# Compute the Work: Pressure analogous to Volume

Tags:
1. Jul 30, 2017

### Techno_Knight

1. The problem statement, all variables and given/known data

One mole of an ideal gas is warmed slowly, so that the pressure and volume go from (Pi, Vi) to (3Pi, 3Vi), in such a way that the pressure is analogous to the volume.

a) What's the Work (W)?
b) What is the correlation between the temperature and the volume during that process?

2. Relevant equations

W = - ∫Vf ViPdV

3. The attempt at a solution

In this case, I don't have a stable pressure or volume, so I'm at the third case. So, I need a function of P that contains V. I know PV = nRT, and I know that n = 1 mole. Plus, R is a known quantity. Problem is, T is not a constant, so I can't integrate P = nRT/V.

Any help is appreciated!

Last edited: Jul 30, 2017
2. Jul 30, 2017

### haruspex

I think this statement
means you are to take P and V as being in a constant ratio.

3. Jul 30, 2017

### Techno_Knight

So kinda like this:

Pi/Vi = c = P/V

WW = - ∫Vf ViPdV = - ∫Vf VicVdV = - c ∫Vf ViVdV = -c[V2/2] |Vf Vi = -c * (9Vi2/2 - Vi2/2) = -4cVi2 = -4(Pi/Vi)*Vi2 = -4PiVi which is the book's answer.

As for (b):

PV = nRT <=> T = PV/nR = cV2/nR <=> T = (Pi/nrVi)*V2 which is the book's answer.

So technically I just use the "formula" that says that the pressure, divided by the volume, of any instance, is equal to the initial pressure, divided by the initial volume, since these two quantities are analogous, correct?

4. Jul 30, 2017

### haruspex

Yes, but it it is a rather unusual use of the word "analogous". Proportional would have been clearer.

5. Jul 30, 2017

### Techno_Knight

Okay, I'll keep that in mind for next time (translations sometimes have these problems). Thanks for the help!