First-Order Reactions

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:

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

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

Hydrogen peroxide

The decomposition of hydrogen peroxide to form oxygen and hydrogen is a first-order reaction.

Using the Method of Initial Rates to Determine Reaction Order Experimentally

$2\;N_{2}O_{5}(g)\rightarrow 4\;NO_{2}(g)+O_{2}(g)$

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:

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

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 N₂O₅ in a flask, and record the rate at which the N₂O₅ decomposes. We can then run the reaction a second time, but with a different initial concentration of N₂O₅. We then measure the new rate at which the N₂O₅ 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 N₂O₅ is 1.4×10^-3 M/s when the initial concentration of N₂O₅ 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 N₂O₅ , which is 0.010 M. On this second trial, we observe that the rate of decomposition of N₂O₅ is 7.0×10^-4 M/s. We can now set up a ratio of the first rate to the second rate:

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

$\frac{1.4\times 10^{-3}}{7.0\times 10^{-4}}=\frac{k(0.020)^{m}}{k(0.010)^{m}}$

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:

$2.0=2.0^m$

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:

$rate=k[N_2O_5]^1=k[N_2O_5]$

Substituting in our first set of values, we have

$1.4\times 10^{-3}=k(0.020)$

$k=0.070\;s^{-1}$