The Integrated Rate Law: The Dependence of Concentration on Time

A differential rate law expresses reaction rate as a change in reactant concentration over a specific time interval. An integrated rate law, derived by integrating the differential form, expresses the relationship between a reactant's initial concentration and its concentration after a specific duration. Both describe reaction kinetics but at different mathematical levels.

Plot the experimental kinetic data using different integrated rate law forms for zero-order, first-order, and second-order reactions. Only the plot that generates a straight line corresponds to the correct overall reaction order. Once the linear plot is identified, you can determine the rate constant and predict reactant concentration at any given time.

For a zero-order reaction, plotting reactant concentration against time produces a straight line. The slope equals the negative value of the rate constant, and the y-intercept represents the initial reactant concentration. This linear relationship reflects the constant reaction rate independent of reactant concentration.

Integration of the first-order differential rate law naturally produces a logarithmic form. Plotting the natural log of reactant concentration versus time yields a straight line, making it easier to identify first-order kinetics and extract the rate constant from the slope and initial concentration from the y-intercept.

Integrated rate laws relate reactant concentration to elapsed time, allowing calculation of how long a radioactive material must decay to reach safe levels. By knowing the initial radioactivity, the rate constant, and the target safe concentration, you can use the appropriate integrated rate law to estimate the required storage duration.

The integrated rate law depends on the overall reaction order, varying for each reaction type. However, all integrated rate laws take the form of a standard linear equation with distinct y, m, x, and b components. This universal linear structure allows plotting to generate straight lines for kinetic analysis across different reaction orders.

In a second-order reaction, plotting the inverse of reactant concentration versus time produces a straight line. The slope of this line equals the rate constant, while the y-intercept represents the inverse of the initial reactant concentration. This relationship allows direct determination of kinetic parameters from experimental data.