Re: exponential cooling

[palmer@sfu.ca (Leigh Palmer)]

> >   >           T(t) = To + (Ti-To) exp (-t/tau)
>   >
>   > where tau is the time constant for the approach to equilibrium.
>
> The above formula also describes the "law" of exponential decay. Just
> replace T(t) by the counting rate due to radioactive atoms, To by the
> background rate and tau by the mean life of an atom. IS THIS SIGNIFICANT?
> The law of decay can be derived from the assumption that atoms transform
> independently of each other and that the probability of decay (per unit
> time), 1/tau, is constant.

Yes, this is significant. The functional time dependence reappears in the
discharge of a capacitor through a resistor, or the decay of current in an
inductor, or the relaxation of any other nonequilibrium system in which
first order homogeneous linear ordinary differential equation models the
physical system well, a common occurence.

Nature seems to function in many ways. The few we can model well recur for
precisely that reason, but the wonder of it all is that they are so often
apt. That is why I start my noncalculus class with that "Big Crunch"
problem - exponential growth. That, too, is a first order homogeneous
linear ordinary differential equation, but the solution at first glance
looks very different (like 2^n). It is not, and it sets up the later cases
of RC decay and radioactive decay which are curricular items in the course.

Leigh