First-Order Reaction

Theorem
Let a substance decompose spontaneously in a first-order reaction.

Let $x_0$ be a measure of the quantity of that substance at time $t = 0$.

Let the quantity of the substance that remains after time $t$ be $x$.

Then:
 * $x = x_0 e^{-k t}$

where $k$ is a positive constant called the rate constant.

Half-Life
The rate of decay of a radioactive element is often given in terms of its half-life, or the time it takes for the material to be reduced to half its quantity.

The half-life $T$ of a material whose rate constant is $k$ is given by:
 * $T = \dfrac {\ln 2} k$

Proof
From the definition of a first-order reaction, the rate of change of the quantity of the substance is proportional to the quantity of the substance present at any time.

As the rate of change is a decrease, this rate will be negative.

Thus the differential equation governing this reaction is given by:


 * $-\dfrac {\mathrm d x}{\mathrm d t} = k x, k > 0$

This is an instance of the Decay Equation, and has the solution:
 * $x = x_0 e^{-k t}$

Proof of Half-Life
At time $t = T$ the material has been reduced to $x = \dfrac {x_0} 2$.

This gives: