10.2 Gas Laws Including the Ideal Gas Law | General Chemistry

Summary

Highlights

• Diving deep into Boyle's, Charles's, and Avogadro's laws—pressure, volume, and temperature relationships made fun! 🎉
• Understanding the Combined and Ideal Gas Laws with relatable examples like balloons and hot air balloons! 🎈
• Mastering the math: From PV=nRT to calculating conditions, learn to solve real-world chemistry problems! 📊
• Exploring kinetic molecular theory and discovering why ideal gases are like unicorns—perfect yet mythical! 🦄
• RMS speed gives insights into molecular motion and distributions; it’s fast physics fun! 🚀

Key Takeaways

• Understanding gas laws requires knowing how pressure, volume, and temperature interact. 🤓
• Boyle's Law states pressure and volume are inversely proportional—squeeze your balloon, and feel the pressure! 🎈
• Charles's Law shows that temperature and volume are besties—they rise and fall together! ☀️📉
• Avogadro's Principle explains that more gas moles mean more volume, under constant pressure and temperature! 🎈➡️📏
• Ideal Gas Law (PV = nRT) is your best friend when conditions are ideal—it’s all about the conditions! 📚
• Root Mean Square Speed: Want to know how fast gas molecules are? Check out RMS for the deets! 🏎️
• Remember: Conditions impact behavior! Higher temperatures and lower pressures make gases behave more 'ideally'. 🥶🔥

Overview

Chapters

• 00:00 - 01:30: Introduction to Gas Laws The chapter 'Introduction to Gas Laws' covers various fundamental gas laws in chemistry. It begins with a discussion on Boyle's Law, Charles's Law, and Avogadro's Law or Avogadro's Principle. The lesson progresses to cover the Combined Gas Law and finally, the Ideal Gas Law, represented by the equation PV=nRT, along with the associated Kinetic Molecular Theory. The chapter concludes with a brief explanation of the concept of root mean square speed for gases.
• 01:30 - 14:00: Boyle's Law, Charles' Law, and Avogadro's Law The chapter discusses three fundamental gas laws: Boyle's Law, Charles' Law, and Avogadro's Law, also known as Avogadro's Principle. It outlines the fact that four variables—pressure, volume, number of moles, and temperature—are essential for describing a gas system. In a multivariable system, if one needs to compare any two variables, the other two must be held constant. For instance, Boyle's Law compares pressure to volume while keeping the number of moles and temperature constant.
• 14:00 - 20:00: Combined Gas Law The chapter titled 'Combined Gas Law' explains the relationship between pressure, volume, and temperature of gases, referencing key scientific figures such as Charles, Avogadro, and Boyle. It discusses how maintaining certain variables constant allows for exploration of relationships between others, like volume and temperature, or volume and the number of moles of gas. The chapter underscores the importance of keeping such considerations in mind during scientific analysis. Boyle's principle that pressure is inversely proportional to volume is a focal point, hinting at the foundational aspects of gas laws.
• 20:00 - 30:00: Ideal Gas Law This chapter covers the Ideal Gas Law, introducing the relationships among pressure, volume, temperature, and the number of moles of gas. It discusses the inverse relationship between pressure and volume, as well as the direct proportional relationships between volume and temperature, and volume and the number of moles, assuming all other variables are held constant. It references how different scientists, such as Boyle, Charles, and Avogadro, contributed to understanding these relationships.
• 30:00 - 41:00: Root Mean Square Speed The chapter humorously imagines a scenario featuring historical figures Mr. Boyle and Mr. Avogadro at a child's birthday party in the 17th or 18th century. During the party, Mr. Avogadro finds an uninflated balloon and decides to blow it up, leading to him exclaiming 'Eureka!' upon realizing that the volume of the balloon increases as he adds more moles of gas. This narrative illustrates the concept of volume expansion with the addition of gas moles.

10.2 Gas Laws Including the Ideal Gas Law | General Chemistry Transcription

• 00:00 - 00:30 gas laws including the ideal gas law going to be the topic of this lesson and we're going to talk about a whole host of different gas laws we'll talk about boyle's law and charles law avogadro's law or avogadro's principle we'll talk about the combined gas law and then we'll finally get into the ideal gas law pv equals nrt along with kinetic molecular theory that goes with it and finally we'll top off this lesson really briefly talking about root mean square speed for gases all right so we're going to start off with a trio of gas laws here we're going to start off with boyle's law charles law
• 00:30 - 01:00 and avogadro's law or avogadro's principle as it's more commonly called and basically what you need to realize here is that for a gas to describe any system of a gas you really have got four variables that are affecting the system here so you've got pressure volume number of moles of the gas and then temperature and so in a multivariable system like this if you want to compare two variables and that's what each of these three guys did you're going to have to hold the other two variables constant that's kind of implied here so if you want to compare pressure to volume like mr boyle did you're going to have to hold the number
• 01:00 - 01:30 of moles and the temperature constant if you're going to want to compare volume to temperature so like mr charles here did you're going to have to hold the pressure and the number of moles of gas constant and if you want to compare the volume to the number of moles of gas well then you're going to have to hold the pressure and the temperature constant like mr avogadro did there so that's kind of implied in there so just keep that in mind that'll be important in one key place i'll bring up all right so we'll start with mr boyle here and mr boyle here said that pressure is inversely proportional to volume so he says as your volume goes up
• 01:30 - 02:00 your pressure goes down or if your as your volume goes down and gets smaller on the gas then your pressure goes up so there's an inverse relationship there whereas charles here compared volume to temperature and he said they're directly proportional again assuming we hold the other variables constant then volume and temperature are proportional as temperature goes up volume goes up and then finally mr avogadro said volume is proportional to the number of moles of gas as you add more moles of gas the volume of that gas is going to increase again assuming the other two variables are held constant now i like to kind of
• 02:00 - 02:30 envision mr mr boyle over here mr avogadro over here at a little kids birthday party way back in the 17 or eighteen hundreds and uh while at this little kid's birthday party so mr avogadro finds a balloon lying on the ground that has not been inflated and he picks it up and he blows it up and he's like eureka my name is amadeo avogadro and i have just discovered that as i put more moles of gas in the balloon it gets bigger it has more volume
• 02:30 - 03:00 way to go sherlock so but that's exactly kind of what mr avogadro discovered so and truth be told he said you know what he really said is that you know if you have two gases under exactly the same conditions of pressure and temperature then equal volumes will have an equal number of particles that's what he really said well what we translate that into is they'll have equal number of moles and that really under the identical conditions of pressure and temperature that your volume is really proportional therefore to the number of moles of gas in that sample so we're kind of extrapolating it out a little
• 03:00 - 03:30 bit from what he said but i like to really simplify it but really he he really did make a profound discovery i just kind of like to make light of what he actually discovered well mr boyle not to be undone is at the same party and so he picks up a balloon off the ground that's not been inflated and he blows it up and ties it off and then he starts squeezing on the balloon and squeezing on the balloon and then it pops and he's like eureka i'm a genius too so as i made the balloon smaller and smaller and smaller the volume got smaller the pressure went up high enough to make it pop and so pressure and
• 03:30 - 04:00 volume are inversely proportional put my name on it so we have another sherlock here it used to be a lot easier to get your name on something here so but that was boyle's conclusion here that pressure and volume are inversely proportional and mr charles is the only one really gonna give credit to i don't i don't think he was really at this birthday party that i'm making up anyways that didn't exist so this fictitious birthday party but he said volume and temperature are directly proportional now i live in arizona here and if you go out on a hot day in august from your air-conditioned house there is
• 04:00 - 04:30 a huge temperature increase say from like you know 75 degrees in your house to like 110 or 115 degrees outside your house and if you take a balloon that's just filled with air not a helium balloon or anything like that although you can do it with that too but just a regular air filled balloon and you know rather full tied off and you take it from inside at 75 degrees to outside at like 110 degrees you will actually notice it actually expand so as the temperature of the molecules goes up the volume goes up
• 04:30 - 05:00 as well and so it's actually noticeable enough with that big of temperature difference to see that difference and you can also see this kind of you know in in a hot air balloon i like to envision mr charles kind of you know riding in a hot air balloon and again that's completely fictitious but i like to envision that he's riding this hot air balloon and he sees some power lines up ahead which also didn't exist when he was alive but hey it's all fictitious anyway so he sees these power lines and so what should he do and don't think jump that's not his option here so what he's going to do is pull the jet so and heat up the gas inside the
• 05:00 - 05:30 balloon and that's going to cause that gas inside the balloon to expand and as it expands keep in mind that density equals mass over volume as that volume inside the balloon expands the density of that gas in the balloon goes down so and all of a sudden the gas inside the balloon is less dense than the gas the air around it and when you've got you know something that's less dense than the fluid around it it's going to rise like putting you know a ping-pong ball you know and carrying it to the bottom of your pool so if you let
• 05:30 - 06:00 it go it's going to rise to the surface because it's less dense than the water around it well same thing when you heat up the air inside that balloon it's going to be less dense that air inside the balloon is less dense than the air around it and so it's going to want to rise there's a buoyant force acting on it it's really what's happening there and so that's what i like to think of mr charles figuring out that you know temperature and volume are directly proportional as you heat up a gas it will expand again assuming you're keeping p and n constant so there's our three gas laws there you should know what they're comparing
• 06:00 - 06:30 volume to and you should know if they're directly proportional or inversely proportional and again it's pressure that's inverse proportional volume but both temperature and moles of gas there's a direct proportional relationship but again keep in mind all three of these are dependent upon the other two variables being held constant so if i said does boyle's law say that as you know volume goes up pressure goes down or his volume goes down pressure goes up is that true always well that's true if you're holding the number of moles of
• 06:30 - 07:00 gas and temperature constant so if i make the volume smaller does that mean the pressure has to go up in all situations for everything well no but it would mean that it has to go up if the number of moles of gas and temperature are being held constant now one thing you should know and i'll just throw this vocab word in there some of you guys are going to get this one but isothermal is a word we use iso means the same and isothermal just means the same temperature so constant temperature so we might say that you know if we carry
• 07:00 - 07:30 out a process isotherm for a specific sample of gas well if it's a specific sample of gas well then we're not adding more gas to it or removing gas to it just for that set sample of gas so if i'm doing a process isothermally for a specific sample of gas that would be one way of indicating to you that yeah it's at constant n and constant t and under those conditions that's when pressure and volume are inversely proportional so and again it's not just an inverse relationship but inversely proportional so if i double the volume the pressure's
• 07:30 - 08:00 cut in half if i triple the volume the pressure is going to be a third of its original value there's that inversely proportional relationship now there's a couple other ways you can actually you know explain this you could say that p times v equals a constant so this way you know if p times v equals a constant well then if your volume goes up by a factor of two well then your pressure would have to go down by a factor of two to
• 08:00 - 08:30 multiply to equal that same constant you could also say p1 v1 equals p2 v2 another way to express it and so you're just if you're you know say changing and altering the conditions of a gas from one set of conditions to a second set of conditions so i got four variables here if all i'm changing is pressure and volume not again molar gas and temperature there's your boils law another way to express it and if i give you you know say the initial pressure initial volume and final pressure i could say what's the final volume so essentially i could give you any three
• 08:30 - 09:00 of these four variables and say what's the fourth one so you know if i start off with say a pressure of one atmosphere and a volume of one liter and let's say i change that pressure to 0.5 atmospheres well the question might be what's that final volume well in this case i've taken that pressure and cut it in half and if you cut the pressure in half the volume needs to double and we would anticipate that it would be two liters and if in lo and behold you solve
• 09:00 - 09:30 for v2 there you will indeed get two liters okay so let's go on to charles law here and charles again compared volume to temperature and they're directly proportional now one thing that's super duper important here is that whenever you deal with temperature in this chapter it is going to be on the absolute scale on the kelvin scale not celsius so if you notice celsius you know mr celsius that's or centigrade or i don't know if that's his real name or where where the nature of that name comes from but uh the celsius scale does
• 09:30 - 10:00 not have a true zero so he just you know mr celsius saw where water froze and boiled and made those the zero and hundred marks on his scale and then just divided the difference up into a hundred little degrees so but you can go colder than zero that's what i mean by it's not a real zero it's not the absolute zero that's what we get on the kelvin scale though so absolute zero kelvin is the complete absence of heat you can't have a lower temperature than that and so that's what i mean by having a real zero so it's kind of like i
• 10:00 - 10:30 decided let's say that i'm the most important person in the universe and i'm the standard by which all other things are measured so you know i'm five foot eight and so five foot eight is the on the new chad scale is the new zero and so if you're five foot nine well on the chad scale you'd only be one inch tall if you're six foot you'd only be four inches tall if you're five foot two you'd be negative six inches tall on this new chad scale well obviously five foot eight is not a real zero so and in this case if i told you somebody
• 10:30 - 11:00 you know who is you know say two inches on the chad scale versus four inches on the chad scale would it be accurate to say that the person who's four inches tall on this fictitious chad scale is twice the height of the person that's only two inches tall well no because again two inches versus four inches on the chad scale we would be five foot ten versus six foot tall and that's definitely not a difference by a factor of two and so when you don't have a real zero you can't actually use that scale to talk about like doubling and tripling and that's why in this particular
• 11:00 - 11:30 chapter we're talking about like proportionalities and stuff like that we have to use the kelvin scale so volume is direct proportional to temperature but make sure you're doing it in kelvin so if i told you that you know i increased the temperature from 10 degrees celsius to 20 degrees celsius you're supposed to realize oh that's celsius that's not a doubling of temperature 10 celsius to 20 celsius would be you know 283 kelvin to 293 kelvin which is like less than a 10 increase and a less than 10 increase in temperature should give you a less than 10 percent increase in volume so be
• 11:30 - 12:00 careful on you know uh temperature in this chapter everywhere in this chapter you should definitely use it in kelvin alright another way to express this though is to say that volume over temperature equals a constant and so if this ratio always gives you the same constant well then if you double the temperature again in kelvin you'd also have to double the volume so that this equals the same exact ratio the same exact constant and another way to say that would be v1 over t1
• 12:00 - 12:30 equals v2 over t2 and similar to what we had over here so if i give you you know any three of these conditions so i'm gonna have a gas under two sets of conditions in this case where the number of moles of gas and the pressure are held constant so but if i give you three of the four conditions here you could solve for the fourth so if i told you that i had a gas at an initial temperature of 50 kelvin and then i'm going to change it to being at a final temperature of 100 kelvin and if its initial volume
• 12:30 - 13:00 is 1 liter and then i say what's the final volume going to be well again in this case on the kelvin scale i doubled the temperature which means the volume should also double and therefore go up to 2 liters that way 1 over 50 is the same as 2 over 100 maintain that equality so that's charles law and then finally we'll make our way over here to avogadro's principle and again we're going to say that volume is proportional to the number of moles of gas assuming pressure and temperature are held constant another way to say
• 13:00 - 13:30 this is to say that v over n that ratio is equal to a constant once again if you double the number of moles of gas you'd have to double the volume so this ratio equals the same constant and similar fashion v1 over n1 equals v2 over n2 just analogous to what we saw with charles law with temperature instead of with moles of gas but pretty much exactly the same so these are the different ways you can express these three lovely glass laws you should
• 13:30 - 14:00 understand all three of them for each because you could get a you know question on the test that just says which of the following is an expression of charles law well you've got three possible correct answer choices that could show up there uh and so you should realize that all three of these are an expression of charles law you should also be prepared to do some basic calculations with these however we'll see in just a sec that you know more like more than likely you're probably just going to take a look at using the combined gas law instead of one of these individual ones because that combined gas law can essentially reduce to any
• 14:00 - 14:30 one of these once you realize which variables are being held constant all right so now we're going to move on to that combined gas law and the combined last gas law just essentially combines boyle's law charles law and avogadro's principle into one and technically actually doesn't even combine avogadro's but we will so but if you combine boils and charles into one you get p1 v1 over t1 equals p2 v2 over t2 and now you can actually look at changing not just two variables but now three well technically if we throw avogadro into there as well now you
• 14:30 - 15:00 actually get p1v1 over n1t1 equals p2 v2 over n2t2 and i'm really disappointed that in the actual combined gas law they don't usually include the number of moles well what if you actually add more moles of gas or remove mosal gas how do you account for that well you don't with the combined gas law so this is what i call the better combined gas law you're probably not going to find it in your textbook or anything like that so but it does allow us to account for the number of moles now one thing to note because this is not going to be in your book it's probably not going to be any summary equation sheet or anything like
• 15:00 - 15:30 that for you so one thing to note if you look at your ideal gas law which we have not covered yet but we will get to eventually it's pv equals nrt and if you rearrange this you can get pv over nt equals r and it turns out r is a constant and so if this ratio pv over nt equals a constant so well then under one set of conditions that ratio is going to equal the same constant as it will under a second set
• 15:30 - 16:00 of conditions and you can derive this better combined gas law from your ideal gas law that we'll learn about later on in this lesson so i just want to pull that out i realize we haven't covered it yet but most of you are probably actually seeing this material for the second time watching this video uh and hopefully you've a good chance you've seen it in class already so but we'll we'll get here to this ideal gas law in just a little bit all right so for this combined gas law essentially you've got two sets of
• 16:00 - 16:30 conditions and you could have this you know we've really got four initial conditions four final conditions as many as eight variables and i give you seven you solve for the eighth well odds are they're probably going to hold at least something constant you know if i gave you a specific sample of a gas at a pressure of one atmosphere okay so initial pressure is one atmosphere with a volume of one liter okay so and a temperature of let's just say zero celsius well again on this you got
• 16:30 - 17:00 to be careful temperature's always got to be in kelvin everything you do in this chapter now it turns out pressure and volume as long as you use the same units on both sides it doesn't matter what you choose but temperature because again we need that absolute skill has to has to has to be in kelvin and so if i say 0 celsius you're supposed to know oh that's got to be 270 3 kelvin so and then i'm just going to call this n1 i don't tell you how many moles of gas we have but we're going to alter the conditions for a specific
• 17:00 - 17:30 sample of gas and so we're just going to change the conditions we're not going to add any more gas to it we're not going to remove any gas from it and so we'll find out that n1 and n2 are going to be equal and they'll just cancel out of the equation and so in this case i'm going to change the pressure to 2 atmospheres and the question's ultimately going to end up being what is the volume so but we've got the same sample of gas i'll come back to n2 in a minute and then we're going to change the temperature up to 5 273 celsius with just 546 kelvin
• 17:30 - 18:00 and so once again in this case though n1 and n2 it's the same sample of gas i haven't added to it or removed anything from it so those are going to be equal we'll get rid of those and the question is what is the final volume of this gas well we can kind of look at this and look at things individually here what did i do to the pressure well i doubled the pressure if i isolated that alone what impact would that have on the volume well pressure and volume are inversely proportional so if i double the pressure that should cut
• 18:00 - 18:30 the volume in half okay well let's go take a look at temperature for temperature here i went from 273 kelvin to 546 kelvin and that is an actual doubling of the temperature on the absolute scale and double the temperature it should double the volume charles law well based on what we did with the pressure the volume should be cut in half based on what we did with the temperature the volume should double and factoring both those in actually means we're going to get no volume change and if you actually solve for v2 here you're
• 18:30 - 19:00 gonna get that it's still equal to one liter so but that's how you could plug in all the variables you're provided with and and again had the let's say we just hadn't changed the temperature well then you'd have the same temperature on both sides and it would have canceled out of the equation cool so ultimately you can get boyle's lot of this if you keep n and t constant well then these cancel and you're just left with p1 v1 equals p2 v2 boyle's law if you hold p and n constant well then p1 and n1 and p2 n2 cancel and you're just left with charles law
• 19:00 - 19:30 and if you hold pressure and temperature constant then you're going to be left with avogadro's law and so all those other laws can be derived back from here but the truth is again we built all those other laws up into this better combined gas law cool now one thing to note we're going to talk about the ideal gas law soon so and again that's going to involve an equation pv equals nrt and a lot of students don't know well one way to use the combined gas law versus you know the ideal gas law well how do you know
• 19:30 - 20:00 well again the big thing is this the ideal gas law you have one set of conditions you don't have two different pressures like an initial pressure and a final pressure you don't have two volumes you just have a single pressure a single volume a single number of moles of gas a single temperature and then that's a constant and you're gonna get three out of the four provided typically and solve for the fourth but the combined gas law deals with one set of conditions and then you're altering it to give a second set of conditions and you're given everything except one variable under either the initial or final set of conditions so the real question comes down to just one set of conditions you're probably doing
• 20:00 - 20:30 the ideal gas law calculation if you've got parts of two different sets of conditions you're probably doing the better combined gas law in this case instead cool so let's go take a look at that ideal gas law all right so now we're going to deal with the ideal gas law so pv equals nrt and we call it the ideal gas loss some books you'll call it the perfect gas law so either way so but it's the ideal gas law and the reason we call it this is that only a gas that's behaving ideally
• 20:30 - 21:00 will follow this law and so it turns out not all gases exhibit ideal behavior or at least more commonly what you're really going to see is that not under all conditions and so an ideal gas it turns out is just like an ideal man and they don't exist there's no such thing as an ideal man there's no such thing as an ideal gas however there are conditions where both gases and men are more likely to behave ideally now we'll learn about what those conditions are for a gas here in a little bit but for a man those ideal conditions would be like a first date
• 21:00 - 21:30 so a man is much less likely to do rude things uh and things of that sort on a first date as opposed to say after being married for 20 years and things of his sort so the ideal condition might be a first state as opposed to being married for 20 years or something like this so ideal gases again turns out they don't exist however we'll find out a little bit we'll talk more about it in a second that under conditions of low pressures and high temperatures that's when gases are most likely to obey this lovely mathematical expression the ideal gas law and we'll review that
• 21:30 - 22:00 again a little bit so it turns out this is all part of what we call kinetic molecular theory and i'll put it up on the board over here and so kinetic molecular theory has a few different tenets or postulates that are part of it and so first one is that the molecules themselves take up negligible volume so it turns out most of an empty gas is made of empty space under conditions of it turns out low pressure uh that'll kind of be what governs it here so in a gaseous phase the molecules and atoms again are separated by a fair
• 22:00 - 22:30 amount of empty space so much so that the percentage of the volume of an entire gas taken up by the little molecules that are spread out themselves is essentially we're going to round it down to zero is it really zero well no but it's going to be so small a fraction of the total volume that's actually getting taken up by the molecules instead of empty space that we're just going to round it down to zero and say that the gas molecules have negligible volume that's tenant number one now the second thing we're going to say is that the gas molecules have no attractive forces so i mean technically
• 22:30 - 23:00 you might say they have no repulsive forces either but it's the attractive forces we're worried about because it turns out all molecules are at least a little bit sticky and so to say that gas molecules have zero attractive forces is not a 100 correct statement but it is one of the tenants or postulates of kinetic molecular theory here and it is one of the tenants that needs to hold true for this equation to describe the behavior of a gas and so it turns out it is most accurate a statement though at high temperatures
• 23:00 - 23:30 and the idea is that at high temperatures molecules are moving faster it turns out kinetic energy and temperature are directly related as temperature goes up kinetic energy goes up and molecules are on average moving faster and we'll study this in a little more detail towards the end of this lesson so um but the idea is that you know when molecules are moving fast then they're going to collide faster too and have less of an opportunity less time to experience any kind of attractive forces and stuff like this so imagine having two magnets and i just bring these
• 23:30 - 24:00 magnets ever so close together until they stick okay now imagine instead that i hurl these magnets at each other at 500 miles an hour so that they just bounce right off each other and the idea is that they had enough kinetic energy to overcome any kind of stickiness they had well the same thing is true for for gas molecules as long as they're moving fast enough they're not really going to get much of an opportunity to feel their attractive forces even though all molecules at least have a little bit of attractive forces some more than others
• 24:00 - 24:30 and so the second tenet here is that there are no attractive forces and again never 100 true but it is most true at high temperatures and so now with the first two postulates of kinetic molecular theory we understand now why the ideal gas law is most accurate at conditions of low pressure and high temperature postulate number one is most accurate a statement at low pressures the when the molecules are really spread out uh and then postulate number two here is most accurate at high temperatures when the molecules are on average moving very
• 24:30 - 25:00 fast all right number three postulate or tenet for kinetic molecular theory is that all collisions are elastic so all collisions are elastic and what that means that no kinetic energy is lost during collision so you know when the molecules collide with each other or with the walls of the container and stuff like this uh we just say that there's no loss in kinetic energy so and that's what actually an elastic collision is if you've had physics when you talk about collisions that's the definition of an elastic collision one in which no kinetic energy
• 25:00 - 25:30 is lost and the idea is though you know this is really kind of related to the previous postulate in the theory so it turns out that an inelastic collision is one in which molecules lose kinetic energy and typically they're losing kinetic energy because they have these two objects that collide in this case molecules would have some propensity to stick to each other well because we already said that they are not going to stick to each other they have no attractive forces that's actually what's kind of guaranteeing that the nest next postulate here is true uh in that there's all collisions are
• 25:30 - 26:00 elastic and there's no loss of kinetic energy finally we say that you know a gas is compound composed of a large number of molecules that are in random motion that is the last major postulate we'll talk about one minor one here but postulate of kinetic molecular theory and so uh the last one though really is and this one's a little more minor and some of you guys might not even get this one so but it says that the kinetic energy on average is proportional to temperature so it turns out if you double the temperature of a gas you'll actually
• 26:00 - 26:30 double its average kinetic energy and we have to say average though so because it turns out in a sample of a gas not all the molecules or atoms are going to be moving at the same speed some will be faster some will be slower what you can say though is that if you double the temperature then the average speed will double as well and therefore i shouldn't say the average speed the average kinetic energy will double we'll see what the speed actually does uh towards the end of this lesson okay so now we know the conditions at which pv equals nrt needs to be true you should know those postulates memorize
• 26:30 - 27:00 them for kinetic molecular theory the first two are the most important so again gas molecules have negligible volume so and uh that there are no attractive forces between the molecules and again those are the ones we use to describe why this lovely mathematical expression p equals nrt the ideal gas law is most true again at low pressures and high temperatures and you should know that as well okay now actually using this it's a plug and chug kind of equation here and a couple things you should know
• 27:00 - 27:30 so r here is what we call the universal gas constant and it has a whole host of values expressed in different units and you're probably only likely to see it in one of two units and this is the one we'll be using here so if you take like a more advanced class though sometimes you'll see it in a general chemistry majors class but more commonly like a physical chemistry class they might give you are expressed in like 10 or 12 different units however you're likely
• 27:30 - 28:00 only going to see it in two and this is one of them and the other one is this guy so it turns out if you're doing any kind of calculations with you know pressure and atmospheres and volume and leaders kind of stuff you're probably using this value right here but this one down here that with it which is in joules per mole kelvin instead if you're doing anything in terms of energy then you're probably using this value instead
• 28:00 - 28:30 now it turns out like i said you might you know actually be provided with r in a more advanced class in 10 or 12 different units that way you could you know have the pressure expressed in any units you want you know instead of atmospheres you get a version where it's in pascals or kilopascals or or tor you know things of a sort and same thing with the volume instead of liters it might be meters cubed and stuff well it turns out that liter atmosphere is actually a unit of energy that combination just like joules of energy notice the mole kelvin parts are the same and so of joules unit of energy so
• 28:30 - 29:00 is a liter atmosphere well it turns out for this to come out in joules your pressure would have to use the si unit of pascals and your volume would have to use the si unit of meters cubed so to use this value and so what you're going to find though is that if you're doing pv equals nrt calculations students will never use this one here otherwise they'd have to remember how to convert you know their pressure into pascals and their volume into meters cubed and they'd have to remember that that's what they even need for this to work and so they almost never use that
• 29:00 - 29:30 one and i say almost never again maybe an advanced class maybe but probably not but yeah this one here because they know how to get the volume in liters and the pressure and atmospheres so and life is good but this also tells you then that if you use this value of the constant which is most likely one you're going to use it tells you all the units you need your volume's got to be in liters your pressure's got to be in atmospheres your temperature has to be in kelvin but again we said temperatures always got to be in kelvin in this chapter and then obviously you got moles of gas here all right we will use this
• 29:30 - 30:00 really briefly at the end of this lesson but i'm going to get rid of it for now and now we'll just take a look at a couple of different calculations you might see using the ideal gas law here and so question here on your on the study guide says what is the volume of two moles of argon gas behaving ideally at a temperature of 298 kelvin and a pressure of 2.0 atmospheres so in this case we're solving for volume so i'm just going to rearrange the expression here to solve for volume and we'll get volume equals n
• 30:00 - 30:30 r t over p and just brought p over here and divided through and in this case we were given everything we need in the units we need them so we're told that it's two moles of argon gas and so in this case it didn't matter that it was argon at all so the equation just calls for the number of moles of gas whether it's argon or carbon dioxide or nitrogen or oxygen it does not matter it just needs whatever the number of moles of gas is so then we'll plug in r here
• 30:30 - 31:00 sometimes you'll see the shortened down to 0.0821 use whatever you're provided with here all right so there's oh i still need to get t in there and in this case we were told it was at a temperature of 298 kelvin so it's given in kelvin right what we need and then finally we'll divide by the pressure in again atmosphere so the units cancel and we're told that it's at a pressure of 2.0 atmospheres
• 31:00 - 31:30 cool and you can see all your units are going to work out here the moles here is going to cancel the kelvin here is going to cancel the atmosphere here is going to cancel and our volume is going to come out in liters and let me grab my calculator all right so let's do some plugging in chugging here in fact the twos will cancel so really we just got to do 0.08206 times 298 and we're going to get
• 31:30 - 32:00 24.5 liters that is not in proper scientific notation and i don't care not worried about scientific notation it's not improper sig figs that i don't care not worried about sig figs in this particular lesson so i really should have rounded this probably down to 24 because it was really 24.45 and i've really got uh actually maybe you have to get down to one sigfig and i don't really want to do that so i'm just going to keep a decimal place and not worry about sig figs
• 32:00 - 32:30 all right so that's the answer to this question it was fairly straightforward you know there are four variables in this equation and one constant now those four variables i gave you three you plug them in you solve for the one you don't know life is good so however we can make this harder and that's what i do in the next question the question says what is the volume of 80 grams of argon gas behaving ideally at a temperature of 25 degrees celsius and a pressure of 1520 torr and what makes this more difficult is that you know you're told that it's 80
• 32:30 - 33:00 grams of argon gas you're told that it's a temperature of 25 degrees celsius and you're told that it's a pressure of 1520 torr and the problem is is none of these are actually expressed in a way in which it will get plugged into that equation i don't need the mass of a gas i need the moles of gas and so you'd have to do a conversion here right so 80 of grams of argon gas you look it up on the periodic table and it turns out that one mole of argon weighs not that
• 33:00 - 33:30 weighs 40 grams and we find out that we've got two moles of argon great just like we a minute ago were provided that we had two moles of argon so 25 degrees celsius we just need to simply add 273 to find out that we're at 298 kelvin just like we had 298 kelvin in the calculation above and finally 1 520 torr we want to convert that to atmospheres to plug it in again all based on the units of r
• 33:30 - 34:00 and one atmosphere equals 760 torr and we'll find out that we're at a pressure of two atmospheres and so in this case we had to do a conversion for all three of these before we got them into the units required for plugging them in here based on the value of that universal gas constant we're using but notice all these came out to the same values we had just a second ago and so if you solve for volume you're still going to get this 24.5 liters
• 34:00 - 34:30 so be prepared for it to be a little more challenging and have to do some conversions more than likely on a typical exam question all right so now we're going to really briefly talk about rms speed that's root mean square speed and so you typically talk about something that's a root mean square when there's actually no overall value for that particular quantity because it's going in all directions so in this case like in this you know room right now the gas molecules in the air they have just as high a likelihood of moving in any particular direction and there's no net movement you know flow of
• 34:30 - 35:00 air in this room and so as a result the average velocity if we we actually you know calculated it and tabulated for all the molecules would be zero uh because you know anything moving to the left would cancel out you know things moving to the right if they're at equal magnitude and stuff and so you find out lo and behold that the average velocity is zero but that's not a very informative number because i want to know like what what speed like how fast are the molecules in this room actually traveling around i mean they're bouncing into each other and stuff and i realized bouncing off the walls and stuff but but how fast are they really moving
• 35:00 - 35:30 so and that's what root mean square uh speed will allow us to do in this case so it's it's kind of like an average it's not the same thing by any stretch but it's kind of like an average when you don't want direction to kind of make uh you know just result in all of them canceling out and giving you a value of zero for something like this so uh in this case it turns out we have a lovely formula for this and we'll derive that real quick we said with uh kinetic molecular theory that the kinetic energy on average was proportional to temperature so and if you look back up on your study guide i actually gave you this expression right here so it turns
• 35:30 - 36:00 out for a monatomic ideal gas here so your kinetic energy your average kinetic energy actually equals three halves r t and if you've taken a physics class you know that kinetic energy is one half mass times velocity squared so in this case that's going to equal three halves r t and so we can get rid of a couple of things here here's the halves part we'll get rid of and so we get v squared if we solve for v squared move the m over it's going to equal 3 rt over m and
• 36:00 - 36:30 then we just take the square root of both sides and you find your velocity would equal the square root of 3r t over m well instead of velocity here again we're actually using root mean square speed u here is going to stand for speed but you can kind of see where this equation comes from a couple things we really got to be careful here though is that if we're trying to get root mean square speed well the si unit for speed is the meter per second and if we're trying to get this to come out in s i units well then we better make sure we're using si units for the entire
• 36:30 - 37:00 calculation and that sucks this is the one place in this chapter where instead of the 0.08206 liter atmospheres per mole kelvin value of r you're going to preferentially use the 8.314 joules per mole kelvin because the joule is the s i is r expressed in si units whereas the liter atmosphere is definitely not s i so one more thing here is that your molar mass here is what mu stands for and that molar mass has to be in si
• 37:00 - 37:30 units as well and the si unit for mass is not the gram it's the kilogram and so you're going to want to express that not in grams per mole but kilograms per mole instead so you got to remember both of those things you got to use the proper value of r and get your molar mass in kilograms per mole rather than grams per mole after that though you'll be good to go so the question we're going to take a look at here is what is the root mean square speed of o2 gas molecules at 273 kelvin so uh and in this case it's just going to be a plug and chug kind of situation
• 37:30 - 38:00 here and so we're going to get our rms speed it's going to equal square root of 3 times 8.314 joules per mole kelvin times temperature of 273 kelvin all over the molar mass well again for o2 for oxygen the molar mass just o is 16 grams per mole but for o2 it's going to be 32 grams per mole but i don't want grams from all i want kilograms per mole so that's going to be instead of 32 grams well it's going to be 0.032
• 38:00 - 38:30 divided by 1000 kilograms per mole and from here then it's just plug and chug so we're going to take 3 times 8.314 times 273 divided by .032 and then i'm going to take the square root of that answer and get 461 meters per second
• 38:30 - 39:00 so that's rather fast the average speed if you will again it's not really an average the root mean square speed of one of the o2 molecules in this room would be a little faster than this because we're not at 273 kelvin we're at 298 kelvinish in this room and so but it's going to be on the order of this though just a little bit faster than this that's actually rather fast if you think about it cool and that's root mean square speed so you know faster than you'd think now one last thing i want to talk about
• 39:00 - 39:30 here with root mean square speed here is this lovely uh diagram here which gives you the maxwell distribution of speeds and we we've said we have to talk about you know average kinetic energy and stuff like this because not all molecules are moving at the same speed or velocity well in this case this gives you a graph of kind of what it looks like and so here i've got actually you know three different graphs on here i'm going to focus on one of them for a second so let's just focus on the red and that's going to be the maxwell distribution of speeds for a gas
• 39:30 - 40:00 at a particular you know particular gas at a particular temperature and what you'll find is that you know they're not all moving at the same velocity but there is this nice lovely looks like fairly gaussian distribution it turns out it's not perfectly gaussian but looks that way and obviously your average would be you know right somewhere up towards the top here and stuff like that life is good so well then what are these other two graphs well they're one of two things you can look at this maxwell distribution of speeds either as a function of temperature or as a function of molar mass and what you
• 40:00 - 40:30 find out is that as you get higher and higher temperatures you're going to get higher root mean square speeds and so you could look at this as what you're going to find is your distribution is going to flatten out and you're going to get higher velocities and so the green here would be the highest temperature the blue would be the second highest temperature and then the red here would be the lowest temperature and you know that way most of your distribution is going to be at a lower velocity so you could look at this as three different temperatures but you could also look at it as three
• 40:30 - 41:00 different molar masses and what you'll find out is that you know your rms speed here is inversely proportional to the molar mass so a larger molar mass would have a lower speed on average or or root mean square speed in this case and so what you could find out is that these could also instead could be three different gases at the same temperature with different molar masses and the red one would represent the largest molar mass having the smallest velocity so and then the green one would represent the lowest molar mass representing therefore the highest velocity and so some of you are
• 41:00 - 41:30 going to see this maxwell distribution of speeds kind of like equation or i shouldn't say equation but uh distribution on a graph and maybe you have to identify a difference either in temperature or a difference in molar mass kind of a thing so but also again helps to you know just to help us realize that uh the gases when we talk about the average kinetic energy here being proportional temperature and stuff like that the reason we have to you know talk about an average because there really is a distribution so for which of these are there gas molecules moving the slowest well they all start over here at zero so
• 41:30 - 42:00 they're all gonna have some slow molecules it's just the one in red here's gonna have more slow molecules than say the one in blue or the one in green and so on and so forth so that is that maxwell distribution of speeds