Compute the Work: Pressure analogous to Volume

In summary: Okay, I'll keep that in mind for next time (translations sometimes have these problems). Thanks for the help!
  • #1
Const@ntine
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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!
 
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  • #2
Darthkostis said:
So, I need a function of P that contains V.
I think this statement
Darthkostis said:
the pressure is analogous to the volume.
means you are to take P and V as being in a constant ratio.
 
  • #3
haruspex said:
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?
 
  • #4
Darthkostis said:
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.
 
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  • #5
haruspex said:
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!
 

FAQ: Compute the Work: Pressure analogous to Volume

1) What is the formula for calculating work using pressure analogous to volume?

The formula for calculating work using pressure analogous to volume is W = PΔV, where W represents work, P represents pressure, and ΔV represents change in volume.

2) How is this formula derived?

This formula is derived from the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. By rearranging this equation, we get ΔV = nRΔT/P. Since nRΔT is constant, we can substitute it for a single variable, k. Therefore, ΔV = k/P, and when multiplied by P, we get W = PΔV.

3) What are the units for work in this formula?

The units for work in this formula are Joules (J) in the SI system and foot-pounds (ft-lb) in the English system.

4) Can this formula be used for any type of gas?

Yes, this formula can be used for any type of gas as long as the gas follows the ideal gas law. However, it is important to note that this formula is an approximation and may not be accurate for gases that deviate significantly from ideal behavior.

5) How is this formula related to the work-energy theorem?

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In the case of pressure analogous to volume, work is done by changing the volume of a gas, which in turn changes its pressure. This change in pressure results in a change in kinetic energy, making the two concepts closely related.

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