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Thread: 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 $\displaystyle 100\%$ oxygen.
    In the room there is a machine which takes in $\displaystyle 5$ 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 $\displaystyle V$, the proportion of oxygen at time $\displaystyle t$ (in minutes) is $\displaystyle x(t)$, so $\displaystyle x(0)=1$ and probably $\displaystyle x(t)\to_{t\to\infty} 0$.

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

    Stated differently, $\displaystyle \frac{dx}{dt}=-\frac{5}{V}x(t)$, so that $\displaystyle x(t)=e^{-\frac{5}{V}t}$. 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 $\displaystyle y(t)$ be a function which represents amount of oxygen at time $\displaystyle t$ in minutes.
    Notice that $\displaystyle y'(t)$ is the rate of change.
    Thus, $\displaystyle y'(t) = \text{ rate out } - \text{ rate in }$.
    The rate in is amount of oxygen coming in per minute is $\displaystyle 5\cdot \tfrac{y}{V}$.
    The rate out is $\displaystyle 0$.
    Thus, $\displaystyle y'(t) = - \tfrac{5}{V}y$ with $\displaystyle y(0) = V$.
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