Compute the Work: Pressure analogous to Volume

[Const@ntine]

Homework Statement

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?

Homework Equations

W = - ∫^Vf _ViPdV

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!

 

[haruspex]

So, I need a function of P that contains V.

I think this statement

the pressure is analogous to the volume.

means you are to take P and V as being in a constant ratio.

 

[Const@ntine]

I think this statement

means you are to take P and V as being in a constant ratio.

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?

 

[haruspex]

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?

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

 

[Const@ntine]

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

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