Practice: An engine operates with the following heat flow diagram. How much entropy is generated in each cycle?
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Concept #1: Introduction to the Second Law of Thermodynamics
Hey guys, in this video we're going to introduce the second law of thermodynamics in two forms known as the engine form and the refrigerator form. Alright let's get to it. Remember guys that real engines and real refrigerators have an efficiency. For an engine we call it the efficiency for a refrigerator we call it the coefficient performance but the efficiency is what determines how much work is going to be produced by the refrigerator sorry by the engine for a given heat flow from the hot to the cold reservoir. The coefficient of performance for a refrigerator determines how much heat is going to flow through the refrigerator against the normal flow for a given input of work.
Real engines can never have a 100 percent efficiency, real refrigerators can never have an infinite coefficient of performance and these are restricted by the first law sorry by the second law of thermodynamics. There are a lot of forms of the second law and we'll see a few of them but the first two very important forms are what we're going to discuss here. The one useful form is the engine form. The engine form of the second law says it's impossible for any cyclic system to convert heat absorbed from a reservoir completely into work. So this statement of the second law of thermodynamics is nothing new we already know this that no engine which does cyclic processes on gas, no cyclic system can entirely convert, can completely convert heat input from reservoir into work. That's impossible, we've already known this. Similarly there's a refrigerator form of the second law of thermodynamics which says that it's impossible for any system, this says specifically a cyclic system but this is for any system, it's impossible for system to transfer heat from a cold reservoir to a hot reservoir without work and this once again is nothing new we already know that this was impossible but there is a particular form of the second author of thermodynamics that says it's impossible. So why is it impossible? Because of the second law of thermodynamics specifically the refrigerator form. Both of these impossibilities can be described in terms of heat flow diagrams. We've seen these types of diagrams before we just didn't really name them but now I just got sick of writing diagrams that show the heat and whatever heat flow diagrams.
This is an impossible on the left, an impossible perfect engine. This engine takes however much heat is absorbed by it from the hot reservoir and converts it entirely into work with nothing sent to the cold reservoir. There's no heat that goes to the cold reservoir. This is absolutely impossible, it's a violation specifically of the engine form of the second law of thermodynamics. On the right we have a perfect refrigerator. This can transport heat from a cold reservoir into a hot reservoir without any work being put in. There's no work required in a perfect refrigerator. A refrigerator with a coefficient of performance of infinity and this is absolutely impossible given the refrigerator form of the second law of thermodynamics. So obviously from these two forms of the second law, we didn't really learn anything new we didn't gain any new insights into the underlying physics so we're going to remedy that with a more general form of the second law of thermodynamics but it's important to start here with the engine and refrigerator form all your books are going to show it your professors are probably talking about it in class so make sure that you know the refrigerator and the engine form. Alright guys that wraps up this introduction to the second law of thermodynamics. Thanks for watching.
Concept #2: Introduction to Entropy and the Second Law of Thermodynamics
Hey guys, in this video we're going to define a new energy-like quantity called entropy and formulate a single and concise second law of thermodynamics in terms of entropy. Alright let's get to it. We want to formulate a concise single form of the second law of thermodynamics. We saw previously the engine and the refrigerator form but that didn't really tell us anything new plus those forms of the second law of thermodynamics didn't explain why that was true, it just said this is true. The new form will actually fully explain both the engine and the refrigerator forms and it'll give us new insight into thermal processes and specifically why do they occur and under what circumstances can thermal processes occur?
In order to formulate the single concise form though we need to define an energy-like quantity called entropy now you've probably heard the word entropy before, it's used a lot, thrown around in a lot of everyday conversations. Sometimes people talk about entropy and time, sometimes people talk about entropy and at the state of the universe but it's very common to talk about entropy in terms of evolution and specifically how it pertains to the formulation of complex cellular machines inside of living organisms but that's like a biology problem we're not going to talk about that but we will talk about what exactly entropy is. The entropy of a system is an energy-like measurement, very important understand not energy, energy-like, that measures the amount of disorder in a system. This may seem very vague and that's because it is but we're going to continue evolving our definition of entropy as these videos go on so don't worry if it doesn't seem to be very precise yet but conceptually the best way to think about energy is it tells us how disordered a system is. It's very very important understand entropy is not an energy it's energy-like and it's energy-like because has very similar properties as energy for instance if you have two systems you haven let me draw this box a little bigger. You have system A, you have system B. System A has an internal energy UA and an entropy SA. Entropy is always given by a capital S. System B has UB and SB. If you bring them together into thermal contact right now they are not in thermal contact then you'll see that the new system A plus B has an internal energy of UA plus UB and it has an entropy of SA plus SB. So this ability to add entropy is when you combine two systems together is why it acts like an energy but it's only energy-like. The easiest way to see that it's not an energy is it has units of joules per Kelvin, it doesn't have units of joules so it's not an energy, units of joules per Kelvin energy-like. The more disorder that a system has the higher the entropy of that system and that's what it means to be dependent on the amount of disorder. If you have two states one that has more disorder whatever that is than the other states that will have more entropy and it turns out that the absolute entropy of a system isn't really all that important what I mean by absolute entropy is like this number SA and SB or this number SA plus SB. That value of the actual amount of entropy that a state has isn't really all that important what's important is the change in the entropy between two states. This is what the second law of thermodynamics concerns itself with it wants to talk about the change in entropy and this is actually pretty common with energy specifically potential energies. Physics doesn't care how much potential energy a system has, it only cares about the change in potential energy from one state to another. Now our concise singular form of the second law of thermodynamics says a system can only undergo processes that lead to no decrease in entropy. No system can undergo a process that decreases the entropy. You can also write delta S for a system can never be less than zero or the entropy, change in entropy for a system has to be greater than or equal to zero. The change in entropy at a constant temperature is a very important equation to remember. It's just the amount of heat that is exchanged during this process divided by the temperature that the process is occurring at. This only works for constant temperature processes which we know as isothermal processes. So only isothermal processes this is applied to. Now read the second law of thermodynamics very carefully. The second law is typically misstated as saying that the entropy has to increase, the second law doesn't say the entropy must increase, it just says the entropy cannot decrease. The best situation you can hope for is one that results in no change in entropy. These are called the isentropic processes by the way. Isentropic means the same entropy. It's weird because all the other ones that mean constant something like isothermal, isobaric, isochoric they all are iso this one's isentropic. There's no O it's probably because there's two vowels but still weird. For an engine there are three components to every system so there are three entropy changes. You have the hot reservoir, you have the cold reservoir which is often referred to as simply the environment and you have the engine itself. Those three components of the engine each contribute a change in entropy to the change in entropy of the system. The change in entropy for the system is going to be the sum of the change in entropies for each of the components of the system. This is another reason why entropy is always called energy-like because changes in energies for a system are always the sum of the change in energy of each of the components of that system so this equation looks just like the equation for the amount of work done for a system, the amount of heat transfer into or out of a system, the total change in internal energy for a system, etc. Lastly something that's very very important to remember about entropy is entropy is a state function just like internal energy is a state function. Changes for state functions depend only on the states, only on changes in the state and that means that changes in state functions are what we call path independent. Just like the internal energy doesn't, the changes in internal energy doesn't depend upon the path taken from an initial state to a final state, the changes in entropy also doesn't depend upon the path taken between an initial and a final state.
The change in entropy for an ideal gas is going to depend only on changes in temperature and volume or changes in pressure. These are interchangeable technically they're all interchangeable but rarely do you describe the state of a ideal gas without discussing the temperature. It's most common to always discuss the temperature and to discuss one more, either volume or pressure dealer's choice. It's usually volume. Let's do a quick example. This is something that we want to show. Where will heat flow if you connect a hot reservoir at TH to a cold reservoir at TC with some thermal conductor? So we have TH, we have TC. Now the heat is going to follow, I know it's going to flow out of the hot reservoir and into the cold reservoir so I'm going to use that fact to help me show why this is true so that's QH, that's QC and because of conservation of energy because energy cannot be created or destroyed however much heat leaves the hot reservoir has to enter the cold reservoir so I know that the amount of heat entering the cold reservoir will be a positive number because heat entering a system is always positive that's going to be the negative of the amount of heat leaving the hot reservoir. That hot reservoir is going to be a negative number and I'll just call them Q. They're both gonna have the same magnitude Q, the cold reservoir QC is going to be positive Q and QH is going to be negative Q. This states right here that QH is -Q, if we consider Q to be positive. So what I want to see is is this process allowed by the second law of thermodynamics and more specifically is the reverse process forbidden? So what I'm interested in is how the entropy changes for the system. This is going to be the change in entropy for the hot reservoir plus the change in entropy for the cold reservoir and remember that the change in entropy at constant temperature is Q over T. A reservoir by definition is something that can give out an infinite amount of energy or take in an infinite amount of energy without changing temperatures. So energy exchanges between reservoirs are never going to change the temperature. They're always at constant temperature so we can absolutely use this equation for reservoirs.
So what's the change in entropy for the system? Well what's Q for the reservoir? QH. What's T for the reservoir? TH plus QC over TC. Now I want to use this definition right here from the conservation of energy. So this is going to be -Q for the hot reservoir where Q is a positive number and positive Q for the cold reservoir. Now I want to combine the denominators that means that I have to multiply TC up, I have to multiply TH up and then the denominator becomes TH times TC. So this is -QTC over TH plus QTH over, sorry, this is THTC in the denominator. THTC, THTC. And now I can combine them based on their denominators. Notice that both of them share Q over THTC, the only thing that's different is this guy has -TC, this guy has positive TH that's the only difference. So this is going to be Q over THTC times TH minus TC. I just reordered the addition of the positive TH and the -TC. This by the second law, by second law, has to be greater than or equal to zero. When is this inequality greater than or equal to zero? That's only true if TH is greater than TC and I started this process out by assuming that this was the hot reservoir and this was the cold reservoir and I showed that heat can flow in this direction heat can absolutely flow from TH to TC if and only if TH is greater than TC so only if TH is truly the hot reservoir and TC is truly the cold reservoir. Notice that if I say that TH is less than TC then the number is negative. That violates the second law of thermodynamics. So you can never ever transport heat without any work being done because that changes the entropy. You can never transfer heat from the cold reservoir to the hot reservoir by itself, that violates the second law of thermodynamics. That's why the refrigerator form of the second law exists as it exists. The refrigerator form says that this cannot happen, that cannot happen that's what the refrigerator law the refrigerator form of the second law says and the reason is because of this result right here. Alright guys, that wraps up this introduction to entropy and specifically how it applies to the second law of thermodynamics. Thanks for watching guys.
Example #1: Why Do Perfect Engines Violate the Second Law?
Hey guys, lets do an example show that a perfect engine will in fact violate the second law of thermodynamics alright so here's a figure describing a perfect engine right the second law of thermodynamics says that the change in entropy for the system must be greater than 0 greater than or equal to 0 right that's the requirement for the second law where the change in entropy for the system is the change in entropy for the hot reservoir plus the change an entropy for the engine plus the change in entropy for the cold reservoir now right off the bat you only ever have a change in entropy when a heat is being transferred there is no heat transfer in the engine because however much heat it gains it loses, it loses either as work or loses as heat going into the cold reservoir so always the change an entropy for an engine is 0 another way that you can think about is engines are always what kind of processes, engines are always cyclic processes right what's the change in entropy for a cyclic process remember that entropy is a state function the change entropy for a cyclic process always zero because you start and end in the same state because you start in the same state the change in entropy is always going to be 0 for a cyclic process right that's because entropy is a state function so the only entropy changes that you are going to have in this case is the hot and the cold reservoir. So remembering our equation for the change in entropy at a constant temperature. If we say that the hot. Reservoir loses an amount of heat Q.H. which is what the diagram shows at the temperature H H sorry T H and the cold reservoir gains an amount of heat Q.C at a temperature T.C that's going to be the change in entropy for the system.
However in a perfect engine cold reservoir doesn't gain any heat right. That's the whole point of a perfect engine for a perfect engine whatever heat would normally go into the cold reservoir isn't there it can convert all of the heat into work what does this say though about the change in entropy for the system. That the change in entropy is negative and what is this, this violates the second law this violates the second law and therefore it cannot happen this perfect engine absolutely cannot exist because it violates the second law of thermodynamics the change in entropy for the system is negative and there is absolutely no way to make this positive unless some amount of heat is transferred to the cold reservoir. Alright guys that wraps up this problem. Thanks for watching.
Practice: An engine operates with the following heat flow diagram. How much entropy is generated in each cycle?
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