 In previous videos, we've talked about how derivatives relate to problems that arrive in physical science, particularly in physics. We've talked about how derivatives are used in the motion problem with respect to velocity and acceleration. We've talked about how derivatives relate together the idea of mass, length, and density of maybe like a rod of some kind, the linear density of that rod. The examples do not stop there, of course. Derivatives occur all over the place in physics. Other examples include the idea of power. Power is the rate at which work is done with respect to time. You have ideas about the rate of heat flow, temperature gradients, that is the rate of change of temperature with respect to position. But it's not just physics, right? We can also do other physical sciences like chemistry, which we're going to see in this video right here. Some examples that you could see in chemistry would be about the rate at which a radioactive isotope is decaying over time. You could talk about, for example, the chemical idea of reaction and concentration, and how this introduces the idea of instantaneous rate of reaction. In this video, I want to talk in a little bit more detail about the idea of isothermal compressibility. So what does that even mean? So one of the quantities of interest in thermodynamics, thermodynamics is about essentially the physics and chemistry of heat and temperature and related topics, right? Thermodynamics, among other things, is concerned about the idea of compressibility. I like to imagine in the movie, the Studio Ghibli movie Ponyo, which is kind of like a Japanese anime version of Little Mermaid, right? At the beginning of the movie, Ponyo, who's the fish, she wants to turn into a human girl, and her dad is trying to squish her back into a fish. So you see her in this magic ball, and she's going to squish, squish, squish, and she really doesn't want to do it, but her magic's not powerful enough to resist her. Her dad compressing her back into a small little fish, right? I mentioned if you haven't seen the movie, this dialogue might not make any sense right now, but the compressibility is the idea of how, given like say a gas, how easily we can compress the gas into a smaller volume. So we have the same amount of mass, but we want to compress it down into a smaller volume. How easily can that be done? Is the ability to compress hydrogen the same as oxygen, or nitrogen, or argon? I don't know. I'm not a chemist, but these are things that a chemist would be interested in, or people in thermodynamics would be interested in. How easily can a substance like a gas, or other fluids, how easily can we compress downward? That seems to be something that's a property of that substance, of that gas. So if a given substance is kept at a constant temperature, then its volume V depends on its pressure P. So this is a very important principle from chemistry here, that the volume, pressure, and temperature of a gas, we'll just say gas here, is they're all related to each other, but it's not exactly the same thing. So if we take our, for example, our gas, and we decrease its volume so we compress it down, then these little particles inside are pushing back, right? That's where the pressure comes from. If we decrease the volume, the pressure will increase. That is to say that the volume and pressure of a gas are inversely proportional with each other. On the other hand, if we take our gas right here, and we add a little bit of heat to it, so this is my flame going on there, very clever here. If we heat up the gas, this is going to excite the particles, which then they'll vibrate more. This will then cause an increase of pressure. Let's say that the volume is restricted. We have a rigid container of some kind. It's not flexible at all. If we increase the temperature of the gas, then that'll increase the pressure, right? The object will get, the particle could get more excited. That's to say that pressure and temperature are directly proportional to each other. If we don't have a solid container, like maybe our gas isn't an expandable container, like a balloon, if we increase the temperature, then these things are going to expand so the volume will increase as we increase the temperature. There is a direct relationship between temperature and volume. All of these things are related to each other by direct or inverse proportions whatsoever. When it comes to isothermal compressibility, the idea is isothermal means that we keep the same temperature. Iso means same thermal, kind of like your thermal underwear, it means temperature here, so the same temperature. If we keep a gas at the same temperature, how does the change of volume affect the change of pressure? That's the question we're trying to consider right now. We can consider the rate of change of volume with respect to pressure because if we increase the pressure, if we take our gas and we start increasing the pressure, we're squishing it, we're compressing it, how does that change the volume? We do know the volume will decrease, but how rapidly does the volume decrease? We're asking about the derivative of volume with respect to pressure. As pressure increases, volume decreases. We know that the derivative is going to be negative. This is again given by the fact that pressure and volume are inversely connected to each other. The compressibility is going to be defined with an extra minus sign in it. We know the derivative is going to be negative. The compressibility quantity we're going to come up with, we want to be positive, so we're going to slap a negative sign in front of it. Isothermal compressibility, which is commonly denoted by the Greek letter beta, this is the quantity negative one over the volume times the derivative of volume with respect to pressure. The negative sign comes from the fact that we know the derivative is going to be negative, so we want a double negative to be positive. We're optimistic here. What's with this one over volume right here? The amount of pressure it takes to decrease a gallon of say atmosphere by one. Let me say that again. Let's say we have a gallon of just regular air that's on here on planet Earth. If we want to decrease that from a gallon to a half a gallon, that takes a certain amount of pressure, increase of pressure to do that, but what if we take a half gallon of air and we want to decrease that to one fourth of a gallon, right? So you cut the volume in half. Well, the fact that you have so much more over here versus here, that original volume has an effect, right? And so to make this thing comparing apples to apples, we divide by the volume here so that everything is, it's not based upon the specific size. We actually get a quantity that's resemblant of the molecule, the element that's in play right here and not on the volume of it whatsoever. So what this tells us is that beta is going to measure how fast per unit of volume the volume of a substance decreases as the pressure on it increases at constant temperature. So that's the idea behind isothermal compressibility. This is a valuable quantity to measure how easily one can compress a certain gas. So let's look at a, for instance, so for instance, the volume V in cubic meters of a sample of air, if we keep it at 25 degrees Celsius, which is a normal temperature for air to be here on planet Earth, let's say that that's found, let me read that again, the volume V in cubic meters of a sample of air at 25 degrees Celsius was found to be related to the pressure in kilopascals by the following equation volume being measured in cubic meters is equal to 5.3 divided by kilo by something the amount of kilopascals. So this is an inverse proportion between the two, the two quantities of volume and pressure right here. So Pascal's is a measurement of pressure, which is honestly not very large. And so oftentimes you have to measure pressure in kilopascals like so. Well, the derivative of volume with respect to pressure when the pressure is at 50 kilopascals is given by the following. We can compute the derivative of this by using the usual power rule, right 5.3 divided by P, we can write that as 5.3 times P to negative one. You could use the quotient rule if you prefer, but the power rule is going to be a lot easier here. Taking the derivative of 5.3 times P to negative one, you're going to end up with negative 5.3 over P squared or P to the negative second power right there. If we evaluate the derivative at 50 kilopascals, you're going to end up with 50 squared in the denominator, which is 2500. You can negative 5.3 over 2500, which is going to compute out to be negative 0.00212 cubic meters per kilopascal. Now, as these are scientific questions, we probably should be worried about significant digits, but this is a math video. I don't care about significant digits. Sorry, everyone, we're just going to crunch numbers and not worry about measurement error and things like that. So we get that. We get that the derivative at 50 kilopascals is going to be this negative 0.002. Okay, so we plug that into our formula for isothermal compressibility. So the volume is, what is the volume? It never actually tells you what the volume is here, but we do have a formula that relates volume and the pressure together. So volume is equal to 5.3 over P, which P we know to be 50 kilopascals. So we see that volume is going to be 5.3 over 50, or more importantly, one over volume is going to be its reciprocal 50 over 5.3, which we see that right here in the formula. So we get negative 50 over 5.3 times that by negative 5.3 over 2500. And so when you simplify that, you end up with this number 0.02. And so this gives you the isothermal compressibility of air, just standard earth air at the temperature 25 degrees Celsius. And you can compare that with other quantities of isothermal compressibility to show you how easily 25 degree air compresses compared to other gases at other temperatures.