Integrated Rate Equation
Definition and meaning of Integrated Rate Equation in chemistry.
An integrated rate equation is a special mathematical formula used in chemical kinetics. It shows the exact concentration of a reactant at any specific moment in time. Chemists create this formula by using calculus to integrate a standard differential rate law.
In more detail
A normal differential rate law shows how the reaction speed depends on the reactant concentration. This rule is usually written mathematically as rate equals k times concentration to a power. However, that differential form is very hard to use during a real laboratory experiment.
An integrated rate equation is much more useful for tracking real chemistry in the lab. It directly links the exact amount of reactant left in the flask to the time passed. Chemists use these equations to graph their experimental lab data in several different ways.
They might plot the raw chemical concentration against the total elapsed time on a graph. Alternatively, they might plot the natural log of that concentration against the elapsed time. The specific graph format that creates a perfectly straight line reveals the true reaction order.
The steep slope of that straight line also gives the exact rate constant for the reaction. Each reaction order has its own unique integrated equation and its own special half-life formula. Zero-order, first-order, and second-order reactions all require different math to track their specific progress. These mathematical equations are the most important tools used in the study of chemical kinetics.
Key facts
| Zero-order equation | [A] = [A]0 - kt |
|---|---|
| First-order equation | ln[A] = ln[A]0 - kt |
| Second-order equation | 1/[A] = 1/[A]0 + kt |
| Main purpose | Find concentration at a specific time |
| Graphical use | Determines reaction order from linear plots |
| Field | Physical Chemistry |
Imagine a simple first-order reaction where reactant A turns into products. The differential rate law simply says the rate equals k times the concentration of A. Using calculus, integration turns this into the formula ln[A] equals ln[A]0 minus kt. A chemist can measure the concentration of A at several different times during the experiment. Plotting the natural log of those measured concentrations against time will yield a straight line. The mathematical slope of this resulting line will be equal to negative k. This linear result completely confirms that the process is a first-order chemical reaction.
Frequently asked questions
How do integrated rate laws help determine the reaction order?
Chemists test different concentration-time plots to see which one creates a perfectly straight line. The specific plot that yields a straight line identifies the order, and its slope gives the rate constant.
How is an integrated rate law related to a differential rate law?
The integrated rate law is derived by using advanced calculus to integrate the differential rate law. This math converts a rate-concentration relationship into a much more useful concentration-time relationship.
What does the symbol [A]0 mean in an integrated rate equation?
The symbol [A]0 represents the initial concentration of the reactant at the very start of the experiment. It is the amount of chemical present when the stopwatch is at exactly zero seconds.