# 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*t*_{1 / 2}).

Approach:

- 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...

*l*

*n*(0.5) = −

*k*

_{d}*

*t*

_{1 / 2}

- Which is the same as...

*k*

_{d}*

*t*

_{1 / 2}