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Math Help - Changing Atmosphere

  1. #1
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    Changing Atmosphere

    I came up with my own problem.
    There is nothing complicated.

    Suppose a room (air tight) is filled with 100\% oxygen.
    In the room there is a machine which takes in 5 liters/minute of air and gives up pure nitrogen.

    Find a way to model the amount of oxygen at any given time.
    (Of course use obvious assumption necessary).
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  2. #2
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    Nice problem indeed, thank you. I guess we assume that nitrogen and oxygen mix immediately in the whole volume.

    The whole volume is V, the proportion of oxygen at time t (in minutes) is x(t), so x(0)=1 and probably x(t)\to_{t\to\infty} 0.

    During an infinitesimal time interval \Delta t, the initial volume of oxygen Vx(t) becomes Vx(t+\Delta t)=Vx(t)-5\Delta t x(t) (since 5\Delta t is the volume that was taken away, among which a proportion x(t) was composed of oxygen). So:
    \frac{x(t+\Delta t)-x(t)}{\Delta t}=-\frac{5}{V}x(t).

    Stated differently, \frac{dx}{dt}=-\frac{5}{V}x(t), so that x(t)=e^{-\frac{5}{V}t}. The proportion hence decreases exponentially fast.

    Laurent.
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  3. #3
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    Here is how I would do it.
    Let y(t) be a function which represents amount of oxygen at time t in minutes.
    Notice that y'(t) is the rate of change.
    Thus, y'(t) = \text{ rate out } - \text{ rate in }.
    The rate in is amount of oxygen coming in per minute is 5\cdot \tfrac{y}{V}.
    The rate out is 0.
    Thus, y'(t) = - \tfrac{5}{V}y with y(0) = V.
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