Air pump physics problem

  1. Air pump is pumping out a gas from a vessel (vessel's volume is V).
    After each cycle i it pumps out dV.
    Gas is ideal. T=Const.
    After how much cycles pressure P will decrease k times?

    Here is my work.

    PV = nRT
    Initially n=n(0)

    After each cycle n decreases (1-dV/V) times.
    So, after x cycles n(x)=n(0)(1-dV/V)^x
    k=p(0)/p(x)=n(0)/n(x)
    n(x)/n(0)=(1-dV/V)^x
    So x =ln(1/k)/ln(1-dV/V)

    But the answer is x=ln(k)/ln(1+dV/V)

    As I understand, they assume that after each cycle n decreases 1/(1+dV/V) times. In this way i really receive x=ln(k)/ln(1+dV/V)
    I don't understand it. Help me please.
     
  2. jcsd
  3. Has anybody any ideas?
     
  4. They have done some mathematics with x =ln(1/k)/ln(1-dV/V) to arrive at a more nice looking answer.

    x= -ln (k) / ln (1- dV/V)
    = ln (k) / ln [(1-dV/V)] ^-1

    now, [(1-dV/V)] ^-1 = 1 + dV/V (approximately)

    so, x = ln (k) / ln [(1+dV/V)
     
  5. Each cycle corresponds to a T=const. process. (in another version of this problem the pumping process is very fast and a the transformation would correspond to an adiabatic process - all you have to do is to add the exponent [tex]\gamma[/tex] to the equations below). After the first cycle the air volume increases from V to V+dV and the pressure decreases at [tex]p_1[/tex]. You have
    [tex]p_0\cdot V=p_1\cdot (V+dV)[/tex]
    After the first cycle, the pump takes out the air from dV, and you get again
    [tex]p_1=p_0 \frac{V}{V+dv}[/tex]
    In the second cycle,
    [tex]p_1\cdot V=p_2\cdot (V+dV)[/tex]
    and
    [tex]p_2=p_1 \frac{V}{V+dv}=p_0 \frac{V^2}{(V+dv)^2}[/tex]

    and so on.....

    [tex]p_n=p_{n-1} \frac{V}{V+dv}=p_0 \frac{V^n}{(V+dv)^n}[/tex]
     
    Last edited: Feb 9, 2005
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