First Order Decay (of anything)
- A pile of some thing, X.
- A first-order chemical process by which X is destroyed (or transformed into something else).
- 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 t1 / 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 t1 / 2, X(t = t) / X(t = 0) = 0.5 by definition. So we can substitute and get...
- Which is the same as...