Integrated Rate Equation for First Order Reactions

A reaction is first order in a reactant $R$ if its rate is directly proportional to

$[R]$ raised to the first power:

Deriving the integrated rate equation. Rearranging so that all $[R]$ terms are on one side:

Integrating from $[R]_0$ at $t = 0$ to $[R]$ at time $t$:

Converting from natural to common (base-10) logarithms ($\ln x = 2.303\log x$) gives the form most

commonly used for calculation:

The linear plot. Since $\ln[R] = -kt + \ln[R]_0$ is a straight-line equation, plotting $\ln[R]$ (or

equivalently $\log[R]$) against $t$ gives a straight line with slope $-k$ (or $-k/2.303$ for the

$\log_{10}$ form) and $y$-intercept $\ln[R]_0$. This linearity of a logarithmic concentration plot,

in contrast to zero order's linear plot of raw concentration, is the standard experimental test for

first order kinetics.

Units of $k$. Because the ratio $[R]_0/[R]$ inside the logarithm is a pure (dimensionless) number,

the units of a first order rate constant are simply $\text{time}^{-1}$ (e.g. $\text{s}^{-1}$ or

$\text{min}^{-1}$), with no concentration units at all -- a useful, immediate way to recognize a first

order rate constant when reported.

Examples. First order kinetics is extremely common and includes every radioactive decay process

(where "concentration" is replaced by the number of undecayed nuclei), the thermal decomposition of

$\text{N}_2\text{O}_5$, and the decomposition of $\text{H}_2\text{O}_2$ in aqueous solution. A special

and very common case is the pseudo-first-order reaction: a reaction that is genuinely second order

overall, but in which one reactant is present in such large excess (often the solvent itself, such as …