# 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).

• 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).

Approach:

• Write differential equation for change in X over time.
$\frac{dX}{dt} = -k_d * [X]$
• Solve equation for [X] as a function of time, t.
$\frac{dX}{[X]} = -k_d * dt$
• Integrating from X(t = 0) to X(t = t)
$[lnX]_{X_{(t=0)}}^{X_{(t=t)}} = [-k_d*t]_{(t=0)}^{(t=t)}$
• Solving at the limits produces...
$ln \Bigg( \frac{X_{(t=t)}}{X_{(t=0)}} \Bigg) = -k_d*t$
• Which provides a general analytical solution for X as a function of time, t
$X_{t=t} = X_{t=0} * e^{-k_d*t}$
• Now, note that at t1 / 2, X(t = t) / X(t = 0) = 0.5 by definition. So we can substitute and get...
ln(0.5) = − kd * t1 / 2
• Which is the same as...
0.69 = kd * t1 / 2