 Hi, I'm Zor. Welcome to Unisor Education. We continue talking about heat as it's related to molecular movements. Today's lecture is about the connection between temperature, pressure, pressure and volume of the ideal gas. Now, this lecture is part of the course called Physics 14, presented on Unisor.com. You might have found this particular lecture somewhere else, like on YouTube or some other website. I do recommend you, however, to go to this website to Unisor.com because every lecture has a very detailed notes. All the lectures are logically connected to each other and presented in the set of menus. Plus, there are exams for those people who would like to challenge themselves. Plus, there is a prerequisite course called Mass 14, which is definitely needed for this course. This is very much theory-oriented, mass-oriented kind of a course of physics. And Unisor.com is completely free, no advertisement. You don't even have to sign in unless you would like to save your exams for your teachers or parents or whatever else. Okay, so we'll talk about the temperature, the pressure and the volume of ideal gas. Now, before that, let me divert a little bit your attention back to what actually temperature is. I did address this issue in another earlier lecture. Temperature is a macro characteristic of some kind of an object. What's important is how we measure it. We measure it using the thermometers, different kinds of thermometers. But for the purposes of this particular lecture, I'm talking about thermometers which are most frequently used in your day-to-day life. Like, for instance, alcohol-based thermometer which measures the temperature in the room or outside the window. Or mercury-based thermometer which is used to measure the temperature of the body. Used to, actually. Nowadays there are some more advanced instruments. But anyway, just concentrate on the thermometer which measures temperature of the air in the room. So it goes up and down on a certain scale. So basically we are relating the temperature in the room which we have already discussed as being very much connected to kinetic energy of the molecules. Their speed, the more intense they're moving, the more intensely they bombard the surface of the thermometer. Which in turn, bombards the liquid inside the thermometer. Which in turn causes the inside molecules to start moving faster. And that's what actually the process of equalizing temperature from the air molecules which are moving at certain speed with certain kinetic energy of each of them. Through the walls of the thermometer to the liquid in the thermometer which also becomes more agitated. Or less agitated, depending on what exactly the difference of the temperature was in the beginning. But eventually the kinetic energy, average kinetic energy of the molecules of the air is equalized with average kinetic energy of the molecules inside the liquid. And as we explained in one of the earlier lectures, this more intense or less intense movement of molecules inside the liquid of the thermometer causes increasing or decreasing of the volume. And so the level of the liquid in the thermometer goes up or down. And now all we need to do is just to properly mark temperature on certain scale. So this is a very much, very brief explanation of how the temperature is measured. So most importantly is that eventually the temperature of the thermometer and the liquid inside the thermometer and the air. Or any other object, whatever we are measuring doesn't really matter. We can measure the temperature of the water in the river. So they equalize. Now what's also important is that if it's a closed system, let's say you have an isolated aquarium with water. Isolated which means there is no outside energy coming in or going out. And now you are inserting the thermometer inside that aquarium. So there is no external source of energy. Which means that whatever the kinetic energy existed before, maybe it was a little bit more in the water of the aquarium and a little bit less in the thermometer. Or vice versa. Eventually it equalizes but the total amount of kinetic energy remains the same because it's basically preserved. There is nothing outside, there is no outside source of energy or consumer of energy. Now that means that measuring the temperature of the object actually changes the temperature of the object itself. Because now the energy of the object should really be somehow equalized with energy of the thermometer. So if energy of the aquarium was greater, average kinetic energy of the molecule was greater, then it will heat up the thermometer to equalize, right? Which means certain amount of energy will be transferred from the aquarium to the thermometer. Or vice versa, depending on what is warmer. So measuring changes the temperature of the object which we are measuring. However, we have to always think about this as a non-important factor because in most of the circumstances the energy of the thermometer is really very small relative to the energy, the total energy of the all molecules of the object whatever we are measuring. In all our practical lives. I mean if it's not the case, then obviously we should use different kind of thermometers. But in our everyday life, again back to the measuring of the temperature of the room using a regular alcohol based thermometer. Obviously it's negligible loss of energy whenever we are introducing thermometer into the room. So that's okay, I mean we are changing but not very significantly and negligible actually. Okay, now what else? In one of the previous lecture we were talking about how the temperature is related to average speed. Now in most, the most important issue of that particular lecture was that if we are talking about molecules they have certain speed then the square of this speed is proportional to temperature if counted from absolute zero. Now the temperature of absolute zero is let's say the temperature in the space outside of any kind of a planetary system far from the stars. And in this case the average kinetic energy speed basically of all the molecules will be zero. So that's what absolute zero actually means and that's why whenever we are talking about proportionality we are talking about this starting with zero and this starting with zero. So whenever we have an absolute zero temperature we will have no movement of the molecules of the object. Then as the speed of the molecules is increasing then the average of the squares of the speeds of all the molecules will be proportional to the level of liquid in the theoretical thermometer whatever is measuring this particular temperature. So that's very important. So this is an absolute temperature starting from zero and this is velocity well it's square so it's basically speed averaging called the squares of the speeds throughout the whole molecules of the object. So this is very important. Now at the same time we can say that if our molecules are of the same mass then it's obviously proportional to kinetic energy, average kinetic energy of all the molecules because kinetic energy as we know is equal to m v square over two. So it's very important right now to connect this and this. So kinetic energy is proportional to the temperature. And again we are talking about absolute temperature and kinetic energy obviously also starting from zero whenever the speed of the molecules is zero. Now obviously we are talking about something which we have already established. The scientists already established the scale of this absolute temperature. It's degrees of Kelvin where absolute zero means absolute zero. There is nothing negative in the Kelvin scale there is no negative degrees as we have in Fahrenheit or Celsius. Now so zero means approximately it's minus 273 degree of Celsius. Now and one particular degree is measured as in case of Celsius as 100 of the difference between the temperature of the freezing water and temperature of the boiling water. So these are degrees of Kelvin and that's how we measure the temperature. And the kinetic energy obviously is measured in system C as Joules. So this is degrees of Kelvin and this is Joules. And proportionality obviously means that there is some kind of a coefficient between them. But let's not talk about this right now we will come to this a little bit later. Now next again I will refer you to the previous lecture where we were connecting to third of kinetic energy average kinetic energy. Let me forget about this and I will put average here. So E average means average kinetic energy of molecules where N is the number of molecules. And if you will divide it by volume then we will have the pressure. Now this is the result of the previous lecture and I do refer you to refresh your memory if you don't remember this formula. It's very important to know that the pressure on the walls of the reservoir which contains gas, right now we are talking about the gas, is equal to two thirds of average kinetic energy times number of molecules. So this is the total amount of energy because this is average per molecule and this is number of molecules. So this is the total kinetic energy. So if you will divide it by volume and coefficient is two thirds. Now it's obviously that the pressure should be proportional to the total energy. And it's also obviously that pressure should be inversely proportional to the volume. Because if the same molecules which are moving with the same speed are spread in a bigger volume, obviously the bigger the volume the less the pressure. Now what's also important is this coefficient and I'm not talking about how I derived it. I derived it in the previous lecture. So in this particular case I'm just taking this as granted. And I can express it slightly differently. I can resolve it for average energy which is equal to what? Three seconds P times volume divided by N. So this is average as basically resolved from here. Now I have already mentioned that it's proportional to absolute temperature. So our purpose of this lecture is to connect temperature, pressure and volume. Temperature, pressure and volume. What is N? Well that's not such an easy thing. N is number of molecules in the reservoir. How can we actually relate to this? Well apparently it's not easy. However here is what I can actually suggest you to do. Now from this let's just write it over there. That the P times volume divided by N is proportional to T. Coefficient doesn't really matter right now. It's too proportional. So that's what we have. Now we have to really find out what kind of proportionality we're talking about. Let me write it differently. K times where K is some kind of coefficient. T is the temperature in absolute temperature in Kelvin's degree. Now P is the pressure. Obviously we all know what it is like Newton's per square meter and this is the number of molecules in the reservoir and volume is again like cubic meters whatever it's measured. Alright so we got that. Now let's talk about the ideal gas. And usually I'm not researching very much to say something like experiment shows. But in this particular case I have to do it. Here is what's important. Molecules of different gases are obviously different. Like hydrogen is different from oxygen and oxygen is different from carbon dioxide etc. And obviously they have different masses, these molecules. Which means that since we are talking about kinetic energy as the most important characteristic related to the temperature masses are important as well as the speeds. And by the way we will determine the speed of the molecules of oxygen traveling in the air as the result of this lecture. Which is kind of interesting to know what's the speed of molecules, how they really move. We will be able to do it. So mass is important. Now what's also important is that gases are such substance when molecules are small relatively to the distance between molecules in the gas. So they are really flying colliding with each other but not very often. I mean they do have a relative freedom of movement. Well obviously when they hit the wall of reservoir that what creates the pressure obviously. But at the same time they are relatively freely moving. Now what's important is that every molecule has certain weight. And here again I'm resorting to our experiments. There are some theory and there are some experiments which leads us to certain knowledge about the molecular weight. So there is something which we call a unit of molecular weight which was some time ago decided to be one-twelves of the mass of the carbon atom. Now in these units we can measure the mass of all other molecules smaller or bigger. And in this particular case the molecule of hydrogen which consists of two atoms of hydrogen. We will talk about atomic structure of different elements later on but right now just take it as a face value that the molecule of hydrogen consists of two atoms and each one atom is approximately one unit approximately equal to one atom of hydrogen. So the mass of this would be two. It's called atomic unit of mass. Atomic mass unit. Now the oxygen which is also the molecule of oxygen which contains two atoms of oxygen. It's 16 plus 16 is 32. Now the chemical formula for carbon dioxide which is 12 plus 32 that's 44. So we do know the masses on a certain scale. Now I don't know how really many molecules are in any particular reservoir but to make experiments with the same number of molecules of different gases I can do it because if I know that for instance this is molecular atomic mass of hydrogen and this is atomic weight of oxygen if I will take a mass of hydrogen to be 16 times greater than mass of oxygen which I take so this is amount of hydrogen and I'm taking this mass of hydrogen I can weight it in some way right? I can put it in the reservoir and weight it so the mass will be proportional to the weight. Now I can take certain amount of oxygen put in the reservoir of the same weight let's say and if I will be able to match this mass of hydrogen if mass of hydrogen is 1.16 of mass of oxygen again mass of hydrogen is 1.16 of mass of oxygen then the number of molecules will be the same right? Am I right or not? If I will have the same number of molecules then the total weight of hydrogen will be 1.16 of total weight of oxygen whatever I said before that's the right thing which means I can take two reservoirs one with oxygen and one is hydrogen or hydrogen and oxygen whatever it is make sure that they have exactly the same number of molecules and make experiments so if my N1 is equal to N2 then I can say that I can make experiments with the same number of molecules now what happens is that if this is the same and volume 1 is equal to volume 2 and T1 is equal to T2 then as a consequence P1 is equal to P2 this is the result of the experiment what does it mean? it means that all gases are behaving in a similar fashion and that's why the concept of ideal gas actually came to being because people saw that different gases provided similar conditions like the same number of molecules enclosed in the same volume reservoir under the same temperature they have the same pressure on the walls so these are experiments or experiment can be done differently for instance equalize the pressure and then see the temperature so it looks like all the gases really behave exactly the same thing which means this coefficient although it's different maybe for liquids in case of we are talking about liquids but in case of gases this particular coefficient is relatively the same for all the gases I mean the difference probably will be so negligible we can't really even detect it using our instruments so that's what's very important which means that instead of having this I will write Kb where Kb is a constant called the Boltzmann Boltzmann's constant Boltzmann Boltzmann, I think that's how it's spelled so it's a Boltzmann's constant and it has specific value and specific value I have to really get somewhere 1.381 times 10 to the minus 23 joules per degree of Kelvin so this is a Boltzmann constant it is a constant for gases and that's why we have this particular equation which represents behavior of all the gases well, within certain limits I mean there are some extreme temperatures or maybe extreme pressures or something like that where it's distorted but again for the purposes of physics of 19th and 20th, beginning of the 20th century you can assume it's a constant we don't really go into extremes okay, so that's very important now, what follows from this is that average energy as we were saying is 3 seconds of P volume divided by N which is equal to 3 seconds KB times temperature now, this is known this we can measure it's just a macro characteristic so using the macro characteristic we can have the micro characteristic of the gas the average kinetic energy of the molecules and that's very, very important you see, measuring something on a macro scale having the constant basically and just measuring the temperature in degrees of Kelvin we can find out the average energy of, average kinetic energy of the of the molecules of the gas okay, fine now, what follows from this is that the P times volume divided by G is equal to KB times N now, what does it mean if you try to interpret it well, if you take a certain amount of gas which is a certain number of molecules of gas in the reservoir then this is constant if we are changing one of these parameters right, so no matter how we change these parameters pressure, temperature, volume if we are talking about the same amount of gas then this pressure times volume divided by temperature is constant because this is constant because we don't change the number of molecules so, for instance if we have some kind of a cylinder okay and it has certain volume, right for instance, it's fixed and then we are applying temperature we are changing the temperature well, if we are changing the temperature while volume remains constant then P should also change proportionally because otherwise the whole thing will not be constant so if I double the temperature my pressure should be double so if I heat it up from one degree, let's say from 100 degree Kelvin to 200 degree Kelvin which means double my temperature my pressure should also double okay whatever the gas is doesn't really matter and this has a specific name, this particular law that if the volume is constant and if the volume is constant the law is called the law of Gay-Lussac so if my if my V equals constant volume, I mean then P and T are proportional to each other then P divided by T equals constant that's my first law now, at the same time I can establish another experiment when I maintain, let's say the temperature if my temperature is constant and I'm changing the volume how can it be done? well, for instance we don't change temperature temperature is always the same temperature is constant but I have some kind of a plunger here whatever it is so initially it's in the upper position and gas molecules are everywhere but then I press it down so if I, let's say decrease my volume by factor of 2 what happens with pressure? if my volume is increasing temperature is constant then my pressure should my volume is decreasing then the pressure should increase so the pressure times volume is constant and finally if my pressure is constant if my pressure is constant then volume divided by T should be constant right? so this is Gauss-Sack law now if temperature is constant that's the Boil-Marriott law and this is the Schar's law so each of these laws were discovered separately and later on everything was combined into something which is kind of a combined law the pressure times volume divided by temperature for a particular amount of gas is constant no matter how you change you can change all three characteristics by the way but this product of P times volume divided by T should be constant alright, so these are three laws and then I promise to find the speed of the molecules of oxygen and that's how we will do it so we will do it using this and also I know that the molecule of oxygen has mass again I have to resort to some experiments whatever it was provided 5.31 times 10 to the minus 26 kg that's mass of one molecule one molecule of oxygen now knowing the mass knowing temperature well let's say room temperature is like 20 degrees Celsius which is 293 degrees Kelvin knowing Boltzmann's constant which is this one I can establish my mv square average divided by 2 that's what kinetic energy average kinetic energy is so if I will multiply this time this time this I will get average kinetic energy knowing the mass I can derive average speed and average speed then I will take the square root of that to get the speed to be 478 meters per second which is pretty fast so the molecules of oxygen in the air which we are breathing are moving with average speed 478 well maybe my calculation is not exactly correct somewhere around this number between 400 and 500 meters per second that's all we are not talking about a concrete molecule it can be faster or slower this is very much around average which is pretty fast however you have to take into consideration that as molecules are moving no matter how small they are they are still colliding with each other so the direction is changing so that's why the molecules maybe are moving very fast but it doesn't mean that the molecules are flying away from us somewhere else or there is some kind of a significant wind there is no wind unless there is a wind from outside so if you are in a room you don't feel this because no matter how fast they are it's still within certain microscopic dimensions so you don't feel it as a wind but anyway that's interesting kind of an equation which we can make using theory well not only theory we did use for instance the mass of oxygen which was determined completely outside of the scope of this lecture and the Boltzmann constant which also was calculated somehow in any case using pure theory and this is very very important this is very very important equations this is dynamics this is the ideal gas so using this theory and certain experimental facts we have derived to some very unexpected result about the speed of the molecules of oxygen in the air which we are bracing with okay I do recommend you to read the notes for this lecture they are on Unisor.com notes basically contain exactly the same information presented in a textbook-like format and well that's basically it that's it thank you very much and good luck