#### Learning Objective

- Manipulate experimentally determined second-order rate law equations to obtain rate constants

#### Key Points

- A second-order reaction will depend on the concentration(s) of one second-order reactant or two first-order reactants.
- To determine the order of a reaction with respect to each reactant, we use the method of initial rates.
- When applying the method of initial rates to a reaction involving two reactants, A and B, it is necessary to conduct two trials in which the concentration of A is held constant, and B changes, as well as two trials in which the concentration of B is held constant, and A changes.

#### Terms

- second-order reactionA reaction that depends on the concentration(s) of one second-order reactant or two first-order reactants.
- reaction mechanismThe step-by-step sequence of elementary transformations by which overall chemical change occurs.

A reaction is said to be second-order when the overall order is two. For a reaction with the general form [latex]aA+bB\rightarrow C[/latex], the reaction can be second order in two possible ways. It can be second-order in either A or B, or first-order in both A and B. If the reaction were second-order in either reactant, it would lead to the following rate laws:

[latex]rate=k[A]^2[/latex]

or

[latex]rate=k[B]^2[/latex]

The second scenario, in which the reaction is first-order in both A and B, would yield the following rate law:

[latex]rate=k[A][B][/latex]

## Applying the Method of Initial Rates to Second-Order Reactions

Consider the following set of data:

If we are interested in determining the order of the reaction with respect to A and B, we apply the method of initial rates.

## Determining Reaction Order in A

In order to determine the reaction order for A, we can set up our first equation as follows:

[latex]\frac{r_1}{r_2}=\frac{k[A]_1^x[B]_1^y}{k[A]_2^x[B]_2^y}[/latex]

[latex]\frac{5.46}{12.28}=\frac{k(0.200)^x(0.200)^y}{k(0.300)^x(0.200)^y}[/latex]

Note that on the right side of the equation, both the rate constant *k* and the term [latex](0.200)^y[/latex] cancel. This was done intentionally, because in order to determine the reaction order in A, we need to choose two experimental trials in which the initial concentration of A changes, but the initial concentration of B is constant, so that the concentration of B cancels. Our equation simplifies to:

[latex]\frac{5.46}{12.28}=\frac{(0.200)^x}{(0.300)^x}[/latex]

[latex]0.444=\left(\frac{2}{3}\right)^x[/latex]

[latex]ln(0.444)=x\cdot ln\left(\frac{2}{3}\right)[/latex]

[latex]x\approx 2[/latex]

Therefore, the reaction is second-order in A.

## Determining Reaction Order in B

Next, we need to determine the reaction order for B. We do this by picking two trials in which the concentration of B changes, but the concentration of A does not. Trials 1 and 3 will do this for us, and we set up our ratios as follows:

[latex]\frac{r_1}{r_3}=\frac{k[A]_1^2[B]_1^y}{k[A]_3^2[B]_3^y}[/latex]

[latex]\frac{5.46}{5.42}=\frac{k(0.200)^2(0.200)^y}{k(0.200)^2(0.400)^y}[/latex]

Note that both k and the concentrations of A cancel. Also, [latex]\frac{5.46}{5.42}\approx 1[/latex], so everything simplifies to:

[latex]1=\frac{(0.200)^y}{(0.400)^y}[/latex]

[latex]1=\left(\frac{1}{2}\right)^y[/latex]

[latex]y=0[/latex]

Therefore, the reaction is zero-order in B.

## Overall Reaction Order

We have determined that the reaction is second-order in A, and zero-order in B. Therefore, the overall order for the reaction is second-order [latex](2+0=2)[/latex], and the rate law will be:

[latex]rate=k[A]^2[/latex]