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Thermodynamics problem involving adiabatic vessels

  1. Jan 25, 2016 #1

    Titan97

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    1. The problem statement, all variables and given/known data
    Two adiabatic vessels containing ##n## moles of mono atomic and diatomic gas respectively are connected by a rod of length ##l##, cross sectional area ##A## and thermal conductivity ##k##. The surface of the rod is insulated.
    The initial temperatures of the vessels are ##T_1## and ##T_2## respectively. Find the time taken for the temperature difference to be half the initial temperature difference.

    2. Relevant equations
    $$\Delta U=nC_v\Delta T$$
    ##C_{v1}=\frac{3nR}{2}##
    ##C_{v2}=\frac{5nR}{2}##

    3. The attempt at a solution
    The gases will undergo an isochoric process. The heat from one vessel moves to the other along the rod.

    Heat lost by vessel 1 = heat gained by vessel 2

    $$\frac{3}{2}nR(T_1-{T'}_1)=\frac{5}{2}nR({T'}_2-T_2)$$
    If the heat energy lost by vessel 1 is ##dQ## in a time ##dt## and the temperature difference is ##\Delta T##,
    $$\frac{dQ}{dt}=\frac{\Delta T\cdot kA}{L}$$
    But I am not able to continue.
     
    Last edited: Jan 25, 2016
  2. jcsd
  3. Jan 25, 2016 #2
    Here's a hint:
    $$\frac{dU_1}{dt}=-kA\frac{(T_1'-T_2')}{L}$$
    $$\frac{dU_2}{dt}=+kA\frac{(T_1'-T_2')}{L}$$
     
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