Brain Twister For the Day

self | 6/6/02 | lafroste

[lafroste]

Here's a good old fashioned math problem. Thanks for the help!

A chemical in an aqueous solution decays exponentially with a half life of t minutes. At what rate must additional chemical be added to the solution in a tank so that the bulk concentration of the chemical remains constant? Fresh solution is NOT added to the tank or withdrawn from the tank, just the chemical is added. Volume of the tank = V (liters). The desired steady state concentration of the chemical = C (mg/l). Assume perfect mixing.

[WhyisaTexasgirlinPA]

If your household is anything like mine, the solution would never make it to the aqueous liquid.......half would have been spilled on the floor, and the other half poured on the dog............ ;^)

[IronJack]

You could calculate this using a slope intercept, y=mx+b, and know the concentration at any given time x. The first-order derivative with respect to time would give you the change in concentration.

That is, if I remember calculus as Sir Isaac Newton taught it to me lo these many years ago.

[John Locke]

The formula for exponential decay is

x(t) = x(0) exp(-kt)

If the half-life is T, then we also have

x(T) = (1/2) x(0)

whence k = loge(2)/T.

Differentiating the first equation,

dx/dt = -k x(0) exp(-kt)

If we start adding more chemical immediately, ie at t=0, this simplifies to

dx/dt = - k x(0) = - (loge(2)/T) x(0)

So you need to add chemical continuously, at a rate such that in each unit of time the amount added is a fraction loge(2)/T of the original amount.