Average vs Instantaneous

The average rate over a time interval is calculated by dividing the change in concentration of a species by the elapsed time. For a reactant A, its concentration decreases, so we write: $\text{average rate} = -\dfrac{\Delta[A]}{\Delta t}$. The negative sign makes the rate positive, since $\Delta[A] = [A]_{\text{final}} - [A]_{\text{initial}}$ is negative for a reactant. For a product P, the rate is positive without a sign change: $\text{average rate} = +\dfrac{\Delta[P]}{\Delta t}$. Average rate is easy to measure experimentally using spectroscopy or titration at two time points, but it is only an approximation of what is happening at any given moment — it smooths over the fact that most reactions slow down as reactants are consumed.

The instantaneous rate is the rate at one specific instant in time. Mathematically, it is the limit of the average rate as the time interval approaches zero: $\text{rate} = -\dfrac{d[A]}{dt}$. On a concentration-vs-time graph, the instantaneous rate at any point equals the slope of the tangent line to the curve at that point. Because reaction rates generally decrease over time (as reactants are used up), the tangent line becomes less steep as the reaction proceeds. The instantaneous rate at $t = 0$ is called the initial rate, which is especially useful because the concentrations of all species are precisely known at the start of the experiment.

Consider the general reaction $\ce{aA + bB -> cC + dD}$. Each species changes concentration at a different speed determined by its stoichiometric coefficient. To assign a single, unambiguous rate of reaction $r$ to the whole process, we divide each individual rate by the corresponding stoichiometric coefficient: $r = -\dfrac{1}{a}\dfrac{d[A]}{dt} = -\dfrac{1}{b}\dfrac{d[B]}{dt} = +\dfrac{1}{c}\dfrac{d[C]}{dt} = +\dfrac{1}{d}\dfrac{d[D]}{dt}$. This ensures that no matter which species you choose to monitor in the laboratory, you calculate the same value for $r$. For example, in $\ce{N2 + 3H2 -> 2NH3}$, hydrogen is consumed three times faster than nitrogen, and ammonia is produced twice as fast as nitrogen is consumed. The unique rate is $r = -\dfrac{d[\ce{N2}]}{dt} = -\dfrac{1}{3}\dfrac{d[\ce{H2}]}{dt} = +\dfrac{1}{2}\dfrac{d[\ce{NH3}]}{dt}$.

Reaction rate has units of concentration per time. In aqueous chemistry, concentrations are measured in mol/L (molarity, M), so the standard unit of rate is mol L⁻¹ s⁻¹ (or equivalently M s⁻¹). Sometimes mol L⁻¹ min⁻¹ is used for slower reactions. It is important to note that rate is always a positive quantity — the sign conventions (negative for reactants, positive for products) are incorporated into the definition of rate, not reported as part of the numerical value.

On a concentration-time graph, the average rate between two time points corresponds to the slope of the secant line (the straight line connecting the two data points). The instantaneous rate at a given time corresponds to the slope of the tangent line at that exact point on the curve. As the two time points of the secant line are brought closer and closer together, the secant line approaches the tangent line, and the average rate approaches the instantaneous rate. This graphical understanding helps make the abstract calculus definition intuitive and is directly testable in AP Chemistry experiments where students draw tangent lines on plotted data.

• r — Rate of reaction (mol L⁻¹ s⁻¹): A single positive quantity describing how fast the overall reaction proceeds, independent of which species is monitored.
• a, b, c, d — Stoichiometric coefficients (dimensionless): The integer coefficients from the balanced chemical equation for reactants A, B and products C, D respectively.
• \Delta[A] — Change in molar concentration of A (mol L⁻¹): Negative for a reactant since its concentration decreases as the reaction proceeds.
• \Delta t — Time interval (s (seconds)): The elapsed time between the initial and final concentration measurements.

Problem. The decomposition of dinitrogen pentoxide follows the equation $\ce{2N2O5(g) -> 4NO2(g) + O2(g)}$. In an experiment, $[\ce{N2O5}]$ drops from 0.0200 mol/L to 0.0120 mol/L over a period of 400 seconds. (a) Calculate the average rate of reaction $r$ during this interval. (b) What is the average rate of appearance of $\ce{NO2}$ during the same interval?

Answer. The average rate of reaction r = 1.00 × 10⁻⁵ mol L⁻¹ s⁻¹; the average rate of appearance of NO₂ = 4.00 × 10⁻⁵ mol L⁻¹ s⁻¹.

• Rate Law and Order of Reaction — Once you can calculate the rate of reaction, the next step is expressing how rate depends on concentration through a rate law: rate = k[A]ⁿ[B]ᵐ. Understanding average vs. instantaneous rate is essential for determining rate laws from experimental data.
• Rate Constant and the Arrhenius Equation — The rate constant k links the rate of reaction to concentrations. Its temperature dependence is described by the Arrhenius equation, extending the kinetic concepts introduced here.
• Integrated Rate Laws — Integrated rate laws are obtained by mathematically integrating the instantaneous rate expression (d[A]/dt) over time, directly connecting the calculus definition of instantaneous rate to practical concentration-time equations.
• Collision Theory and Activation Energy — Collision theory provides the molecular explanation for why reaction rates have the values they do — understanding that rate reflects the frequency and energy of effective molecular collisions gives physical meaning to the mathematical rates defined here.
• Derivatives and Rates of Change — The instantaneous rate of reaction is formally defined as a derivative d[A]/dt. Understanding derivatives from calculus provides the rigorous mathematical foundation for the concept of instantaneous rate.