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The work done by a perfect gas

  1. Feb 22, 2014 #1
    1. The problem statement, all variables and given/known data
    An ideal gas at initial temperature T1 and pressure P1 is compressed by a piston to half its original volume. The temperature is varied so that the relation P=AV always holds and A is a constant. What is the work done n the gas in terms of n (moles of gas) R (gas constant) and T1?


    2. Relevant equations

    Using ideal gas law, PV= nRT

    and assuming the work done is [itex]W = \int^{V_b}_{V_a} P dV[/itex]

    3. The attempt at a solution

    OK, simple enough, right?

    Vb = (1/2) Va

    So V = nRT/P and [itex]V_a = \frac{nRT}{P_1}[/itex] and [itex]V_b = \frac{nRT}{2P_1}[/itex]

    Plug this into the integral above. Since P=AV (in the givens) it should be

    [itex]W = \int^{V_b}_{V_a} (AV) dV = A \int^{V_b}_{V_a} V dV = A\frac{V^2}{2}|^{V_a}_{V_b}[/itex]

    Which leaves me with

    [itex]=\frac{A}{2} \left[\left(\frac{nRT}{2P_1}\right)^2 - \left(\frac{nRT}{P_1}\right)^2\right] = \frac{A}{2}\left(\frac{nRT}{P_1}\right)^2(-3/4) = \frac{-3A}{8}\left(\frac{nRT}{P_1}\right)^2[/itex]

    Yet the answer is listed as [itex]\frac{-3A}{8}\left(\frac{nRT}{P_1}\right)[/itex]

    So I am trying to figure out how I got the extra factor in there. I think I did everything right, but is there some stupid mathematical error I made?

    Anyhow, it's possible there's a typo in the book's answer too. But...

    any help is appreciated, even though I anticipate it will be trivial...
     
  2. jcsd
  3. Feb 22, 2014 #2

    Andrew Mason

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    Homework Helper

    The correct answer is 3nRT1/8, which can be derived from your answer.

    The given answer is not dimensionally correct. The answer has to have dimensions of energy (work) ie. PV or nRT. Since A has dimensions of P/V, the correct answer has to have dimensions of AV2. The given answer has dimensions of AV = P.


    AM
     
  4. Feb 22, 2014 #3
    Thanks lots, I knew there was something stupid I was missing. I looked at it again and the given answer was yours, but I forgot about just plugging P/V in for A. D'OH!!!
     
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