containing 1.85 times 10 to the third liters

of helium gas at 23 degrees Celsius and 765 Torr

is launched into the atmosphere.

The balloon travels for two hours

before bursting at an altitude of 32 kilometers,

where the temperature is negative 44 degrees Celsius

and the pressure is 6.51 Torr.

What is the volume of the balloon just before it bursts?

So pause this video and see if you can figure that out.

All right, so you might already have an intuitive sense

that this has something to do with the ideal gas law,

because they're giving us a bunch of pressures,

volumes, and temperatures,

and the ideal gas law deals with that,

it tells us that pressure times volume

is equal to the number of moles

times the ideal gas constant times temperature.

Now, what's different about this example

is that they aren't just giving us

several of these variables and asking us

to solve one of them,

they're talking about these variables changing,

and how that might affect other variables.

And so one way to think about it is if we divide

both sides by T, you get PV over T is equal to NR.

And in this example, as this balloon goes

to higher and higher altitudes,

the number of moles does not change,

and the ideal gas constant does not change.

So one way to think about it

is that PV over T has to be constant.

So our volume and our temperature could change,

but because this whole expression on the left

has to be constant, that could then determine our pressure.

Or another way to think about it,

you could say your starting pressure

times your starting volume over your starting temperature

is going to be equal to the number of moles

times the ideal gas constant,

which also needs to be equal to your pressure

right before it bursts times the volume

right before it bursts, divided by the temperature

right before it bursts,

or you could just say that P one times V one

over T one is equal to P two times V two over T two.

And so what are these different variables?

Well, let's first think about P one,

so pressure at time one is what, it's 765 Torr.

765 Torr, and what's P two?

That's the pressure just before it burst,

and they tell us it's 6.51 Torr, much lower pressure,

which makes intuitive sense, we're at a higher altitude.

6.51 Torr, now, what is V one?

Well, they tell us that right over there,

that is 1.85 times 10 to the third liters.

Now, what is V two?

Well, that's what they want us to figure out,

what is the volume of the balloon just before it bursts?

So I'll put a little question mark there.

And then, last but not least, what is T one?

Well, they tell us the starting temperature

is at 23 degrees Celsius,

but you have to think on more of an absolute scale,

and deal with temperatures in terms of Kelvin,

so to convert 23 degrees Celsius into Kelvin,

you have to add 273, so this is going to be 296 Kelvin,

and then what is T two?

Well, T two is negative 44 degrees Celsius,

if we add 273 to that, let's see,

that's going to be, if we subtract, it's going to be

in my head, 229 Kelvin.

And so we have everything we need

in order to solve for V two,

in fact, we can solve for V two

before we even put in these numbers,

if we multiply both sides of this equation

times T two over P two,

and the reason why I'm multiplying it times this

is so that this cancels with this, this cancels with that,

so I have just V two on the right-hand side.

Of course, I have to do that on both sides.

T two over P two, I am going to get,

and I'll now color code it,

I'm going to get that T two times P one

times V one, over P two times T one, T one,

is equal to V two, V two.

So we just have to calculate this right now,

let me give myself a little bit more real estate

with which to do it,

and so we could write that V two is equal to T two,

which is 229 Kelvin, times P one, which is 765 Torr,

times V one, which is 1.85 times 10 to the third liters,

all of that over P two, which is 6.51 Torr,

times T one, which is 296 Kelvin,

and we can confirm that the units work out,

Torr cancels with Torr, Kelvin cancels with Kelvin,

so we're just gonna have a bunch of numbers, a calculation,

and the units we're left with is liters,

which is good, because that is what we care about

when we care about volume.

So this is going to be equal to 229 times 765,

times 1.85 times 10 to the third,

divided by 6.51, divided by 296

is equal to this business right over here.

Let's see, we have three significant digits here,

three significant digits here, three here,

three here, and three there,

so our answer's going to have three significant figures,

so it's going to be, if we round,

it's gonna be 168,000, and so we could just write that

as 168,000 liters, or if we wanna write that

in scientific notation, we could write that

as 1.68 times 10 to the one, two, three, four, five,

so let me write it that way,

so this is going to be equal

to 1.68 times 10 to the fifth liters.

And I always like to do a nice intuition check,

does that makes sense?

So our starting volume was 1,850 liters,

and then our volume got a lot larger,

because we're going to a much higher altitude,

and that does make intuitive sense to me.