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.

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:


 * $$-\frac {dx}{dt} = k x, k > 0$$

It can be solved thus:

$$ $$ $$

The initial condition $$x = x_0$$ when $$t = 0$$ gives $$C = \ln {x_0}$$.

Thus:

$$ $$ $$ $$