some process happen, and that process can be entirely reversed where the system and the surroundings can be return to their original conditions. That's an ideal process, reversible process is ideal and can not be achieved in real life. So now let's think of some practical examples of irreversible systems. Take for example the breaks that you have on your bicycle or your car. When we apply the breaks, and that's a friction process, where the breaks grab tire, grab the rims of the wheels, those wheels are heated by friction. And we can't return the wheels or the brakes to their original condition. Friction causes heating within the wheel and within the brakes. We can cause some damage for example, physical deformation of the materials. We can't return the process to its original state. That process, that real process is irreversible. And the amount of friction heating is an indication of the magnitude of the entropy generation. If we could measure the entropy of the breaks before and after that process, we would be able to quantify the amount of entropy that was generated by that breaking process. Now our observations tell us something like the second law of thermodynamics should exist. So we're going to have a conceptual introduction to the second law of thermodynamics right now. So we're going to consider three systems. One where we have a hot and a cold material. That are placed in contact with each other initially. And we know over time that this system will equilibrate to the same temperature. And that's thermal equilibrium. We can consider a high pressure and a vacuum system that let's say, are initially separated by some wall or a membrane. So we have high pressure on one side and we have low pressure on another side. Then we remove that barrier, the membrane between these two fluids and we know that those fluids will mix and now end up with the system which has uniform pressure. And that's mechanical equilibrium. And then in this last system, we could consider a box where we have oxygen on one side, again separated by a wall from nitrogen. And if we remove that wall, we know that the gases will mix. And I'll have a uniform mixture. Of gases. We know intuitively from observation that for everyone of these systems, they perceive from the system initial condition described on the left to the final condition described on the right. We know intuitively that we can't take a bundle of materials and have them spontaneously separate into a hot group of materials and a cold group of materials. Conversely or similarly I should say, we can't take a fluid that's at a uniform pressure and then have it stratify itself into a low pressure section and then a high pressure section. And we can't unmix gases spontaneously. Let me be careful here. We could accomplish any one of these processes from right to left, if we put energy into the system. But we have never seen that occur spontaneously. So what's key here is that we know that spontaneously the system will equilibrate. Now, the conservation of energy tells me either direction is perfectly possible. It's the second law of thermodynamics that says, no, it's this direction that matters. So I want to spend a minute or two just discussing and presenting some proofs for entropy, and the second law of thermodynamics that can maybe whet your appetite for all of the interesting conversations you can have with entropy. Once classic example that is described in thermodynamics literature is that the entropy of the universe is always increasing. So for this little proof what we'll start with is this extremely large system which will refer to as an isolated system. So we'll define this very specifically. So our isolated system consists of a system. Let's make it a really tiny heat engine which we've discussed in the lectures. So here's my very, very tiny heat engine with a high temperature source and a low temperature source. So that's my system, and it's surrounded by some environment. And together we're going to consider the system and the surroundings as an isolated system. And no energy transfer crosses my isolated system boundaries. So there's no energy transfer. So while there's energy transfer within my system, the system is within the boundaries. And so all of this energy transfer is internal to the overall isolated system. So if we apply the conservation of energy to the system. Again, no mass transfer, no energy transfer, we have the change in energy of the isolated system has to be equal to the net heat transfer and the net work transfer of the isolated system, except I just said, there's no energy transfer. So these are equal to 0. So it says hey, the change in energy of this isolated system has to be equal to 0, the energy has to be conserved within the isolated system. So now, we look at what the second law tells us, the variation of entropy. And we have the change in the entropy of the isolated system has to be balanced by the heat transfer of the isolated system across the boundaries plus the generation of entropy that's caused by the process. So again, we've stipulated there's no heat transfer, no work transfer, no energy transfer. So that has to be identically equal to 0. So what we have is the change in entropy of the isolated system which consists of that little system. My little power plant that I have within this plus the change in energy of the surroundings, the change in entropy of the surroundings and that has to be identically equal to the amount of entropy that is generated by this interaction. And we know that this term from the statement of the second law has to be greater than or equal to 0. So if we look at this system from this perspective, the entropy in the isolated system always has to be increasing. So this is where we hear conversations about the entropy of the universe is always increasing. Because we can imagine this isolated system, or the universe, where we've drawn the boundaries so large that all the energy transfer mechanisms are entirely internal, that we end up with an isolated system. And then the second law says that has to lead to, in a real world it has to lead to an increase in the entropy of the universe. So that's kind of a fun quick little proof. And it shows you some of somewhat esoteric but very interesting conversation you can have using the second law statements. So let's consider another one, in this example what I want to consider is a process. Which we're going to look at on any old state diagram will do, but we'll just choose a PV diagram, just to give us a reference frame. And we'll consider an initial state one and a final state two. And we're going to consider three process paths. Process path A, which goes from 1 to 2, which is internally reversible. Process path B, which we will also consider internally reversible, and it also goes from 1 to 2. And then we'll consider sorry, that's process path C, which is also internally reversible, but the direction is now from 2 to 1. There are many different prose descriptions of the second law of thermodynamics. And most of those prose statements have mathematical definitions as well. The one we want to invoke for our discussion is referred to as the Clausius Inequality Statement of the second law. So this is called Clausius Inequality. And the Clausius Inequality says. That the heat transfer. The net heat transfer and this circle over the integral means, it's the net heat transfer throughout all boundaries and throughout the entire cycle has to be less than or equal to 0. So I'm just going to stipulate the Clausius inequality has been proven. And we're going to use this definition to show that the property of entropy has to exist based on these little process examples that are shown in my pv diagram. So again, I'm not going to prove the Clausius Inequality. Although there are some really neat discussions that show this has to be true. We're just going to invoke this definition for my discussion of why property entropy has to exist. So if we look at this process here, I can take path A and combine it with path C and I can make a cycle. So I have for that state then I can apply the Clausius Inequality for that cycle. So I have path A from 1 to 2 plus the return path which is from 2 to 1. For path C and that is equal to 0, because I've already stipulated in my little proof here that all processed paths are internally reversible. So we know that that equal to 0, otherwise the statement would have to be less than 0. If the processes weren't internally reversible. I can create another cycle, where I use process path B combined with return path. C. And again, because the system is internally reversible, we know that this has to be an equality sign on the right hand side from the definition of the Clausius Inequality. I'm going to take these two cycles and mathematically, I'm just going to sum the two or excuse me, difference the two. So I'm going to take equation 1, and this is equation 2. And I'm going to take equation 1 minus equation 2. And what I'm left with is A and B are still present. The integrals. But these two integrals are identically equal, and so they cancel. So 0, that's the 2 integrals cancelling identically to be 0. And then the right hand side is equal to 0. So if I look at this, what this equation now says to me is that I can take a process variable, we've talked about this in the class. The difference between a variable that's associated with a process path versus a variable which is associated with a state. And this tells me. This equation tells me wait a minute, I can take any arbitrary path that's internally reversible between states 1 and 2. And I end up with identically equal values for this integrals. That's the very fundamental definition of a property. A property only cares about the endpoints, not about the process path. So the Clausius Inequalities tells us there has to be some property associated with states 1 and 2 that are identically described by this integral. So it says to us hey, there has to be some property, the change in some property let me be very precise here, that is described by this integral value. So what we're seeing is essentially the justification for the existence of the thermodynamic property entropy. So with that, I hope these couple of examples that are fairly esoteric and very, very much based on the theory of the second law of thermodynamics give you some ideas of areas that you might be interested in exploring. It's very difficult to get a physical and tangible, kind of tactile understanding of entropy. But these examples may give you some ideas of concepts where you can apply and understand entropy. For example, a really fun thing to think about is if you drop a rubber ball, we know that the arc of that ball as it rebounds as a function of time will get smaller and smaller and smaller. It's the second law that tells us why does it get smaller and smaller. If we drop a ball it will never spontaneously bounce higher and higher as a function of time. That would be a violation of the second law. So I'd encourage you to think about these subjects. And explore, and there's a lot of good literature on both the proofs and the concepts, and the equalities and inequalities, and all sorts of outcomes of the second law of thermodynamics. Much more than what we've discussed here, or that we've discussed when we were doing second law analysis of devices. There's a huge amount of material and I encourage you to explore that.
