Integrated Rate Equation for Zero Order Reactions

A reaction is zero order in a reactant $R$ if its rate does not depend on $[R]$ at all:

Deriving the integrated rate equation. Separating variables and integrating from the initial

concentration $[R]_0$ at $t=0$ to the concentration $[R]$ at time $t$:

This is a straight-line equation: a plot of $[R]$ (on the vertical axis) against $t$ (on the horizontal

axis) is a straight line with slope $-k$ and $y$-intercept $[R]_0$. This linearity is the standard

experimental signature used to confirm that a reaction is genuinely zero order -- a curved

concentration-time plot would immediately rule zero order out.

When zero order kinetics arises. A true zero order reaction is comparatively unusual, because it

requires the rate to stay completely constant even as the reactant is progressively used up. It

typically arises when the reaction actually occurs on a fixed, limited number of active sites -- for

instance, the catalytic decomposition of a gas such as $\text{NH}_3$ or $\text{HI}$ on the surface of a

hot metal (platinum or gold) is zero order in the gas as long as the metal surface remains fully

saturated with adsorbed reactant molecules: adding more gas increases only the pressure above the

surface, not the number of molecules actually reacting at any instant, since every available surface

site is already occupied. Certain enzyme-catalyzed reactions behave the same way once the enzyme's

active sites are all saturated with substrate (see the section on enzyme catalysis later in this

chapter), and certain photochemical reactions, whose rate is limited by the fixed intensity of light

absorbed rather than by reactant concentration, are zero order for the same underlying reason: the

rate-limiting resource (surface sites, enzyme active sites, or photons absorbed per second) is fixed …