0:00:01.289,0:00:03.731
- Something you see a lot[br]when doing thermodynamics

0:00:03.731,0:00:06.316
especially problems[br]involving the first law

0:00:06.316,0:00:08.208
are what are called PV Diagrams.

0:00:08.208,0:00:09.810
Now, the P stands for Pressure

0:00:09.810,0:00:11.458
and the V stands for Volume.

0:00:11.458,0:00:12.794
And this gives you a diagram of what

0:00:12.794,0:00:15.556
the pressure and volume[br]are in any given instant.

0:00:15.556,0:00:16.475
So what does this mean?

0:00:16.475,0:00:17.612
Well, imagine you had a container

0:00:17.612,0:00:20.433
full of a gas and there's[br]a movable piston on top.

0:00:20.433,0:00:22.337
Piston can move up or down, changing

0:00:22.337,0:00:23.915
the amount of volume, right?

0:00:23.915,0:00:25.744
This is the volume we're talking about,

0:00:25.744,0:00:27.885
is the volume within here.

0:00:27.885,0:00:30.939
So that movable piston can[br]change that amount of volume.

0:00:30.939,0:00:33.063
And that would change the[br]amount of pressure inside,

0:00:33.063,0:00:35.965
depending on what heat is[br]added, how much work is done.

0:00:35.965,0:00:37.964
So say we started with a[br]certain amount of volume, right?

0:00:37.964,0:00:39.427
Let's say we start with that much volume.

0:00:39.427,0:00:41.590
And the pressure inside[br]is probably not zero.

0:00:41.590,0:00:43.827
If there's any gas[br]inside, it can't be zero.

0:00:43.827,0:00:45.602
And so we come over to here,

0:00:45.602,0:00:47.529
let's say we start at[br]this point right here.

0:00:47.529,0:00:48.505
Now, what do we do?

0:00:48.505,0:00:53.505
I know if I push the piston[br]down, my volume decreases.

0:00:53.705,0:00:56.550
And if I pull the piston[br]up, my volume increases.

0:00:56.550,0:01:01.550
So if I push the piston down,[br]I know volume goes down.

0:01:01.728,0:01:05.082
That means on this graph,[br]I'm going that way.

0:01:05.082,0:01:07.846
Piston going down means decreasing volume.

0:01:07.846,0:01:09.343
What about piston going up?

0:01:09.343,0:01:12.709
Well, if the piston goes up,[br]then my volume's increasing

0:01:12.709,0:01:15.647
and I know on my graph I'd[br]better be going to the right.

0:01:15.647,0:01:17.864
Now maybe I'm going up and right.

0:01:17.864,0:01:19.443
Maybe I'm going down and right.

0:01:19.443,0:01:21.347
All I know is, my volume[br]better be increasing,

0:01:21.347,0:01:24.911
so this is increasing volume,[br]that's increasing volume,

0:01:24.911,0:01:26.387
that's increasing volume.

0:01:26.387,0:01:28.406
This is not increasing volume,

0:01:28.406,0:01:29.810
so I know if my piston goes up,

0:01:29.810,0:01:32.051
my volume increases, I gotta be going

0:01:32.051,0:01:34.942
to the rightward in[br]some way on this graph.

0:01:34.942,0:01:36.405
And if my piston goes down, I better

0:01:36.405,0:01:38.713
be going to the left[br]on this graph somehow.

0:01:38.713,0:01:40.321
Now, what happens to the pressure?

0:01:40.321,0:01:42.625
You gotta know a little[br]more detail about it.

0:01:42.625,0:01:44.578
But just knowing the[br]direction of the piston,

0:01:44.578,0:01:46.366
that lets you know which[br]way you go on this graph.

0:01:46.366,0:01:47.909
So say I push the piston down.

0:01:47.909,0:01:50.046
Say I push it down really fast.

0:01:50.046,0:01:51.706
What do you think's gonna[br]happen to the pressure?

0:01:51.706,0:01:52.786
The pressure's probably gonna go up.

0:01:52.786,0:01:53.981
How would I represent that?

0:01:53.981,0:01:57.706
Well, volume's gotta go down,[br]pressure would have to go up,

0:01:57.706,0:02:01.446
so I might take a path that[br]looks something like this.

0:02:01.446,0:02:03.803
Volume's gotta go down to the left.

0:02:03.803,0:02:05.243
Pressure's gotta go up, so maybe

0:02:05.243,0:02:06.543
it does something like that.

0:02:06.543,0:02:08.637
There's really infinitely many ways

0:02:08.637,0:02:11.024
the gas could get from[br]one state to another.

0:02:11.024,0:02:12.707
It could take any possible range

0:02:12.707,0:02:15.401
and unless you know the exact details,

0:02:15.401,0:02:17.445
it's hard to say exactly[br]what's gonna happen.

0:02:17.445,0:02:20.566
So there's infinitely many[br]possibilities on this diagram.

0:02:20.566,0:02:23.299
You can loop around,[br]it's not like a function.

0:02:23.299,0:02:24.700
You can do something like this.

0:02:24.700,0:02:28.379
This gas can take some crazy[br]path through this PV Diagram.

0:02:28.379,0:02:30.679
There's infinitely many ways it can take.

0:02:30.679,0:02:33.825
But there are four thermodynamic processes

0:02:33.825,0:02:37.424
that are most commonly[br]represented on a PV Diagram.

0:02:37.424,0:02:40.442
Again, these are not the[br]only four possibilities.

0:02:40.442,0:02:42.220
These are just the four that are kind of

0:02:42.220,0:02:44.468
the simplest to deal with mathematically.

0:02:44.468,0:02:46.445
And they're often a good representation

0:02:46.445,0:02:49.763
and accurate approximation[br]to a lot of processes

0:02:49.763,0:02:52.066
so the math's good, they work pretty well,

0:02:52.066,0:02:53.268
we talk about them a lot.

0:02:53.268,0:02:56.391
The first one is called[br]in isobaric process.

0:02:56.391,0:02:59.234
Iso means constant, so[br]whenever you see iso

0:02:59.234,0:03:01.812
before something, it means constant.

0:03:01.812,0:03:05.718
Whatever follows next,[br]and this one's isobaric.

0:03:05.718,0:03:08.488
Baric, well bars, that's[br]a unit of pressure,

0:03:08.488,0:03:10.705
so baric is talking about pressure.

0:03:10.705,0:03:13.491
Isobaric means constant pressure.

0:03:13.491,0:03:15.825
So how do you represent[br]this on a PV DIagram?

0:03:15.825,0:03:18.144
Well, if you wanna[br]maintain constant pressure,

0:03:18.144,0:03:20.248
you can't go up or down, because if I were

0:03:20.248,0:03:22.303
to go up, my pressure would be increasing.

0:03:22.303,0:03:24.592
If I were to go down, my[br]pressure would be decreasing.

0:03:24.592,0:03:28.432
The only option available is[br]to go along a horizontal line.

0:03:28.432,0:03:31.231
So this would be in iso, well, sometimes

0:03:31.231,0:03:35.027
they're called isobars,[br]and isobar for short.

0:03:35.027,0:03:39.171
This is an isobar, this[br]is an isobaric expansion

0:03:39.171,0:03:42.091
if I go to the right, cause[br]I know volume's increasing.

0:03:42.091,0:03:43.873
And if I go to the left it would be

0:03:43.873,0:03:48.029
an isobaric compression because[br]volume would be decreasing.

0:03:48.029,0:03:50.072
But it doesn't have to be[br]in this particular spot.

0:03:50.072,0:03:52.231
It could be anywhere on this PV Diagram,

0:03:52.231,0:03:55.324
any horizontal line is gonna be an isobar,

0:03:55.324,0:03:57.363
an isobaric process.

0:03:57.363,0:03:59.934
Now, I bring up the isobaric process first

0:03:59.934,0:04:02.263
because it allows me to[br]show something important

0:04:02.263,0:04:06.092
that's true of every process[br]that's just easier to see

0:04:06.092,0:04:07.811
for the isobaric process.

0:04:07.811,0:04:10.447
In physics, the area under the curve

0:04:10.447,0:04:13.129
often represents something significant.

0:04:13.129,0:04:15.119
And that's gonna be true here as well.

0:04:15.119,0:04:17.065
Let's try to figure out[br]what the area under this

0:04:17.065,0:04:19.328
curve represents.

0:04:19.328,0:04:20.651
So first of all, to find the area

0:04:20.651,0:04:22.567
of this rectangle, we know it's gonna be

0:04:22.567,0:04:25.005
the height times the[br]width, what's the height?

0:04:25.005,0:04:26.503
The height's just the pressure, right?

0:04:26.503,0:04:28.524
The value of this pressure over here

0:04:28.524,0:04:30.288
is gonna be the height and the width

0:04:30.288,0:04:32.748
is the change in volume so if I started

0:04:32.748,0:04:35.280
with V initial and I ended with V final,

0:04:35.280,0:04:38.539
let's say it was the expansion[br]instead of the compression.

0:04:38.539,0:04:42.129
This V final minus V initial, this delta V

0:04:42.129,0:04:45.624
is going to represent the[br]width of this rectangle.

0:04:45.624,0:04:49.745
So we know area is going to[br]be the value of the pressure

0:04:49.745,0:04:51.823
times the change in the volume.

0:04:51.823,0:04:53.240
Well, what does that mean?

0:04:53.240,0:04:56.026
We know that pressure, we know[br]the definition of pressure,

0:04:56.026,0:04:58.324
pressure is just the force per area.

0:04:58.324,0:05:02.413
So on this gas, even on a[br]force exerted on it per area,

0:05:02.413,0:05:05.208
and the change in volume,[br]what do we know is the volume?

0:05:05.208,0:05:07.228
How could I represent the volume in here?

0:05:07.228,0:05:09.644
I know this piston has some area,

0:05:09.644,0:05:12.604
so there's some area that this piston has.

0:05:12.604,0:05:14.798
And then there's a certain height.

0:05:14.798,0:05:17.303
This inner cylinder of volume in here

0:05:17.303,0:05:19.966
has a certain height[br]and then a certain area

0:05:19.966,0:05:22.637
so we know the volume is[br]just height times area.

0:05:22.637,0:05:25.749
So it would be height times[br]the area of the piston.

0:05:25.749,0:05:27.990
Which of these is[br]changing in this process?

0:05:27.990,0:05:29.406
Well, the area is not changing.

0:05:29.406,0:05:31.333
If the area of this piston changed,

0:05:31.333,0:05:32.807
it either let some of the gas out

0:05:32.807,0:05:35.512
or it would bust through[br]the sides of the cylinder,

0:05:35.512,0:05:37.788
both of which we're[br]assuming is not happening.

0:05:37.788,0:05:40.412
So I can pull area out of this delta sign

0:05:40.412,0:05:42.025
since the area is constant.

0:05:42.025,0:05:46.715
And what I get is F times A over A times

0:05:46.715,0:05:48.692
the change in the height.

0:05:48.692,0:05:51.451
Well the A is canceled, A cancels A

0:05:51.451,0:05:54.561
and I get F times the[br]change in the height.

0:05:54.561,0:05:57.490
But look at, this is just[br]force times a distance.

0:05:57.490,0:06:01.110
Times the distance by[br]which this height changes.

0:06:01.110,0:06:04.010
So delta H will be the amount by which

0:06:04.010,0:06:06.591
this piston goes up or down.

0:06:06.591,0:06:08.866
And we know force times the distance

0:06:08.866,0:06:12.232
by which you apply that[br]force is just the work.

0:06:12.232,0:06:16.511
So now we know the area[br]under this isobaric process

0:06:16.511,0:06:19.686
represents the work done either on the gas

0:06:19.686,0:06:23.226
or by the gas depending[br]on which way you're going.

0:06:23.226,0:06:27.151
So this area is the work, this[br]area, the value of this area

0:06:27.151,0:06:31.667
equals the amount of work[br]done on the gas or by the gas.

0:06:31.667,0:06:32.863
How do you figure out which?

0:06:32.863,0:06:35.046
Well, technically this area represents

0:06:35.046,0:06:38.064
the work done by the gas, because if we're

0:06:38.064,0:06:39.828
talking about a positive area,

0:06:39.828,0:06:42.545
mathematically that means[br]moving to the right,

0:06:42.545,0:06:44.264
like on a graph in math class.

0:06:44.264,0:06:46.631
The area, positive area,[br]you're moving to the right.

0:06:46.631,0:06:49.824
So if we want to be[br]particular and precise,

0:06:49.824,0:06:52.146
we'll say that this is a[br]process moving to the right.

0:06:52.146,0:06:55.049
And we know if the volume is going up

0:06:55.049,0:06:56.628
like this graph is going to the right,

0:06:56.628,0:06:58.043
which means volume is increasing,

0:06:58.043,0:07:00.005
we know that gas is doing work.

0:07:00.005,0:07:03.022
So technically, this area[br]is the work done by the gas.

0:07:03.022,0:07:05.927
You can see that as well[br]since this is P delta V.

0:07:05.927,0:07:08.619
If your delta V comes out positive,

0:07:08.619,0:07:10.872
pressure is always[br]positive, if your Delta V

0:07:10.872,0:07:13.974
comes out positive, the[br]volume is increasing.

0:07:13.974,0:07:16.166
That means work is being done by the gas.

0:07:16.166,0:07:17.269
So you have to be careful.

0:07:17.269,0:07:19.022
If you calculate this P delta V

0:07:19.022,0:07:21.785
and you go to your first law equation,

0:07:21.785,0:07:25.965
which remember, says delta U is Q plus W,

0:07:25.965,0:07:28.251
well you can't just plug[br]in the value of P delta V.

0:07:28.251,0:07:29.726
This is the work done by the gas,

0:07:29.726,0:07:32.064
so you have to plug in negative that value

0:07:32.064,0:07:34.788
for the work done, and[br]also correspondingly,

0:07:34.788,0:07:36.146
if you were to go to the left,

0:07:36.146,0:07:39.246
if you did have a process[br]that went to the left.

0:07:39.246,0:07:42.176
That is to say the volume was decreasing.

0:07:42.176,0:07:44.202
If you find this area and you're careful,

0:07:44.202,0:07:46.525
then you'll get a negative delta V

0:07:46.525,0:07:49.053
if you're going leftward[br]because you'll end

0:07:49.053,0:07:51.968
with a smaller value for the[br]volume than you started with.

0:07:51.968,0:07:54.701
So if you really treat[br]the left one as the final,

0:07:54.701,0:07:57.082
cause that's where you end[br]up if you're going left,

0:07:57.082,0:07:59.123
and the rightward one as the initial,

0:07:59.123,0:08:01.505
your leftward final point will be smaller

0:08:01.505,0:08:04.386
than your initial point, you[br]will get a negative value here.

0:08:04.386,0:08:07.875
So again, you plug in negative[br]of that negative value.

0:08:07.875,0:08:09.408
You'll get your positive work,

0:08:09.408,0:08:11.708
cause positive work is[br]being done on the gas.

0:08:11.708,0:08:13.460
That sounds very complicated.

0:08:13.460,0:08:15.119
Here's what I do, quite honestly.

0:08:15.119,0:08:17.428
I just look at the shape, I find the area,

0:08:17.428,0:08:19.867
I do the magnitude of the height, right,

0:08:19.867,0:08:21.365
the size of it, no negatives.

0:08:21.365,0:08:23.686
The size of the width, no negatives.

0:08:23.686,0:08:26.147
I multiply the two and then I just look.

0:08:26.147,0:08:27.680
Am I going to the left?

0:08:27.680,0:08:31.397
If I'm going to the left,[br]I know my work is positive.

0:08:31.397,0:08:33.868
If I'm going to the right,[br]I know my work is negative

0:08:33.868,0:08:36.665
that I plug into here, so I[br]just add the negative sign in.

0:08:36.665,0:08:38.919
Makes it me easier for me to understand.

0:08:38.919,0:08:42.378
So I said that this works for[br]any process, how is that so?

0:08:42.378,0:08:43.747
If I take some random process,

0:08:43.747,0:08:46.969
I'm not gonna get a nice[br]rectangle, how is this true?

0:08:46.969,0:08:49.901
Well, if I did take a random process

0:08:49.901,0:08:51.805
from one point to another, say I took

0:08:51.805,0:08:53.778
this crazy path here.

0:08:53.778,0:08:55.791
Even though it's not a perfect rectangle,

0:08:55.791,0:08:58.259
I can break it up into small rectangles

0:08:58.259,0:09:01.445
so I can take this, break[br]this portion up into,

0:09:01.445,0:09:03.681
if I make the rectangle small enough,

0:09:03.681,0:09:06.746
I can approximate any[br]area as the summation

0:09:06.746,0:09:09.334
of a whole bunch of little rectangles.

0:09:09.334,0:09:11.771
And look at, each one of these rectangles,

0:09:11.771,0:09:15.475
well, P delta V, that's the[br]area underneath for that one,

0:09:15.475,0:09:17.438
add them all up, I get[br]the total area undeneath.

0:09:17.438,0:09:22.219
So even though it might be[br]difficult to find this area,

0:09:22.219,0:09:25.820
it's always true that if[br]I could find this area

0:09:25.820,0:09:30.820
under any process, this area[br]does represent the work done.

0:09:31.011,0:09:33.703
And again, it's by the gas.

0:09:33.703,0:09:36.087
So in other words, using the formula

0:09:36.087,0:09:39.682
work done by the gas[br]that we had previously

0:09:39.682,0:09:42.956
equals P times delta V, that works

0:09:42.956,0:09:46.859
for one small little rectangle[br]and you can add all those up,

0:09:46.859,0:09:48.796
but it work for the entire process.

0:09:48.796,0:09:51.810
If you tried to use the,[br]say, initial pressure

0:09:51.810,0:09:53.857
times the total change in volume,

0:09:53.857,0:09:55.842
and that's not gonna[br]give you an exact answer,

0:09:55.842,0:09:58.338
that's assuming you[br]have one big rectangle.

0:09:58.338,0:10:01.182
So this formula won't work[br]for the whole process.

0:10:01.182,0:10:04.480
But we do know if you[br]have an isobaric process,

0:10:04.480,0:10:07.624
if it really is an isobaric process,

0:10:07.624,0:10:09.682
then we can rewrite the first law.

0:10:09.682,0:10:13.199
The first law says that delta U equals Q

0:10:13.199,0:10:15.637
plus work done on the gas?

0:10:15.637,0:10:18.479
Well, we know a formula for[br]the work done by the gas.

0:10:18.479,0:10:20.617
Work done by the gas is P delta V.

0:10:20.617,0:10:21.975
So the work done on the gas is just

0:10:21.975,0:10:25.078
negative P times delta V.

0:10:25.078,0:10:27.060
Here's a formula for the first law

0:10:27.060,0:10:31.540
if you happen to have an isobaric process.

0:10:31.540,0:10:34.002
So an isobaric process is pretty nice.

0:10:34.002,0:10:37.020
It gives you an exact[br]way to find the work done

0:10:37.020,0:10:40.201
since the area underneath[br]is a perfect rectangle.

0:10:40.201,0:10:42.082
But how would you physically set up

0:10:42.082,0:10:44.219
an isobaric process in the lab?

0:10:44.219,0:10:45.960
Well, imagine this, let's say you heat up

0:10:45.960,0:10:49.037
this cylinder, you allow heat to flow in.

0:10:49.037,0:10:51.299
That would tend to increase the pressure.

0:10:51.299,0:10:54.133
So the only way we could[br]maintain constant pressure,

0:10:54.133,0:10:58.056
cause an isobaric process[br]maintains constant pressure,

0:10:58.056,0:11:00.774
if I want the pressure to stay[br]the same as heat flows in,

0:11:00.774,0:11:03.119
I better let this piston move upwards.

0:11:03.119,0:11:06.138
While I add heat I can[br]maintain constant pressure.

0:11:06.138,0:11:08.714
In fact, you might think[br]that's complicated.

0:11:08.714,0:11:10.213
How are you going to do that exactly?

0:11:10.213,0:11:11.860
It's not so bad, just allow the piston

0:11:11.860,0:11:14.856
to come into equilibrium with whatever

0:11:14.856,0:11:18.130
atmospheric pressure plus[br]the weight of this piston is.

0:11:18.130,0:11:20.173
So there's a certain pressure[br]down from the outside

0:11:20.173,0:11:21.915
and then there's the weight of the piston

0:11:21.915,0:11:23.854
divided by the area[br]gives another pressure.

0:11:23.854,0:11:25.850
This heat will try to make[br]the pressure increase,

0:11:25.850,0:11:27.579
but if you just allow this system

0:11:27.579,0:11:30.355
to come into equilibrium[br]with the outside pressure,

0:11:30.355,0:11:32.618
the inside pressure is always gonna equal

0:11:32.618,0:11:35.475
the outside pressure[br]because if it's not equal,

0:11:35.475,0:11:37.889
this piston will move[br]up or down accordingly.

0:11:37.889,0:11:39.515
So if this piston can move freely,

0:11:39.515,0:11:41.639
it'll maintain a constant pressure

0:11:41.639,0:11:44.019
and that would be a way[br]to physically ensure

0:11:44.019,0:11:46.016
that the pressure remains constant

0:11:46.016,0:11:49.290
and you have an isobaric process.

0:11:49.290,0:11:52.076
I'll explain the next three[br]thermodynamic processes

0:11:52.076,0:11:53.549
in the next video.