0.18 Reaction rate laws  (Page 3/5)

Therefore, the relationship between the rate of the reaction and the concentration of the reactant in this case is given by

$\mathit{\text{Rate}}=\frac{d\left[{\text{C}}_4{\text{H}}_6\right]}{\text{d}\text{t}}=\text{k}{\left[{\text{C}}_4{\text{H}}_6\right]}^2$

which is the rate law for the reaction in Figure 5. This is a very interesting result when compared to Equation (2). In both cases, the results show that the rate of reaction depends on the concentration of the reactant. However, we now also see that the way in which the rate varies with the concentration depends on what the reaction is. Each reaction has its own rate law, and this rate law must be measured and determined experimentally.

Observation 2: rate laws and the order of reaction

We would like to understand what determines the specific dependence of the reaction rate on the reactant concentration in each reaction. In the first case considered above, the rate depends on the concentration of the reactant to the first power. We refer to this as a “first order reaction.” In the second case above, the rate depends on the concentration of the reactant to the second power, so this is called a “second order reaction.” There are also third order reactions, and even “zeroth order reactions” whose rates do not depend on the amount of the reactant. We need more observations of rate laws for different reactions.

The approach used in the previous section to determine a reaction’s rate law is fairly clumsy and at this point difficult to apply. We consider here a more systematic approach. First, consider the decomposition of N ₂ O ₅ (g).

2N ₂ O ₅ (g) → 4NO ₂ (g) + O ₂ (g)

We can create an initial concentration of N ₂ O ₅ in a flask and measure the rate at which the N ₂ O ₅ first decomposes. We can then create a different initial concentration of N ₂ O ₅ and measure the new rate at which the N ₂ O ₅ decomposes. By comparing these rates, we can find the order of the decomposition reaction. The rate law for decomposition of N ₂ O ₅ (g) is of the general form:

Rate = k[N ₂ O ₅ ] ^m

so we need to determine the exponent m . For example, at 25 ºC we observe that the rate of decomposition is 1.4≥10 ^-3 M/s when the concentration of N ₂ O ₅ is 0.020 M. If instead we begin we [N ₂ O ₅ ] = 0.010 M, we observe that the rate of decomposition is 7.0≥10 ^-4 M/s. We can compare the rate from the first measurement Rate(1) to the rate from the second measurement Rate(2) . From Equation (4), we can write that

$\frac{\mathit{\text{Rate(1)}}}{\mathit{\text{Rate(2)}}}=\frac{k{\left[{\text{N}}_2{\text{O}}_5\right]}_1^m}{k{\left[{\text{N}}_2{\text{O}}_5\right]}_{\text{2}}^m}=\frac{1.4\times {10}^{-3}M/s}{7.0\times {10}^{-4}M/s}=\frac{k{\left(0.020M\right)}^m}{k{\left(0.010M\right)}^m}$

This can be simplified on both sides of the equation to give

2.0 = (2.0) ^m

Clearly, then m = 1, and the decomposition is a first order reaction. We can also then find the first order rate constant k for this reaction by simply plugging in one of the initial rate measurements to Equation (4). We find that k = 0.070 s ^-1 .