0:04 the topic of this video is on relating
0:06 the pressure volume amount and
0:10 temperature of gases ultimately um
0:13 talking about the ideal gas law the
0:14 learning objectives are on your screen
0:16 so you can go ahead and pause the video
0:18 now to write those down I'm going to
0:20 first before jumping into the ideal gas
0:22 law just introduce some some
0:24 observations about gas the behavior of
0:27 gases under different conditions and um
0:30 many of these laws have names so the
0:32 first one is I guess sometimes known as
0:35 amonton's law although I this this is
0:37 honestly the first time I've seen it
0:40 written as amonton's law um uh but uh
0:42 the more common one at least in my own
0:44 experience is that of um uh the
0:48 scientist uh Joseph Louie gay
0:51 lasak and so sometimes or oftentimes uh
0:53 in my own experience I've seen it
0:55 written as gay lao's
0:58 law and really what this is doing is
1:00 exploring the relationship between
1:05 pressure and temperature at constant
1:08 volume all right and um so if we think
1:09 about a fixed volume container we can
1:12 think about like um some sort of rigid
1:14 vessel uh like we have in the figure
1:17 here in this R rigid vessel we have some
1:19 some container with the gas in it we can
1:22 heat that gas in a water bath and and uh
1:25 sort of monitor what's happening to the
1:26 pressure with this pressure gauge as a
1:28 function of temperature and what we see
1:30 is that when we increase the temperature
1:32 of the gas the pressure increases this
1:33 is intuitive and this is actually you
1:35 know you might if you have a pressure C
1:38 cooker an instant pot insta pot in your
1:40 kitchen um that's exactly what's
1:42 happening it doesn't oftentimes uh at
1:44 least mine doesn't have a pressure gauge
1:47 on it but certainly the pressure goes up
1:50 uh quite a bit when you heat this rigid
1:53 vessel so um you can also plot this
1:56 relationship here um and if we explore
1:59 pressure on the y- axis as a function of
2:01 temperature you see that there's a
2:03 linear relationship when temperature
2:05 goes up so too does pressure so
2:07 mathematically what we would say with
2:09 this type of relationship with this
2:11 positive slope um when we plot one
2:13 variable against the other is that
2:17 pressure is proportional to
2:19 temperature when temperature goes up so
2:21 too does pressure if we want to write
2:23 this as as a mathematical expression we
2:26 could say pressure is equal to
2:27 temperature I left a little space here
2:29 on purpose because we can't just leave
2:31 it as pressure equals temperature that's
2:33 clearly not the case they are different
2:35 units and um but they do they are
2:39 related to one another um so what we can
2:40 say here just for the time being is that
2:44 there's must be some constant K uh that
2:46 that's related to the the the slope
2:49 there right um of that of that linear
2:51 plot and so this is our relationship so
2:53 K is just some constant we don't need to
2:55 Define that right now this is just a
2:57 relationship pressure is directly
3:01 proportional to temperature
3:03 the next law that we can talk about is
3:06 referred to as Charles's Law
3:08 Law
3:11 Charles's Law this is relating um this
3:15 is exploring what happens uh uh between
3:17 um volume and temperature so exploring
3:20 that relationship when you actually have
3:23 constant pressure this could be some
3:25 type of vessel where it's not a rigid
3:27 vessel but it can expand to make sure
3:30 that the pressure is always constant you
3:31 could think about an imaginary balloon
3:33 that can expand to Infinity where the
3:35 pressure won't start increasing when
3:36 once you stretch out the material too
3:39 far you could also think about a piston
3:42 that um allows the the the volume to
3:45 fluctuate but it's it's it's set such
3:48 that um it will always adjust the volume
3:50 so that the pressure does not um
3:53 change um so if we look at the plot here
3:55 um in Charles's Law exploring volume and
3:58 temperature where volume as a function
4:00 of temperature um in this plot we see
4:04 again similar to uh amonton's law or gay
4:06 laak's law that there is a direct
4:08 relationship here that is when
4:12 temperature increases so too does the uh
4:15 volume right if we go from 100 to 200
4:18 Kelvin we see that we go from some lower
4:21 volume up to a higher volume this makes
4:23 sense if you heat up a piston that it's
4:27 allowed to have the the the sort of the
4:31 a variable um volume it would expand
4:33 right if you heat up a balloon um it
4:35 will expand but of course the pressure
4:38 will at some point increase um because
4:40 the material will no longer expansion
4:41 but we're talking under this particular
4:44 controlled case of constant of constant
4:47 pressure um so what we can say here is
4:50 similar to the first scenario um these
4:52 are directly proportional to one another
4:56 so we can write v um uh is proportional
4:59 to T and if we want to write this as an
5:02 mathematical expression uh we would say
5:06 that V is equal to some constant
5:09 lowercase k multiplied by T V is not
5:12 equal to T they are different um
5:15 variables but they are related to one
5:16 another and they're directly
5:18 proportional in that
5:22 relationship the next law that we can
5:28 look at is called Bo's law boils
5:30 law Bo
5:34 boils law and boils law explores the
5:37 relationship between volume and pressure
5:39 this time at
5:41 constant temperature the example given
5:45 for boils law is a syringe where we can
5:47 take this syringe plunger that is the
5:50 inner part of the overall device that
5:52 I'm circling in red this is the plunger
5:55 and we can either push that plunger into
5:56 the barrel of the syringe or we can pull
5:58 that plunger out of the barrel of the
6:01 syringe and in this particular case um
6:02 all we're doing is measuring the
6:04 pressure this would not be a very good
6:07 syringe for doing anything of uh like
6:09 biomedical importance but it's good for
6:12 looking at um pressure changes um in boils
6:14 boils
6:18 law so uh what we can say here is that
6:20 if we look um over in this plot where
6:22 we're we're looking at the effect of um
6:25 or the uh uh pressure as a function of
6:27 volume we have a little bit different
6:28 Behavior now actually very different
6:29 behavior from what we've seen in the
6:32 other laws um essentially what's
6:34 happening is that the um when we
6:36 increase the volume okay so we pull that
6:39 plunger out we actually see that the pressure
6:41 pressure
6:45 drops okay conversely okay if we think
6:46 about what happens when you shove that
6:49 plunger into the barrel of the syringe
6:53 let's say we go from 20 uh a volume of
6:56 20 Millers and we uh reduce that volume
6:58 by half by shoving the syringe down so
6:59 it's only 10 Millers of space of
7:01 available to the gas again at constant
7:04 temperature what happens is we see that
7:07 this pressure goes up from about 10 here
7:10 to 20 in this plot this is a very
7:12 different relationship than before so so
7:14 what's happening is when volume
7:17 decreases pressure increases when
7:20 pressure decreases volume increases that
7:22 is an inversely proportional
7:24 relationship not only that but as
7:27 written um here it's not linear so what
7:29 we can say is what if we plot the
7:30 inverse of pressure if this is an
7:32 inverse relationship we plot inverse
7:35 pressure as a function of volume we get
7:38 a linear relationship where we increase
7:40 the volume here from like let's say 10
7:44 to 20 and we see an increase in the
7:47 inverse pressure so what does that mean
7:49 it means that if we want to write out
7:52 boils law as a uh sort of the expression
7:54 here um the the relationship is that
7:57 pressure is inversely proportional to
8:01 volume so we write one over v um and
8:03 then as a mathematical expression we
8:07 could write P equals um K some constant
8:11 K times 1 over V or just you know some
8:12 constant K Over
8:16 V again um uh p is not exactly equal to
8:19 1 over B but they are proportional to
8:20 one another there's an inverse
8:24 relationship there
8:28 okay the uh last law before we get into
8:31 the ideal gas law um is going to be uh avagadro's
8:34 avagadro's
8:36 law and you're wonder you might be
8:38 wondering like avagadro as an avagadro's
8:42 number yes uh and so avagadro's law
8:46 explores um the quantity of gas and um
8:49 volume so n here is not principal
8:52 quantum number anymore um it is actually
8:57 going to stand for a mo quantity okay so
9:00 moles so moles of gas and its
9:02 relationship with volume and this is
9:07 going to be at constant pressure and
9:09 temperature I don't have a plot to show
9:12 you for this um but it's very intuitive
9:14 if you think about when you're inflating
9:17 a balloon let's say with um a noble gas
9:22 like helium um the more helium you add
9:25 the bigger the balloon gets the more
9:28 volume is required for that uh for
9:29 increasing amounts of gas
9:33 so you could imagine then that um that's
9:36 a direct relationship the more gas the
9:38 more moles of the gas you put into um a
9:40 container the larger the volume that gas
9:42 wants to occupy if we allow constant
9:46 pressure and temperature so um what that
9:48 means is that we would predict that uh
9:50 volume is directly proportional to n the
9:53 mole amount of gas or we could write
9:55 volume is equal to some constant K
9:57 multiplied by
10:00 n when you combine all of these laws
10:05 that we uh uh have discussed when you
10:08 combine these you uh we we come up with
10:19 law the ideal gas law when everything is
10:23 combined is p v equals
10:27 nrt this equation uh encompasses all of
10:29 the relationships we've discussed so far
10:31 in those individual laws those individual
10:32 individual
10:35 observations you already know pressure
10:38 volume mole amount and temperature the
10:41 let's just call this some constant thing
10:44 that I was doing throughout all of the
10:47 other laws is now coales into a singular
10:49 constant that we use in the ideal gas
10:52 law that is capital r this is the ideal gas
10:57 constant and depending on the units it
10:59 comes in different you can use different
11:01 values for it the two most common that
11:06 we will see is um r equal
11:09 0.08206 the units here and these are
11:10 very important that you keep track of
11:14 the units liter atmospheres per mole
11:17 Kelvin uh you can also sometimes see
11:19 ideal gas constant I mean there are many
11:21 versions of the ideal gas constant but
11:22 um another one is 8.314
11:25 8.314
11:28 kilopascal liters or liter kilopascals
11:30 uh per mole
11:34 Calin okay so what you clearly uh uh
11:37 what's what should be hopefully um uh uh
11:39 registering now is that your units of
11:41 all the other variables or your units of
11:43 the other variables will depend on which
11:46 ideal gas constant you use so be very
11:47 mindful of which ideal gas constant you
11:49 use because you have to make sure that
11:52 the units cancel out appropriately so
11:54 why don't we go ahead and um do a
11:56 practice problem Oh I want to say one
11:58 more thing for the ideal gas law any gas
12:02 that um obeys this relationship is
12:06 called an ideal gas so any
12:08 gas that
12:11 obeys uh this
12:14 this
12:16 relationship and by this relationship I
12:22 mean the ideal gas law um is called an
12:27 ideal gas so it's often times the case
12:29 that um in in most contexts we just
12:31 assume ideal behavior of gases even if
12:36 they have nonideal behavior um in some
12:38 experiments let's do a quick
12:41 example um so what we're going to do is
12:43 uh oops actually I have it over here I
12:44 forgot about that okay so using the
12:46 ideal gas law methane CH4 is being
12:48 considered for use as an alternative
12:51 Automotive fuel to replace gasoline one
12:53 gallon of gasoline could be replaced by
12:56 655 G of methane what is the volume of
12:59 this much methane at 25° C and 20 and 70
13:05 for 745 T so uh if we remember the ideal
13:08 gas law here PV equals
13:12 nrt um what we can do here is actually
13:14 uh uh solve we want to solve for volume
13:16 so the final expression that we will
13:26 P but we have to be uh let's pick what
13:28 ideal gas constant we want to use uh I'm
13:30 going to use r equal 0.08206
13:32 0.08206
13:36 ler atmospheres per mole
13:38 Kelvin so that means that we need um
13:42 units of uh uh atmospheres moles and
13:46 Calvin in all the other um variables so
13:50 to begin we can convert that 655 gram uh
13:53 methane quantity using the molar mass of methane
13:54 methane
14:00 16.04 G per 1 Mo will give us 4
14:03 4.8 moles of methane so we have moles
14:06 that's good um you'll notice though that
14:07 it gave us temperature in degrees
14:10 celsius but we need it in kelvin um so
14:12 temperature is going to be equal to 25°
14:15 C but we can convert uh celsus to Kelvin
14:21 273 so this gives us a value of
14:25 298 Kelvin and pressure was given to us
14:27 in um
14:32 T 7 45 T but we need atmospheres again I
14:33 know that because atmospheres appears in
14:36 our ideal gas constant units so we can
14:40 use this uh conversion factor 760 t uh
14:42 is equivalent to one atmosphere this
14:45 gives us
14:48 0.980 atmospheres so now if we plug
14:51 everything in volume is equal to
14:56 40.8 moles multiplied by here R again is
14:59 0.08206 liter
15:02 atmospheres over mole Kelvin our
15:03 temperature is
15:09 298 Kelvin all over a pressure of
15:12 0.980 atmospheres so Kelvin will cancel
15:14 out here and here moles will cancel out
15:16 here and here atmospheres cancel out top
15:18 and bottom we're left with units of
15:20 liters that's great because we want the
15:21 volume and that's going to be in liters
15:27 so this gives us 1.02 * 103r l so that's
15:30 how to use the ideal gas law finally
15:34 what I want to mention is um uh some
15:35 conditions that are often times used
15:39 with these types of problems it's called standard
15:41 standard
15:45 temperature and pressure so oftentimes
15:47 we do experiments where we keep the
15:49 temperature and pressure uh constant or
15:51 we we have a well- defined uh uh meaning
15:54 for these so this is oftentimes um
15:56 called STP for standard temperature and
15:58 pressure so you might see something
15:59 referred to
16:03 a problem at STP the temperature at STP is
16:04 is
16:09 273.15 Kelvin exactly and the
16:12 pressure is equal to one atmospheres one
16:14 atmosphere exactly at
16:17 STP um so this now in the molar Highway
16:19 uh I've been telling you know uh my
16:21 section to we have not covered ideal gas
16:25 law yet you cannot convert between moles
16:28 and a volume yet uh prior to chapter 8
16:30 because we did not talk about the ideal
16:34 gas law but now in the molar Highway uh
16:35 uh conversions when you see where you
16:37 can convert from a mole of some some
16:40 gaseous substance to the volume of that substance
16:41 substance
16:44 directly um that's going to be under the
16:45 assumption of standard temperature and
16:47 pressure and under standard temperature
16:52 and pressure at
16:55 STP one mole of
16:58 gas um equals or it occupies a volume of
17:03 22.4 L that 22.4 L is called the standard
17:05 standard molar
17:07 molar
17:09 volume and so the the beauty of the
17:13 ideal gas law and um uh is that if a gas
17:15 behaves ideally it doesn't matter what
17:17 the identity is it doesn't matter if
17:19 it's helium if we have four grams of
17:23 helium if we have 17 grams of ammonia or
17:26 if you have 32 grams of o2 one Mo if
17:29 it's one mole of an ideal gas will
17:33 occupy 22.4 lers uh uh at STP regardless
