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Finding the number of moles of an ideal gas in a capillary

  1. Oct 21, 2016 #1
    1. The problem statement, all variables and given/known data
    The temperature across the capillary with constant cross-sectional area and length L is given by ##T=T_0e^{-kx}##. Assuming an ideal gas and constant pressure show the number of moles to be: $$n=\frac{PV(e^{kL} - 1)}{RkLT_0}$$

    2. Relevant equations
    ##PV=nRT##

    3. The attempt at a solution

    The equation of state can be expressed as ##g(P,V,T) = 0## but since pressure is given to be constant we have ##g(V,T) = 0## therefore we can express the volume as ##V=V(T)## from which we can get the differential for V as $$dV = (\frac{\partial V}{\partial T})dT = \frac{nR}{P}dT\\ V =\frac{nR}{P} \int_{T_i}^{T_f}dT = \frac{nR}{P}(T_f-T_i)$$

    Using ##T=T_0e^{-kx}## it is evident that ##T_f = T_0e^{-kL}## and ##T_i = T_0## therefore $$V = \frac{nRT_0}{P}(e^{-kL}-1)\Longrightarrow n=\frac{PV}{nRT_0(e^{-kL}-1)}$$

    Which clearly isn't the correct answer. I'm curious as to what the mistake is following this reasoning.

    Another method I've attempted is for an ideal gas ##n=n(P,V,T)## but pressure is constant therefore ##n=n(V,T)## and we obtain the differential $$dn = \left(\frac{\partial n}{\partial V}\right)_{P,T} dV + \left(\frac{\partial n}{\partial T}\right)_{P,V} dT$$

    Now I know from the the answer that ##\left(\frac{\partial n}{\partial T}\right)_{P,V}## must equal 0 since the answer is obtained from integrating ##dn = \left(\frac{\partial n}{\partial V}\right)_{P,T} dV ## but I can't seem to justify why ##\left(\frac{\partial n}{\partial T}\right)_{P,V}## should be equal to zero without setting both of them equal to zero.
     
  2. jcsd
  3. Oct 21, 2016 #2

    haruspex

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    You need to understand what your differential equation says. You have taken a constant n and expressed how the volume changes if you change the temperature. Since the change in temperature was negative, you got a negative volume change.
    Consider segments length dx and the number of moles they contain.
     
  4. Oct 21, 2016 #3

    Edit: Ok so that's the logic behind taking ##dn/dV## and then multiplying the ##dV## over and integrating, but shouldn't this also come from my second method in the main post?
     
    Last edited: Oct 21, 2016
  5. Oct 21, 2016 #4

    haruspex

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    Your second method considers how n varies as T and V change independently. I cannot think what that means in the context of the question.
    As I posted, your independent variable should be distance along the capillary.
     
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