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

$\blacksquare$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate