# Definition:First-Order Reaction

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## Definition

Suppose a body has a tendency to decompose spontaneously into smaller bodies at a rate independent of the presence of other bodies.

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

Such a reaction is called a **first-order reaction**.

### Rate Constant

Let $S$ be a substance which decomposes spontaneously in a **first-order reaction**.

By First-Order Reaction, this is governed by the equation:

- $x = x_0 e^{-k t}$

where:

- $x$ is the quantity of $S$ at time $t$
- $x_0$ is the quantity of $S$ at time $t = 0$
- $k$ is a positive number.

The number $k$ is called the **rate constant**.

$k$ is constant for a particular substance, and different substances, in general, have different **rate constants**.

## Also see

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 4$: Growth, Decay and Chemical Reactions