Rate Expression and Rate Constant

Setting up a general rate expression

Take a general reaction

where $a, b, c, d$ are the stoichiometric coefficients of the reactants and products in the balanced equation. Experiment shows that the rate can be written as

• $[A], [B]$ — molar concentrations of the reactants
• $x, y$ — exponents found from experiment; they may or may not equal the stoichiometric coefficients $a, b$
• $k$ — the rate constant, a proportionality constant for the reaction
• $-\dfrac{d[R]}{dt}$ — the differential form of the rate expression, giving the instantaneous rate in terms of how fast a reactant $R$ is consumed

An equation of this kind, linking the rate of a reaction to the concentrations of its reactants, is the rate law: it expresses the reaction rate in terms of the molar concentration of each reactant raised to some power, and that power need not match the reactant's stoichiometric coefficient in the balanced equation.

Checking this against a real reaction

Consider

Its rate can be measured as a function of the initial concentrations of NO and O₂, by holding one reactant's concentration fixed while varying the other, and vice versa.

Doing this experimentally shows two distinct patterns:

• doubling the concentration of NO, while keeping O₂ fixed, raises the initial rate by a factor of four — so the rate depends on the square of [NO];
• doubling the concentration of O₂, while keeping NO fixed, simply doubles the initial rate — so the rate depends on the first power of [O₂].

This gives the rate equation and its differential form:

Here, the exponents obtained from experiment (2 and 1) happen to be identical to the stoichiometric coefficients of NO and O₂ in the balanced equation (2 and 1).

Where the exponents do not match the coefficients

This agreement is not guaranteed. For other reactions:

$CH_3COOC_2H_5 + H_2O \;\rightarrow\; CH_3COOH + C_2H_5OH,\qquad \text{Rate} = k\,[CH_3COOC_2H_5]^{1}[H_2O]^{0}$ …