Question

Consider the reaction $$ \mathrm{A}+\mathrm{B} \longrightarrow \text { products } $$ From these data obtained at a certain temperature, determine the order of the reaction and calculate the rate constant: $$ \begin{array}{ccc} {[\mathrm{A}](\mathrm{M})} & {[\mathrm{B}](M)} & \text { Rate }(M / \mathrm{s}) \\ \hline 1.50 & 1.50 & 3.20 \times 10^{-1} \\ 1.50 & 2.50 & 3.20 \times 10^{-1} \\ 3.00 & 1.50 & 6.40 \times 10^{-1} \end{array} $$

Short answer

The order of the reaction is 1 and the rate constant is \(2.13 s^{-1}\).

Step-by-step solution

Determine the Order of Reaction

First, examine how changes in the concentration of each reactant affect the rate of reaction. From the given data, when the concentration of reactant B changes from 1.5M to 2.5M and the concentration of reactant A is held constant at 1.5M, the rate of reaction remains constant at \(3.20 \times 10^{-1}\ M/s\). This indicates that the order of the reaction with respect to reactant B is zero. Now, observe the change in the rate of reaction when the concentration of reactant A is doubled (from 1.5M to 3.0M) and the concentration of reactant B is held constant. The rate of reaction doubles from \(3.20 \times 10^{-1}\ M/s\) to \(6.40 \times 10^{-1}\ M/s\). This indicates that the order of the reaction with respect to reactant A is one, as a doubling of the concentration results in a doubling of the rate. Therefore, the overall order of the reaction is \(1+0 = 1\).

Calculate the Rate Constant

Having determined that the reaction is first order with respect to reactant A, the rate law for the reaction can be expressed as \[Rate = k[A]\] where k is the rate constant, [A] is the concentration of A. To find the value of the rate constant, input any set of rate and concentration data from the table into this equation. For instance, using the first set of rate and concentration data: \(3.20 \times 10^{-1} M/s = k(1.5M)\). By solving for k, we get \(k = \frac{3.20 \times 10^{-1}M/s}{1.5M} = 2.13 s^{-1}\]

Key concepts

Rate Law

The rate law is a mathematical equation that links the rate of a chemical reaction to the concentrations of its reactants. This law helps us understand how different reactants influence the reaction's speed. In a rate law expression, each reactant is associated with an exponent that reflects its impact on the reaction rate, known as the reaction order. Using these exponents, the rate law can take the form: \( \text{Rate} = k[A]^m[B]^n \), where:

• \(k\) is the rate constant, a proportionality factor unique to each reaction.
• [A] and [B] represent the concentrations of reactants A and B.
• \(m\) and \(n\) are the reaction orders concerning A and B respectively.

In our example, A has a reaction order of one, implying it has a direct, linear effect on the reaction rate. Reactant B, however, has a reaction order of zero, meaning changes in its concentration do not alter the rate. Therefore, our rate law simplifies to \( \text{Rate} = k[A] \), focusing only on A's impact.

Rate Constant

The rate constant, denoted as \(k\), is a crucial component of the rate law, providing insight into the reaction's speed under specific conditions. Its value is essential for making predictions about reaction behavior. The units of \(k\) vary with the reaction order:

• For a first-order reaction like ours, \(k\) has units of \(s^{-1}\).
• Higher-order reactions have correspondingly more complex units.

To calculate \(k\), we use the rate law equation rearranged to express \(k\): \[ k = \frac{\text{Rate}}{[A]} \]Given the data from the exercise, substituting the initial conditions for [A] and Rate, we find \(k\) to be approximately \(2.13 \, s^{-1}\). This constant shows the reaction's speed is fairly consistent per unit concentration of A.

Kinetics

Kinetics is the branch of chemistry that examines the rate of chemical reactions and the mechanisms by which they occur. By studying kinetics, we can understand how various factors, such as temperature, concentration, and catalyst presence, can influence these reaction rates.
Chemical kinetics not only tells us the rate at which a reaction proceeds but also provides details about how the reactants transform into products. Key elements of kinetics include:

• Reaction Rate: This is essentially the speed at which the reactants are consumed and products are formed in a reaction.
• Mechanisms: The step-by-step sequence of elementary reactions by which overall chemical change occurs.
• Effect of Concentrations: Changing the concentration of a reactant alters the rate; this relationship helps deduce the order of the reaction.

In our example, the kinetics study involves determining the impact of concentration changes on the reaction rate, leading to insights about reaction order and the calculation of the rate constant \(k\). This detailed study aids in predicting how reactions proceed under various conditions.