Practice problems for these concepts can be found at:

The rate (or speed) of reaction is related to the change in concentration of either a reactant or product with time. Consider the general reaction: 2A + B → C + 3D. As the reaction proceeds, the concentrations of reactants A and B will decrease and the concentrations of products C and D will increase. Thus, the rate can be expressed in the following ways:

The first two expressions involving the reactants are negative, because their concentrations will decrease with time. The square brackets represent moles per liter concentration (molarity).

The rate of reaction decreases during the course of the reaction. The rate that is calculated above can be expressed as the average rate of reaction over a given time frame or, more commonly, as the initial reaction rate—the rate of reaction at the instant the reactants are mixed.

### The Rate Equation

The rate of reaction may depend upon reactant concentration, product concentration, and temperature. Cases in which the product concentration affects the rate of reaction are rare and are not covered on the AP exam. Therefore, we will not address those reactions. We will discuss temperature effects on the reaction later in this chapter. For the time being, let's just consider those cases in which the reactant concentration may affect the speed of reaction. For the general reaction: a A + b B +… → c C + d D + … where the lower–case letters are the coefficients in the balanced chemical equation; the upper–case letters stand for the reactant; and product chemical species and initial rates are used, the rate equation (rate law) is written:

- Rate =

*k*[

*A*]

^{m}[

*B*]

^{n}…

In this expression, *k* is the **rate constant**—a constant for each chemical reaction at a given temperature. The exponents *m* and *n*, called the **orders of reaction**, indicate what effect a change in concentration of that reactant species will have on the reaction rate. Say, for example, m = 1 and n = 2. That means that if the concentration of reactant A is doubled, then the rate will also double ([2]^{1} = 2), and if the concentration of reactant B is doubled, then the rate will increase fourfold ([2]^{2} = 4). We say that it is first order with respect to A and second order with respect to B. If the concentration of a reactant is doubled and that has no effect on the rate of reaction, then the reaction is zero order with respect to that reactant ([2]^{0} = 1). Many times the overall order of reaction is calculated; it is simply the sum of the individual coefficients, third order in this example. The rate equation would then be shown as:

- Rate =

*k*[A][B]

^{2}(If the exponent is 1, it is generally not shown.)

It is important to realize that the rate law (the rate, the rate constant, and the orders of reaction) is determined experimentally. Do not use the balanced chemical equation to determine the rate law.

The rate of reaction may be measured in a variety of ways, including taking the slope of the concentration versus time plot for the reaction. Once the rate has been determined, the orders of reaction can be determined by conducting a series of reactions in which the reactant species concentrations are changed one at a time, and mathematically determining the effect on the reaction rate. Once the orders of reaction have been determined, it is easy to calculate the rate constant.

For example, consider the reaction:

- 2 NO(g) + O

_{2}(g) → 2 NO

_{2}(g)

The following kinetics data were collected:

There are a couple of ways to interpret the data to generate the rate equation. If the numbers involved are simple (as above and on most tests, including the AP exam), you can reason out the orders of reaction. You can see that in going from experiment 1 to experiment 2, the [NO] was doubled, ([O_{2}] held constant), and the rate increased fourfold. This means that the reaction is second order with respect to NO. Comparing experiments 1 and 3, you see that the [O_{2}] was doubled, ([NO] was held constant), and the rate doubled.

Therefore, the reaction is first order with respect to O_{2} and the rate equation can be written as:

- Rate =

*k*[NO]

^{2}[O

_{2}]

The rate constant can be determined by substituting the values of the concentrations of NO and O_{2} from any of the experiments into the rate equation above and solving for *k*.

Using experiment 1:

- 0.05 M/s =

*k*(0.01 M)

^{2}(0.01 M)

*k*= (0.05 M/s)(0.01 M)

^{2}(0.01 M)

*k*= 5 × 10

^{4}/M

^{2}s

Sometimes, because of the numbers' complexity, you must set up the equations mathematically. The ratio of the rate expressions of two experiments will be used in determining the reaction orders. The equations will be chosen so that the concentration of only one reactant has changed while the others remain constant. In the example above, the ratio of experiments 1 and 2 will be used to determine the effect of a change of the concentration of NO on the rate, and then experiments 1 and 3 will be used to determine the effect of O_{2}. Experiments 2 and 3 cannot be used, because both chemical species have changed concentration.

Remember: In choosing experiments to compare, choose two in which the concentration of only one reactant has changed while the others have remained constant.

Comparing experiments 1 and 2:

Canceling the rate constants and the [0.01]^{n} and simplifying:

Comparing experiments 1 and 3:

Canceling the rate constants and the [0.01]^{n} and simplifying:

Writing the rate equation:

- Rate =

*k*[NO]

^{2}[O

_{2}]

Again, the rate constant *k* could be determined by choosing any of the three experiments, substituting the concentrations, rate, and orders into the rate expression, and then solving for *k*.

Practice problems for these concepts can be found at:

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