#### Learning Objective

- Design initial rate experiments to determine order of reaction with respect to individual reactants

#### Key Points

*k*is the first-order rate constant, which has units of 1/s.- The method of determining the order of a reaction is known as the method of initial rates.
- The overall order of a reaction is the sum of all the exponents of the concentration terms in the rate equation.

#### Term

- first-order reactionA reaction that depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but each will be zero-order.

A first-order reaction depends on the concentration of only one reactant. As such, a first-order reaction is sometimes referred to as a unimolecular reaction. While other reactants can be present, each will be zero-order, since the concentrations of these reactants do not affect the rate. Thus, the rate law for an elementary reaction that is first order with respect to a reactant A is given by:

[latex]r = -\frac{d[A]}{dt} = k[A][/latex]

As usual, *k* is the rate constant, and must have units of concentration/time; in this case it has units of 1/s.

## Using the Method of Initial Rates to Determine Reaction Order Experimentally

[latex]2\;N_{2}O_{5}(g)\rightarrow 4\;NO_{2}(g)+O_{2}(g)[/latex]

The balanced chemical equation for the decomposition of dinitrogen pentoxide is given above. Since there is only one reactant, the rate law for this reaction has the general form:

[latex]Rate= k[N_{2}O_{5}]^{m}[/latex]

In order to determine the overall order of the reaction, we need to determine the value of the exponent *m*. To do this, we can measure an initial concentration of N_{2}O_{5} in a flask, and record the rate at which the N_{2}O_{5} decomposes. We can then run the reaction a second time, but with a different initial concentration of N_{2}O_{5}. We then measure the new rate at which the N_{2}O_{5} decomposes. By comparing these rates, it is possible for us to find the order of the decomposition reaction.

## Example

Let’s say that at 25 °C, we observe that the rate of decomposition of N_{2}O_{5} is 1.4×10^{-3 }M/s when the initial concentration of N_{2}O_{5} is 0.020 M. Then, let’s say that we run the experiment again at the same temperature, but this time we begin with a different concentration of N_{2}O_{5 }, which is 0.010 M. On this second trial, we observe that the rate of decomposition of N_{2}O_{5} is 7.0×10^{-4 }M/s. We can now set up a ratio of the first rate to the second rate:

[latex]\frac{Rate_1}{Rate_2}=\frac{k[N_2O_5]_{i1}^{m}}{k[N_{2}O_{5}]_{i2}^{m}}[/latex]

[latex]\frac{1.4\times 10^{-3}}{7.0\times 10^{-4}}=\frac{k(0.020)^{m}}{k(0.010)^{m}}[/latex]

Notice that the left side of the equation is simply equal to 2, and that the rate constants cancel on the right side of the equation. Everything simplifies to:

[latex]2.0=2.0^m[/latex]

Clearly, then, *m*=1, and the decomposition is a first-order reaction.

## Determining the Rate Constant *k*

Once we have determined the order of the reaction, we can go back and plug in one set of our initial values and solve for *k*. We find that:

[latex]rate=k[N_2O_5]^1=k[N_2O_5][/latex]

Substituting in our first set of values, we have

[latex]1.4\times 10^{-3}=k(0.020)[/latex]

[latex]k=0.070\;s^{-1}[/latex]