PV diagrams - part 1: Work and isobaric processes | Chemical Processes | MCAT | Khan Academy

- Something you see a lot
when doing thermodynamics

especially problems
involving the first law

are what are called PV Diagrams.

Now, the P stands for Pressure

and the V stands for Volume.

And this gives you a diagram of what

the pressure and volume
are in any given instant.

So what does this mean?

Well, imagine you had a container

full of a gas and there's
a movable piston on top.

Piston can move up or down, changing

the amount of volume, right?

This is the volume we're talking about,

is the volume within here.

So that movable piston can
change that amount of volume.

And that would change the
amount of pressure inside,

depending on what heat is
added, how much work is done.

So say we started with a
certain amount of volume, right?

Let's say we start with that much volume.

And the pressure inside
is probably not zero.

If there's any gas
inside, it can't be zero.

And so we come over to here,

let's say we start at
this point right here.

Now, what do we do?

I know if I push the piston
down, my volume decreases.

And if I pull the piston
up, my volume increases.

So if I push the piston down,
I know volume goes down.

That means on this graph,
I'm going that way.

Piston going down means decreasing volume.

What about piston going up?

Well, if the piston goes up,
then my volume's increasing

and I know on my graph I'd
better be going to the right.

Now maybe I'm going up and right.

Maybe I'm going down and right.

All I know is, my volume
better be increasing,

so this is increasing volume,
that's increasing volume,

that's increasing volume.

This is not increasing volume,

so I know if my piston goes up,

my volume increases, I gotta be going

to the rightward in
some way on this graph.

And if my piston goes down, I better

be going to the left
on this graph somehow.

Now, what happens to the pressure?

You gotta know a little
more detail about it.

But just knowing the
direction of the piston,

that lets you know which
way you go on this graph.

So say I push the piston down.

Say I push it down really fast.

What do you think's gonna
happen to the pressure?

The pressure's probably gonna go up.

How would I represent that?

Well, volume's gotta go down,
pressure would have to go up,

so I might take a path that
looks something like this.

Volume's gotta go down to the left.

Pressure's gotta go up, so maybe

it does something like that.

There's really infinitely many ways

the gas could get from
one state to another.

It could take any possible range

and unless you know the exact details,

it's hard to say exactly
what's gonna happen.

So there's infinitely many
possibilities on this diagram.

You can loop around,
it's not like a function.

You can do something like this.

This gas can take some crazy
path through this PV Diagram.

There's infinitely many ways it can take.

But there are four thermodynamic processes

that are most commonly
represented on a PV Diagram.

Again, these are not the
only four possibilities.

These are just the four that are kind of

the simplest to deal with mathematically.

And they're often a good representation

and accurate approximation
to a lot of processes

so the math's good, they work pretty well,

we talk about them a lot.

The first one is called
in isobaric process.

Iso means constant, so
whenever you see iso

before something, it means constant.

Whatever follows next,
and this one's isobaric.

Baric, well bars, that's
a unit of pressure,

so baric is talking about pressure.

Isobaric means constant pressure.

So how do you represent
this on a PV DIagram?

Well, if you wanna
maintain constant pressure,

you can't go up or down, because if I were

to go up, my pressure would be increasing.

If I were to go down, my
pressure would be decreasing.

The only option available is
to go along a horizontal line.

So this would be in iso, well, sometimes

they're called isobars,
and isobar for short.

This is an isobar, this
is an isobaric expansion

if I go to the right, cause
I know volume's increasing.

And if I go to the left it would be

an isobaric compression because
volume would be decreasing.

But it doesn't have to be
in this particular spot.

It could be anywhere on this PV Diagram,

any horizontal line is gonna be an isobar,

an isobaric process.

Now, I bring up the isobaric process first

because it allows me to
show something important

that's true of every process
that's just easier to see

for the isobaric process.

In physics, the area under the curve

often represents something significant.

And that's gonna be true here as well.

Let's try to figure out
what the area under this

curve represents.

So first of all, to find the area

of this rectangle, we know it's gonna be

the height times the
width, what's the height?

The height's just the pressure, right?

The value of this pressure over here

is gonna be the height and the width

is the change in volume so if I started

with V initial and I ended with V final,

let's say it was the expansion
instead of the compression.

This V final minus V initial, this delta V

is going to represent the
width of this rectangle.

So we know area is going to
be the value of the pressure

times the change in the volume.

Well, what does that mean?

We know that pressure, we know
the definition of pressure,

pressure is just the force per area.

So on this gas, even on a
force exerted on it per area,

and the change in volume,
what do we know is the volume?

How could I represent the volume in here?

I know this piston has some area,

so there's some area that this piston has.

And then there's a certain height.

This inner cylinder of volume in here

has a certain height
and then a certain area

so we know the volume is
just height times area.

So it would be height times
the area of the piston.

Which of these is
changing in this process?

Well, the area is not changing.

If the area of this piston changed,

it either let some of the gas out

or it would bust through
the sides of the cylinder,

both of which we're
assuming is not happening.

So I can pull area out of this delta sign

since the area is constant.

And what I get is F times A over A times

the change in the height.

Well the A is canceled, A cancels A

and I get F times the
change in the height.

But look at, this is just
force times a distance.

Times the distance by
which this height changes.

So delta H will be the amount by which

this piston goes up or down.

And we know force times the distance

by which you apply that
force is just the work.

So now we know the area
under this isobaric process

represents the work done either on the gas

or by the gas depending
on which way you're going.

So this area is the work, this
area, the value of this area

equals the amount of work
done on the gas or by the gas.

How do you figure out which?

Well, technically this area represents

the work done by the gas, because if we're

talking about a positive area,

mathematically that means
moving to the right,

like on a graph in math class.

The area, positive area,
you're moving to the right.

So if we want to be
particular and precise,

we'll say that this is a
process moving to the right.

And we know if the volume is going up

like this graph is going to the right,

which means volume is increasing,

we know that gas is doing work.

So technically, this area
is the work done by the gas.

You can see that as well
since this is P delta V.

If your delta V comes out positive,

pressure is always
positive, if your Delta V

comes out positive, the
volume is increasing.

That means work is being done by the gas.

So you have to be careful.

If you calculate this P delta V

and you go to your first law equation,

which remember, says delta U is Q plus W,

well you can't just plug
in the value of P delta V.

This is the work done by the gas,

so you have to plug in negative that value

for the work done, and
also correspondingly,

if you were to go to the left,

if you did have a process
that went to the left.

That is to say the volume was decreasing.

If you find this area and you're careful,

then you'll get a negative delta V

if you're going leftward
because you'll end

with a smaller value for the
volume than you started with.

So if you really treat
the left one as the final,

cause that's where you end
up if you're going left,

and the rightward one as the initial,

your leftward final point will be smaller

than your initial point, you
will get a negative value here.

So again, you plug in negative
of that negative value.

You'll get your positive work,

cause positive work is
being done on the gas.

That sounds very complicated.

Here's what I do, quite honestly.

I just look at the shape, I find the area,

I do the magnitude of the height, right,

the size of it, no negatives.

The size of the width, no negatives.

I multiply the two and then I just look.

Am I going to the left?

If I'm going to the left,
I know my work is positive.

If I'm going to the right,
I know my work is negative

that I plug into here, so I
just add the negative sign in.

Makes it me easier for me to understand.

So I said that this works for
any process, how is that so?

If I take some random process,

I'm not gonna get a nice
rectangle, how is this true?

Well, if I did take a random process

from one point to another, say I took

this crazy path here.

Even though it's not a perfect rectangle,

I can break it up into small rectangles

so I can take this, break
this portion up into,

if I make the rectangle small enough,

I can approximate any
area as the summation

of a whole bunch of little rectangles.

And look at, each one of these rectangles,

well, P delta V, that's the
area underneath for that one,

add them all up, I get
the total area undeneath.

So even though it might be
difficult to find this area,

it's always true that if
I could find this area

under any process, this area
does represent the work done.

And again, it's by the gas.

So in other words, using the formula

work done by the gas
that we had previously

equals P times delta V, that works

for one small little rectangle
and you can add all those up,

but it work for the entire process.

If you tried to use the,
say, initial pressure

times the total change in volume,

and that's not gonna
give you an exact answer,

that's assuming you
have one big rectangle.

So this formula won't work
for the whole process.

But we do know if you
have an isobaric process,

if it really is an isobaric process,

then we can rewrite the first law.

The first law says that delta U equals Q

plus work done on the gas?

Well, we know a formula for
the work done by the gas.

Work done by the gas is P delta V.

So the work done on the gas is just

negative P times delta V.

Here's a formula for the first law

if you happen to have an isobaric process.

So an isobaric process is pretty nice.

It gives you an exact
way to find the work done

since the area underneath
is a perfect rectangle.

But how would you physically set up

an isobaric process in the lab?

Well, imagine this, let's say you heat up

this cylinder, you allow heat to flow in.

That would tend to increase the pressure.

So the only way we could
maintain constant pressure,

cause an isobaric process
maintains constant pressure,

if I want the pressure to stay
the same as heat flows in,

I better let this piston move upwards.

While I add heat I can
maintain constant pressure.

In fact, you might think
that's complicated.

How are you going to do that exactly?

It's not so bad, just allow the piston

to come into equilibrium with whatever

atmospheric pressure plus
the weight of this piston is.

So there's a certain pressure
down from the outside

and then there's the weight of the piston

divided by the area
gives another pressure.

This heat will try to make
the pressure increase,

but if you just allow this system

to come into equilibrium
with the outside pressure,

the inside pressure is always gonna equal

the outside pressure
because if it's not equal,

this piston will move
up or down accordingly.

So if this piston can move freely,

it'll maintain a constant pressure

and that would be a way
to physically ensure

that the pressure remains constant

and you have an isobaric process.

I'll explain the next three
thermodynamic processes

in the next video.