the topic of this video is on relating the pressure volume amount and temperature of gases ultimately um talking about the ideal gas law the learning objectives are on your screen so you can go ahead and pause the video now to write those down I'm going to first before jumping into the ideal gas law just introduce some some observations about gas the behavior of gases under different conditions and um many of these laws have names so the first one is I guess sometimes known as amonton's law although I this this is honestly the first time I've seen it written as amonton's law um uh but uh the more common one at least in my own experience is that of um uh the scientist uh Joseph Louie gay lasak and so sometimes or oftentimes uh in my own experience I've seen it written as gay lao's law and really what this is doing is exploring the relationship between pressure and temperature at constant volume all right and um so if we think about a fixed volume container we can think about like um some sort of rigid vessel uh like we have in the figure here in this R rigid vessel we have some some container with the gas in it we can heat that gas in a water bath and and uh sort of monitor what's happening to the pressure with this pressure gauge as a function of temperature and what we see is that when we increase the temperature of the gas the pressure increases this is intuitive and this is actually you know you might if you have a pressure C cooker an instant pot insta pot in your kitchen um that's exactly what's happening it doesn't oftentimes uh at least mine doesn't have a pressure gauge on it but certainly the pressure goes up uh quite a bit when you heat this rigid vessel so um you can also plot this relationship here um and if we explore pressure on the y- axis as a function of temperature you see that there's a linear relationship when temperature goes up so too does pressure so mathematically what we would say with this type of relationship with this positive slope um when we plot one variable against the other is that pressure is proportional to temperature when temperature goes up so too does pressure if we want to write this as as a mathematical expression we could say pressure is equal to temperature I left a little space here on purpose because we can't just leave it as pressure equals temperature that's clearly not the case they are different units and um but they do they are related to one another um so what we can say here just for the time being is that there's must be some constant K uh that that's related to the the the slope there right um of that of that linear plot and so this is our relationship so K is just some constant we don't need to Define that right now this is just a relationship pressure is directly proportional to temperature the next law that we can talk about is referred to as Charles's Law Charles's Law this is relating um this is exploring what happens uh uh between um volume and temperature so exploring that relationship when you actually have constant pressure this could be some type of vessel where it's not a rigid vessel but it can expand to make sure that the pressure is always constant you could think about an imaginary balloon that can expand to Infinity where the pressure won't start increasing when once you stretch out the material too far you could also think about a piston that um allows the the the volume to fluctuate but it's it's it's set such that um it will always adjust the volume so that the pressure does not um change um so if we look at the plot here um in Charles's Law exploring volume and temperature where volume as a function of temperature um in this plot we see again similar to uh amonton's law or gay laak's law that there is a direct relationship here that is when temperature increases so too does the uh volume right if we go from 100 to 200 Kelvin we see that we go from some lower volume up to a higher volume this makes sense if you heat up a piston that it's allowed to have the the the sort of the a variable um volume it would expand right if you heat up a balloon um it will expand but of course the pressure will at some point increase um because the material will no longer expansion but we're talking under this particular controlled case of constant of constant pressure um so what we can say here is similar to the first scenario um these are directly proportional to one another so we can write v um uh is proportional to T and if we want to write this as an mathematical expression uh we would say that V is equal to some constant lowercase k multiplied by T V is not equal to T they are different um variables but they are related to one another and they're directly proportional in that relationship the next law that we can look at is called Bo's law boils law Bo boils law and boils law explores the relationship between volume and pressure this time at constant temperature the example given for boils law is a syringe where we can take this syringe plunger that is the inner part of the overall device that I'm circling in red this is the plunger and we can either push that plunger into the barrel of the syringe or we can pull that plunger out of the barrel of the syringe and in this particular case um all we're doing is measuring the pressure this would not be a very good syringe for doing anything of uh like biomedical importance but it's good for looking at um pressure changes um in boils law so uh what we can say here is that if we look um over in this plot where we're we're looking at the effect of um or the uh uh pressure as a function of volume we have a little bit different Behavior now actually very different behavior from what we've seen in the other laws um essentially what's happening is that the um when we increase the volume okay so we pull that plunger out we actually see that the pressure drops okay conversely okay if we think about what happens when you shove that plunger into the barrel of the syringe let's say we go from 20 uh a volume of 20 Millers and we uh reduce that volume by half by shoving the syringe down so it's only 10 Millers of space of available to the gas again at constant temperature what happens is we see that this pressure goes up from about 10 here to 20 in this plot this is a very different relationship than before so so what's happening is when volume decreases pressure increases when pressure decreases volume increases that is an inversely proportional relationship not only that but as written um here it's not linear so what we can say is what if we plot the inverse of pressure if this is an inverse relationship we plot inverse pressure as a function of volume we get a linear relationship where we increase the volume here from like let's say 10 to 20 and we see an increase in the inverse pressure so what does that mean it means that if we want to write out boils law as a uh sort of the expression here um the the relationship is that pressure is inversely proportional to volume so we write one over v um and then as a mathematical expression we could write P equals um K some constant K times 1 over V or just you know some constant K Over V again um uh p is not exactly equal to 1 over B but they are proportional to one another there's an inverse relationship there okay the uh last law before we get into the ideal gas law um is going to be uh avagadro's law and you're wonder you might be wondering like avagadro as an avagadro's number yes uh and so avagadro's law explores um the quantity of gas and um volume so n here is not principal quantum number anymore um it is actually going to stand for a mo quantity okay so moles so moles of gas and its relationship with volume and this is going to be at constant pressure and temperature I don't have a plot to show you for this um but it's very intuitive if you think about when you're inflating a balloon let's say with um a noble gas like helium um the more helium you add the bigger the balloon gets the more volume is required for that uh for increasing amounts of gas so you could imagine then that um that's a direct relationship the more gas the more moles of the gas you put into um a container the larger the volume that gas wants to occupy if we allow constant pressure and temperature so um what that means is that we would predict that uh volume is directly proportional to n the mole amount of gas or we could write volume is equal to some constant K multiplied by n when you combine all of these laws that we uh uh have discussed when you combine these you uh we we come up with something called the ideal gas law the ideal gas law when everything is combined is p v equals nrt this equation uh encompasses all of the relationships we've discussed so far in those individual laws those individual observations you already know pressure volume mole amount and temperature the let's just call this some constant thing that I was doing throughout all of the other laws is now coales into a singular constant that we use in the ideal gas law that is capital r this is the ideal gas constant and depending on the units it comes in different you can use different values for it the two most common that we will see is um r equal 0.08206 the units here and these are very important that you keep track of the units liter atmospheres per mole Kelvin uh you can also sometimes see ideal gas constant I mean there are many versions of the ideal gas constant but um another one is 8.314 kilopascal liters or liter kilopascals uh per mole Calin okay so what you clearly uh uh what's what should be hopefully um uh uh registering now is that your units of all the other variables or your units of the other variables will depend on which ideal gas constant you use so be very mindful of which ideal gas constant you use because you have to make sure that the units cancel out appropriately so why don't we go ahead and um do a practice problem Oh I want to say one more thing for the ideal gas law any gas that um obeys this relationship is called an ideal gas so any gas that obeys uh this relationship and by this relationship I mean the ideal gas law um is called an ideal gas so it's often times the case that um in in most contexts we just assume ideal behavior of gases even if they have nonideal behavior um in some experiments let's do a quick example um so what we're going to do is uh oops actually I have it over here I forgot about that okay so using the ideal gas law methane CH4 is being considered for use as an alternative Automotive fuel to replace gasoline one gallon of gasoline could be replaced by 655 G of methane what is the volume of this much methane at 25° C and 20 and 70 for 745 T so uh if we remember the ideal gas law here PV equals nrt um what we can do here is actually uh uh solve we want to solve for volume so the final expression that we will want to use is NR over U nrt over P but we have to be uh let's pick what ideal gas constant we want to use uh I'm going to use r equal 0.08206 ler atmospheres per mole Kelvin so that means that we need um units of uh uh atmospheres moles and Calvin in all the other um variables so to begin we can convert that 655 gram uh methane quantity using the molar mass of methane 16.04 G per 1 Mo will give us 4 4.8 moles of methane so we have moles that's good um you'll notice though that it gave us temperature in degrees celsius but we need it in kelvin um so temperature is going to be equal to 25° C but we can convert uh celsus to Kelvin by adding 273 so this gives us a value of 298 Kelvin and pressure was given to us in um T 7 45 T but we need atmospheres again I know that because atmospheres appears in our ideal gas constant units so we can use this uh conversion factor 760 t uh is equivalent to one atmosphere this gives us 0.980 atmospheres so now if we plug everything in volume is equal to 40.8 moles multiplied by here R again is 0.08206 liter atmospheres over mole Kelvin our temperature is 298 Kelvin all over a pressure of 0.980 atmospheres so Kelvin will cancel out here and here moles will cancel out here and here atmospheres cancel out top and bottom we're left with units of liters that's great because we want the volume and that's going to be in liters so this gives us 1.02 * 103r l so that's how to use the ideal gas law finally what I want to mention is um uh some conditions that are often times used with these types of problems it's called standard temperature and pressure so oftentimes we do experiments where we keep the temperature and pressure uh constant or we we have a well- defined uh uh meaning for these so this is oftentimes um called STP for standard temperature and pressure so you might see something referred to a problem at STP the temperature at STP is 273.15 Kelvin exactly and the pressure is equal to one atmospheres one atmosphere exactly at STP um so this now in the molar Highway uh I've been telling you know uh my section to we have not covered ideal gas law yet you cannot convert between moles and a volume yet uh prior to chapter 8 because we did not talk about the ideal gas law but now in the molar Highway uh uh conversions when you see where you can convert from a mole of some some gaseous substance to the volume of that substance directly um that's going to be under the assumption of standard temperature and pressure and under standard temperature and pressure at STP one mole of gas um equals or it occupies a volume of 22.4 L that 22.4 L is called the standard molar volume and so the the beauty of the ideal gas law and um uh is that if a gas behaves ideally it doesn't matter what the identity is it doesn't matter if it's helium if we have four grams of helium if we have 17 grams of ammonia or if you have 32 grams of o2 one Mo if it's one mole of an ideal gas will occupy 22.4 lers uh uh at STP regardless of what it is