Consider a chemical reaction of the generic type:

$$a\ A + b\ B \rightleftharpoons c\ C + d\ D$$

where $a$, $b$, $c$ and $d$ are the stoichiometric coefficients. Intuitively we could understand that there must be a concept such as reaction rate that is capable of measuring the speed with which a given reaction takes place. But, formally, how could we define the rate of reaction? For example, if we look at the reagent $A$, we can say that the reaction rate is the rate at which its total concentration in the medium decreases . In other words, the reaction rate would be the rate at which the reactant $A$ disappears, which is given by:

$$v_A = -\ \frac{d[A]}{dt}$$

But this form of definition presents a problem. Let us now consider the reaction product $C$. We could define the reaction rate in the same way as the speed with which the total concentration of $C$ increases in the medium, that is, as the amount:

$$v_C = \frac{d[C]}{dt}$$

And what is the problem? Well, for every $a$ moles of reactant $A$ that disappear in a given period of time, $c$ moles of product $C$ will have appeared. That is:

$$-\ \frac{1}{a}\ \cdot\ \frac{d[A]}{dt} = \frac{1}{c}\ \cdot\ \frac{d[C]}{dt }$$

In the same way we could have looked at any of the other two substances that participate in the reaction, so that in the end we would have four different reaction rates, all of them related to each other by the following equalities:

$$-\ \frac{1}{a}\ \cdot\ \frac{d[A]}{dt} = -\ \frac{1}{b}\ \cdot\ \frac{d[B]} {dt} = \frac{1}{c}\ \cdot\ \frac{d[C]}{dt} = \frac{1}{d}\ \cdot\ \frac{d[D]}{dt }$$

In view of these details, we will define the reaction rate for the process described above, as the quantity $v$ given by:

$$v = -\ \frac{1}{a}\ \cdot\ v_A = -\ \frac{1}{b}\ \cdot\ \ v_B = \frac{1}{c}\ \cdot\ v_C = \frac{1}{d}\ \cdot\ v_D$$

At this point we have to reflect. Defining the reaction rate in this way, we see that it is dependent on the volume of the system. If during the study of a reaction the volume varies (due to a dilution effect, for example) that would affect the value that we are defining but not the real speed with which the reactants are transformed into products.

For these reasons, the IUPAC has been for a long time, first in its 1994 recommendations ( Glossary of terms used in physical organic chemistry – IUPAC Recommendations 1994 ) and later in its Golden Book ( Compendium of Chemical TerminologyGold Book – latest version 2.3.2 of August 19, 2012)( i ), recommended in these cases to use the so-called conversion speed that is defined as the following quantity:

$$\dot\xi = \frac{d\xi}{dt} = \frac{1}{\nu_i}\ \cdot\ \frac{dn_i}{dt}$$

where $\xi$ is the so-called extension of the reaction , $\nu_i$ is the stoichiometric coefficient (with positive sign for the products and negative sign for the reactants) of the substance $X_i$ present in the reaction, and $n_i$ is the amount of substance (usually in moles) in $X_i$. The use of the time derivative of $\xi$ has the advantage that it is independent both of the reference substance $X_i$ that is chosen and of the volume of the system. In a system that keeps its volume constant, the conversion rate per unit volume and the reaction rate are the same throughout the reaction.

However, most of the formulas that we will use use the time derivative of the concentration and we will normally call it the rate of reaction (see recommendation below). The reaction rate is measured in $mol\cdot dm^{-3}\cdot s^{-1}$, as corresponds to the definition we have given, in which we use molarity as a measure of concentration.

For reactions that occur in several stages, this definition of reaction rate (and that of reaction extent) is applicable only if there is no accumulation of intermediate products or formation of side products. The IUPAC recommends that the term reaction rate be used only in cases where it has been experimentally proven that these conditions are applicable.

When there are variations in the amounts of reactants or products that are not due to the reaction process, there is no choice but to apply the full mass balance:

$$Input - Output + Generation = Accumulation$$

which in our case, applying it to any of the substances involved in the reaction, $X_i$, would be as follows ( ii ):

$$\left ( \phi_{X_i}\right )_E - \left (\phi_{X_i}\right )_S + \int_0^V v \cdot dV = \frac{dn_i}{dt} = \nu_i \cdot \frac{d\xi}{dt}$$

Obviously, the integration of the equation can be quite complicated if the flow in or out of $X_i$ varies with time...

For general cases in which the use of the term reaction rate may be ambiguous as stated above, the IUPAC recommends using instead the terms rate of disappearance of reactant $X_i$ or rate of appearance of product $X_j$ as appropriate. to avoid ambiguities in the definition.

Although we have a mathematical formula to define the reaction speed in practice, we must resort to experimental data to determine it in each case. The measurement of speed means having to study how the concentration of one or more chemical species varies over time. These measurements can be made using both chemical and physical methods.

Chemical analysis of species concentrations has the advantage of being specific and usually highly accurate, but has the disadvantages of being slow and disturbing the system. In general, we cannot arrest the course of a reaction while we are entertaining ourselves with an analysis. Therefore, any concentration measurement method that we are going to use must be fast enough so that they do not vary significantly while we are taking the measurement.

Physical methods of measurement are faster than chemical ones, and usually do not disturb the progress of the reaction, but are very rarely specific to a given substance. For example, if we measure the refractive index, it is necessary to calibrate it for the concentrations of each of the species present. In addition, the presence of impurities in the medium or the existence of reactions that compete with the one we are studying are problems that make the measurement very complicated.

The factors that can affect the rate of a reaction are, for the most part, the same as those that are important in equilibrium thermodynamics. Among the most important we can highlight:

• medium temperature;
• ambient pressure;
• concentrations (or activities) of the substances present;
• nature and physical state of the reagents;
• presence of catalysts;
• degree of agitation of the system.

Reaction rates, in general, have a great dependence on temperature, which we must carefully control if we want to make accurate measurements. The environment where the reaction takes place is also important, since many reactions have different mechanisms depending on the environment where they take place.

A reaction that takes place in a single phase will be said to be homogeneous . On the other hand, when the reaction takes place through an interface, we will say that it is heterogeneous . This is quite important because there are many reactions that have reaction rates strongly influenced by the presence of solid surfaces, since some of the determining stages of the reaction mechanism take place on the surface and the reaction rate will depend on the rate at which reactants can approximate it.

To complicate matters further, some reactions have competing homogeneous and heterogeneous mechanisms. One way to distinguish between the two is to modify the surface/volume ratio in the container that contains the reaction, for example, by altering its shape or by introducing small fragments of material similar to that of the container walls. The rate of the heterogeneous reaction will increase as the surface on which the reaction proceeds does, while this will not alter the rate of the homogeneous reaction. Another way to distinguish between the two is to modify the temperature. The rate of homogeneous reactions, as we will see later, varies exponentially with temperature, while the rate of heterogeneous reactions, which is limited by the rate of diffusion of the reactants, usually varies approximately proportional to $\ sqrt{T}$.

This differentiation between homogeneous and heterogeneous reactions is not absolute either. There are intermediate cases in which the definitions given here cannot be applied, as would be the case of reactions that take place at certain points of a large biomolecule. The most characteristic example of this is the case of enzymes, which act as catalysts in biochemical processes.

The rates of homogeneous and heterogeneous reactions are both influenced by the presence of catalysts. A catalyst is defined as a substance that modifies the rate of a reaction without affecting the final equilibrium. In heterogeneous catalysis , the catalyst is the solid surface itself, which acts as a substrate on which reactants can interact more quickly. In homogeneous catalysis , the catalyst is somehow part of the sequence of individual reactions, although its total concentration does not change in the course of the reaction.

i International Union of Pure and Applied Chemistry - “Compendium of Chemical Terminology - Gold Book”, version 2.3.2 – 2012/08/19 ( http://goldbook.iupac.org/index.html ) (PDF: http://goldbook.iupac.org/PDF/goldbook.pdf ).