BE.180:FirstOrderDecay

First Order Decay (of anything)
Givens:
 * A pile of some thing, X.
 * A first-order chemical process by which X is destroyed (or transformed into something else).

Tasks:
 * Compute amount of X remaining as a function of time.
 * Compute amount of time until there is half as much X as there is now (this length of time is called the "half-life" of X or $$t_{1/2}$$).

Approach: $$\frac{dX}{dt} = -k_d * [X]$$ $$\frac{dX}{[X]} = -k_d * dt$$ $$[lnX]_{X_{(t=0)}}^{X_{(t=t)}} = [-k_d*t]_{(t=0)}^{(t=t)}$$ $$ln \Bigg( \frac{X_{(t=t)}}{X_{(t=0)}} \Bigg) = -k_d*t$$ $$X_{t=t} = X_{t=0} * e^{-k_d*t}$$ $$ln(0.5) = -k_d*t_{1/2}$$ $$0.69 = k_d*t_{1/2}$$
 * Write differential equation for change in X over time.
 * Solve equation for [X] as a function of time, t.
 * Integrating from $$X_{(t=0)}$$ to $$X_{(t=t)}$$
 * Solving at the limits produces...
 * Which provides a general analytical solution for X as a function of time, t
 * Now, note that at $$t_{1/2}$$, $$X_{(t=t)}/X_{(t=0)} = 0.5$$ by definition. So we can substitute and get...
 * Which is the same as...