Proof: Volume ratios in a Carnot cycle

The goal in this video is to essentially prove a pretty simple result. And that's that the ratio between the volumes-- let me write this down-- that the ratio between the volume at state B and the volume at state A-- so the ratio of that volume to that volume-- is equal to, in our Carnot cycle, is equal to the ratio between the volume at state C. So this volume and that volume. So volume at C to the volume at D. So this is what I'm about to embark to prove. A fairly simple result, that maybe is even, if you look at this, that looks about right. So if you're happy just knowing that, you don't have to watch the rest of the video. But if you are curious how we get there, I encourage you to watch it, although it gets a little bit math-y. But I think the fun part about it, even, it will one, satisfy you. That this is true. But the other thing is, we'll be able to study adiabatic processes a little bit more. So just kind of launching off of that, the whole proof revolves around to this step right here and this step right here. when we go from D to A. So by definition, an adiabatic process is one where there's no transfer of heat. So our heat transfer in an adiabatic process is 0. So if we go back to our original definitions-- let me show you that here. Right here, at the step and this step, we have no transfer of heat. So if we go back to there. Adiabatic-- we're completely isolated from the rest of the world. So there's nothing to transfer heat to or from. So if we go to our definition, almost, or our first law of thermodynamics, we know that the change in internal energy is equal to the heat applied to the system minus the work done by the system. And the work done by the system is equal to the pressure of the system times some change in volume. At least, maybe it's a very small change in volume, while the pressure is constant. But if we're doing a quasi-static process, we can write this. Pressure, you can view it as kind of constant for that very small change in volume. So that's what we have there. Now, if it's adiabatic, we know that this is 0. And if that's 0, we can add P delta V to both sides of this equation, and we will get that-- this is only true if it were adiabatic-- that delta U, our change in internal energy, plus our pressure times our change in volume, is equal to 0. And let's see if we can do this somehow, we can do something with this equation to get to that result that I'm trying to get to. So a few videos ago, I proved to you that U, the internal energy at any point in time-- let me write it here. The internal energy at any point in time is equal to 3/2 times n times R times T. Which is also equal to 3/2 times PV. Now, if I have a change in internal energy, what can change on this side? Something must have changed. Well, 3/2 can't change. n can't change. We're not going to change the number of molecules we have. The universal gas constant can't change. So the temperature must change. So there you have it. You have delta U could be rewritten as delta-- let me do it in a different color-- delta U could be written as 3/2 n times R times our change in temperature. And that's why I keep saying in this-- especially when we're dealing with the situation where all of the internal energy is essentially kinetic energy-- that if you don't have a change in temperature, you're not going to have a change in internal energy. Likewise, if you don't have a change in internal energy, you're not going to have a change in temperature. So let me put this aside right here. I'm going to substitute it back there. But let's see if we can do something with this P here. Well, we'll just resort to our ideal gas equation. Because we're dealing with an ideal gas, we might as well. PV is equal to nRT. This should be emblazoned in your mind, at this point. So if we want to solve for P, we get O is equal to nRT over V. Fair enough. So let's put both of these things aside, and substitute them into this formula. So delta U is equal to this thing. So that means that 3/2 nR delta T plus P-- P is this thing-- plus nRT over V times delta V is equal to 0. Interesting. So what can we do further here? And I'll kind of tell you where I'm going with this. So that tells me, my change in internal energy over a very small delta T-- this tells me my work done by the system over a very small delta V. And we're saying that, you know, over each little small increments, they're going to add up to zero. So let me just go back to the original graph. So this is over a very small delta V right there. Let me do it in a more vibrant color. A very small change, as we go from there to, let's say, there. We're going to have some change in our volume. And you don't see the temperature here. So don't try to even imagine, when we do the integral, that we should think of it in some terms of area. But we're going to integrate over the change in temperature, as well. The temperature changes a little bit from there to there. So what I want to do-- this is, right here, over a small change. I want to integrate eventually over all of the changes that occur during our adiabatic process. So let's see if we can simplify this before I break out the calculus. So if we divide both sides by nRT, what do we get? So let's divide it by nRT, let's divide it by nRT. And we have to do it to both sides of the equation. nRT. Well, on this term, the n's cancel out, the R cancels out. Over here, this nRT cancels out with this nRT. And what are we left with? We're left with 3/2-- we have this 1 over T left-- times 1 over T delta T plus 1 over V delta V is equal to-- well, zero divided by anything is just equal to 0. Now we're going to integrate over a bunch of really small delta T's and delta V's. So let me just change those to our calculus terminology. We're going to do an infinite sum over infinitesimally small changes in delta T and delta V. So I'll rewrite this as 3/2 1 over T dt plus 1 over V dv is equal to 0. Remember, this just means a very, very small change in volume. This is a very, very, very, small change, an infinitesimally small change, in temperature. And now I want to do the total change in temperature. I want to integrate over the total change in temperature and the total change in volume. So let's do that. So I want to go from always temperature start to temperature finish. And this will be going from our volume start to volume finish. Fair enough. Let's do these integrals. This tends to show up a lot in thermodynamics, these antiderivatives. The antiderivative of 1 over T is natural log of T. So this is equal to 3/2 times the natural log of T. We're going to evaluate it at the final temperature and then the starting temperature, plus the natural log-- the antiderivative of 1 over V is just the natural log of V-- plus the natural log of V, evaluated from our final velocity, and we're going to subtract out the starting velocity. This is just the calculus here. And this is going to be equal to 0. Right? I mean, we could integrate both sides-- well, if every infinitesimal change is equal to the sum is equal to 0, the sum of all of the infinitesimal changes are also going to be equal to 0. So this is still equal to 0. See what we can do here. So we could rewrite this green part as-- so it's 3/2 times the natural log of TF minus the natural log of TS, which is just, using our log properties, the natural log of TF over the natural log of TS. Right? When you evaluate, you get natural log of TF minus the natural log of TS. That's the same thing as this. Plus, for the same reason, the natural log of VF over the natural log of VS. When you evaluate this, it's the natural log of VF minus the natural log of VS, which can be simplified this to this, just from our logarithmic properties. So this equals 0. And now we can-- this coefficient out front, we can use our logarithmic properties. Instead of putting a 3/2 natural log of this, we can rewrite this as the natural log of TF over TS to the 3/2. Now we can keep doing our logarithm properties. You take the log of something plus the log of something. That's equal to the log of their product. So this is equal to-- I'll switch colors-- The natural log of TF over TS to the 3/2 power, times the natural log of VF over VS. And this is a fatiguing proof. All right. And all that is going to be equal to 0. Now what can we say? Well, we're saying that e to the 0 power-- the natural log is log base e-- e to the 0 power is equal to this thing. So this thing must be 1. E to the 0 power is 1. So we can say-- we're almost there-- that TF, our final temperature over our starting temperature to the 3/2 power, times our final volume over our starting volume is equal to 1. Now let's take this result that we worked reasonably hard to produce. Remember all of this, we just said, we're dealing with an adiabatic process, and we started from the principle of just what the definition of internal energy is. And then we substitute it with our PV equals nRT formulas. Although this was kind of PV-- this is internal energy at any point is equal to 3/2 times PV. And then we integrated over all the changes, and we said, look, this is adiabatic. So the total change-- the sum of all of our change in internal energy and work done by the system has to be 0, then we use the property of log to get to this result. Now let's do these for both of these adiabatic processes over here. So the first one we could do is this one where we go from volume B at T1 to volume C at T2. Watch the Carnot cycle video, if you forgot that. This was the VB All of these things up here were at temperature 1. All of the things down here were at temperature 2 So we're at temperature 1 up here, and temperature 2 down here, volume C. So let's look at that. So on that right part, that right process, our final temperature was temperature 2. So let me write it down. Temperature 2. Our initial temperature was temperature 1, where we started off at point B to the 3/2. Times-- what was our final volume? Our final volume was our volume at C divided by the volume at B. And that's going to be equal to 1. Neat. That's the result we got from this adiabatic process. We got that formula saying, this is adiabatic, we did a bunch of math, and then we just substitute for our initial and final volumes and temperatures. Let's do it the same way, but let's go from D to A. So when you go from D to A, what's your final temperature? Don't want to get you dizzy going up. Well your final temperature, we're going from D to A. So our final temperature is T1. and our final volume is the volume at A. Go back down. So our final temperature is T1. Our initial temperature is T2. We're going from D to A to the 3/2 power is times-- let me write our form formula there. Our final volume is the volume at A. That's where we moved to. And we moved from our volume at D. And this is going to be equal to 1. OK. We're almost there, if your eyes are beginning to glaze over. But this is interesting. And if anything, it's a little bit of fun mathematics to wake you up in the morning. So let's see. We can almost relate these two things. We could set them equal to each other, but it's not quite satisfying yet. Let's take the reciprocal of both sides of this equation right here. So obviously if we take the reciprocal of this-- and we could just say, this is T2 over T1 to the minus 3/2 power, which is the same thing as T1 over T2, to the 3/2 power. Right? That's just the reciprocal. And we're taking both sides to the negative 1 power, so we're going to have to take this to the negative 1 power. VB over VC. And when you take the reciprocal of 1, that just equals 1, That still equals 1. Which this also equals, so we could say, that equals this thing over here. So that is equal to T1 over T2, to the 3/2 power, times VA over VD. Now, these things are equal to each other. We can get rid of the 1. These two-- actually, let me just erase some of this. I don't want to make it say not equal to. They're equal to each other. They both equal 1. So they both equal each other. This thing and this thing are the same thing. T1 over T2 to the 3/2, T1 over T2 to the 3/2. So let's just divide both sides by that. Those cancel out. And what are we left with? I think you can see the finish line. The finish line is near. We have VB over VC is equal to VA over VD. Now that's not quite the result we wanted, but it takes a little bit of simple arithmetic to get there. Let's just cross-multiply. And you get VB times VD is equal to VC times VA. Now if we divide both sides by VBVC-- actually, let's do it the other way. Let's divide both sides by VDVA. What do we get? These cancel out, and these cancel out. And we are left with VB over VA is equal to VC over VD. All that work for a nice and simple result, but that's better than doing a lot of work for a hairy and monstrous result. So that's what we set out to prove. That VB over VA is equal to VC over VD. And we got it all from the notion that we're dealing with adiabatic process, that our change in heat is 0. and we just went to our formula, or our definition of our change in internal energy, the first law of thermodynamics, that if we have any change in internal energy, it must be equal to the amount of work done by the system. Or at least a negative of the work done by the system. When you add them up, you get to 0. Then we use that result from a few videos ago, where we said the internal energy at any point is 3/2 times nRT. So the change in internal energy is that times delta T, because that's the only thing that can change. We used PV equal nRT. And then we just integrated along all of the little changes in temperature and volume, as we moved along this line. As we moved along the line, we took the integral. We said that had to be equal to 0. And we ended up with this formula over here, and then we just applied it to our two adiabatic processes. And we went from B to C, and we went from D to A. And we got these results. And we got to our finish line. See you in the next video.