First-Order Reaction

Theorem
Suppose an object has a tendency to decompose spontaneously into smaller objects at a rate independent of the presence of other objects.

Then the number of objects that decompose in a single unit of time is proportional to the total number present.

Such a reaction is called 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.

Radioactive Decay
The most important first-order reaction is radioactive decay.

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, it is clear that 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: