Rate of a Chemical Reaction: Average and Instantaneous Rate

Chemical kinetics is concerned with the speed of a chemical reaction -- how quickly reactants

disappear and products appear -- and, later in this chapter, with the molecular-level reasons a

reaction proceeds at the particular speed it does.

Average rate. For a reaction $R \rightarrow P$, the average rate over a time interval $\Delta t$ is

the change in concentration of a reactant or product divided by the time taken:

A negative sign is written for a reactant because its concentration falls with time

($\Delta[R]$ is negative), while rate itself is always reported as a positive quantity; no such sign is

needed for a product, whose concentration rises. The usual units are $\text{mol L}^{-1}\text{time}^{-1}$

(e.g. $\text{mol L}^{-1}\text{s}^{-1}$ or $\text{mol L}^{-1}\text{min}^{-1}$), though for a gas-phase

reaction pressure units such as $\text{atm s}^{-1}$ are sometimes used instead.

Instantaneous rate. The average rate depends on how large an interval $\Delta t$ is chosen, and it

usually changes as the reaction proceeds (fastest at the start, when reactant concentration is

highest, and slowing as reactants are used up). The rate at one particular instant -- the

instantaneous rate -- is obtained by shrinking $\Delta t$ down to an infinitesimally small interval,

i.e. by taking the limit $\Delta t \rightarrow 0$:

Graphically, if concentration is plotted against time, the average rate over an interval is the slope

of the straight line joining the two end-points of that interval, while the instantaneous rate at a

particular time is the slope of the tangent to the concentration-time curve at that one point. In

practice, the instantaneous rate at time $t$ can be closely approximated experimentally by measuring

the average rate over a very short interval centred on $t$.

Relating the rates of different species. For a general reaction $aA + bB \rightarrow cC + dD$, the

various species are not all consumed or formed at the same numerical rate whenever their

stoichiometric coefficients differ -- $C$, for instance, forms twice as fast as $A$ is consumed if

$c = 2a$. To have one single, unambiguous "rate of the reaction" that does not depend on which species

happens to be watched, each rate of change is divided by its own stoichiometric coefficient:

This single expression is what is meant, unambiguously, by "the rate of the reaction" at any instant,

and it is the quantity that appears in a reaction's rate law, introduced in the next section.