BE.180:FirstOrderDecay
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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 [math]\displaystyle{ t_{1/2} }[/math]).
Approach:
- Write differential equation for change in X over time.
- Solve equation for [X] as a function of time, t.
- Integrating from [math]\displaystyle{ X_{(t=0)} }[/math] to [math]\displaystyle{ X_{(t=t)} }[/math]
- Solving at the limits produces...
- Which provides a general analytical solution for X as a function of time, t
- Now, note that at [math]\displaystyle{ t_{1/2} }[/math], [math]\displaystyle{ X_{(t=t)}/X_{(t=0)} = 0.5 }[/math] by definition. So we can substitute and get...
- Which is the same as...