Differential Equation governing First-Order Reaction

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

The differential equation which governs this reaction is given by:
 * $-\dfrac {\d x} {\d t} = k x$

where:
 * $x$ determines the quantity of substance at time $t$.
 * $k \in \R_{>0}$.

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:
 * $-\dfrac {\d x} {\d t} = k x$

for some $k \in \R_{>0}$.