First-Order Reaction/Proof 2

Theorem

Let a substance decompose spontaneously in 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 the rate constant.

Proof

From Differential Equation governing First-Order Reaction, the differential equation governing this reaction is given by:

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

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

This is an instance of the Decay Equation, and has the solution:

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

$\blacksquare$

Also see

• Half-Life of Radioactive Substance