Question

Can someone please explain how work done on a gas is equal to -Pi * Vi * ln(Vf / Vi) ?

Answer 1

This is one of our class slides. I'm confused at how

$$ W = -\int P\ dV $$ turns into $$ W = -P_i V_i \int \frac{dV}{V} $$

Answer 2

Just take a calm look at the question again, you see it gives the gas law pv =nRT ?

Answer 3

Yes, I see that it gives the gas law.
Pressure * Volume is a constant, and is also equal to the number of moles, times the gas constant R, times the temperature.

Answer 4

I know it's right there, and it's probably extremely obvious.

Answer 5

Just having trouble seeing it right away.

Answer 6

Ok, the gas law is true in general for an ideal gas, but your question considers the special case of an Isothermal change, you see that ?

Answer 7

And so that is why we can say, for these isothermal changes, that PV is constant. - ok so far ?

Answer 8

Okay, so it's constant because in an isothermal change - the temperature remains constant, and therefore n, R, and T are all constant.

Answer 9

Right.
So now, suppose we know p and v at the start of the change, and at some other point in the process, so we have Pi and Vi, the initial values, and we have P and V some time later.
What can we say about PV ?

Answer 10

Sorry, i meant we know Pi and Vi only - what can we say about PV a bit later ?

Answer 11

PV is still going to equal the same value, but V will increase from 2m^3 to 3m^3, and therefore P has to drop proportionally

Answer 12

well if as we said above pv is constant, then we know that pv at the start is the same as pv a bit later, in other words PiVi = PV - we agree on that

Answer 13

So now, look at your first integral - just take a look at it for a moment to remind yourself what it looks like - ok ?

Answer 14

Agreed

Answer 15

Ok, the problem is, we don't know what to do with P - but wait ! what's another expression we can use instead of P ?

Answer 16

Hint - we were just talking about it.

Answer 17

I think I see it now!

$$ P_i V_i = PV $$
$$ P = \frac{P_iV_i}{V} $$

Answer 18

yes, i think you got it !

Answer 19

PiVi is just a constant and can come outside the integral when you replace P with PiVi/V

Answer 20

then you have to integrate 1/V, which i am sure you know is natural log of V - problem solved ?

Answer 21

Okay, I understand now that you pointed it out, thank you!

But let me ask you - *why* is it that I needed to replace P with PiVi, since P is independent of the integral? Is it because we needed Vi for the integral?

Answer 22

And yes I have no problems with integration

Answer 23

Correction: "*why* is it that I needed to replace P with PiVi/V"

Answer 24

well to do the integral you need to express p in terms of the variable of integration, which in this case is V

Answer 25

Ahhh, okay, I think I understand now.

Thank you for your help!

Answer 26

welcome