Question

Two isomers \((A \text { and } B)\) of a given compound dimerize as follows: $$\begin{aligned} &2 \mathrm{A} \stackrel{k_{1}}{\longrightarrow} \mathrm{A}_{2}\\\ &2 \mathrm{B} \stackrel{k_{2}}{\longrightarrow} \mathrm{B}_{2} \end{aligned}$$ Both processes are known to be second order in reactant, and \(k_{1}\) is known to be 0.250 \(\mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\) at \(25^{\circ} \mathrm{C} .\) In a particular experiment \(\mathrm{A}\) and \(\mathrm{B}\) were placed in separate containers at \(25^{\circ} \mathrm{C}\) where \([\mathrm{A}]_{0}=1.00 \times 10^{-2} \mathrm{M}\) and \([\mathrm{B}]_{0}=2.50 \times 10^{-2} \mathrm{M} .\) It was found that after each reaction had progressed for 3.00 min, \([\mathrm{A}]=3.00[\mathrm{B}] .\) In this case the rate laws are defined as $$\begin{array}{l} \text { Rate }=-\frac{\Delta[\mathrm{A}]}{\Delta t}=k_{1}[\mathrm{A}]^{2} \\ \text { Rate }=-\frac{\Delta[\mathrm{B}]}{\Delta t}=k_{2}[\mathrm{B}]^{2} \end{array}$$ a. Calculate the concentration of \(\mathrm{A}_{2}\) after 3.00 min. b. Calculate the value of \(k_{2}\) c. Calculate the half-life for the experiment involving A.

Short answer

a. The concentration of A2 after 3 minutes is \(5.62 \times 10^{-3}\ \mathrm{M}\). b. The value of k2 is \(0.104\ \mathrm{L / mol \cdot s}\). c. The half-life for the experiment involving A is \(400\ \mathrm{s}\).

Step-by-step solution

Determine the remaining concentration of A and B after 3 minutes

Using the given relationship \([\mathrm{A}] = 3.00[\mathrm{B}]\) after 3.00 minutes, we can express the concentration of A in terms of B, and vice versa. Given the initial concentrations as \([\mathrm{A}]_{0}=1.00 \times 10^{-2} \mathrm{M}\) and \([\mathrm{B}]_{0}=2.50 \times 10^{-2} \mathrm{M}\). We will solve for the remaining concentration of A and B after 3 minutes.

Apply the integrated rate law for A

Now we will apply the integrated rate law for second-order reactions to isomer A: \[ \frac{1}{[\mathrm{A}]} = k_{1}t + \frac{1}{[\mathrm{A}]_{0}} \] We are given \(k_{1} = 0.250\ \mathrm{L / mol \cdot s}\), \(t = 3.00\ \mathrm{min} \cdot \frac{60\ \mathrm{s}}{1\ \mathrm{min}} = 180\ \mathrm{s}\), and \([\mathrm{A}]_{0} = 1.00 \times 10^{-2}\ \mathrm{M}\). We will substitute these values and solve for the remaining concentration of A after 3 minutes.

Calculate the concentration of A2 after 3 minutes

Now that we have the remaining concentration of A after 3 minutes, we can calculate the concentration of A2. Since A dimerizes to form A2, the amount of A that has reacted will form the same amount of A2: \[ [\mathrm{A}_{2}] = [\mathrm{A}]_{0} - [\mathrm{A}] \] We will plug in the calculated remaining concentration of A and the initial concentration of A to find the concentration of A2 after 3 minutes.

Apply the integrated rate law for B

Similar to Step 2 but now for isomer B: \[ \frac{1}{[\mathrm{B}]} = k_{2}t + \frac{1}{[\mathrm{B}]_{0}} \] Having the value of \([\mathrm{B}]_{0} = 2.50 \times 10^{-2}\ \mathrm{M}\) and \(t = 180\ \mathrm{s}\), we will substitute these values and the remaining concentration of B we calculated in Step 1. Then we will solve for the second order rate constant k2.

Calculate the half-life for the experiment involving A

The half-life for a second-order reaction can be calculated using the following formula: \[ t_{1/2} = \frac{1}{k[\mathrm{A}]_{0}} \] We are given the initial concentration of A as \([\mathrm{A}]_{0}=1.00 \times 10^{-2} \mathrm{M}\) and the first-order rate constant for A as \(k_{1} = 0.250\ \mathrm{L / mol \cdot s}\). We will substitute these values into the half-life formula and solve for the half-life for the experiment involving A.

Key concepts

Second-Order Reactions

In chemical kinetics, second-order reactions refer to processes where the rate of reaction is proportional to the square of the concentration of one reactant, or to the product of the concentrations of two reactants. This can be mathematically expressed as:

Rate = k[A]^{2} or Rate = k[A][B]

where 'Rate' is the speed at which the reaction proceeds, 'k' is the second-order rate constant, and '[A]' and '[B]' are the molar concentrations of the reactants. In the given exercise, both the reactions of the isomers A and B dimerizing follow second-order kinetics. Understanding the behavior of such reactions is critical in predicting how their concentrations change over time, which is essential in fields such as pharmacology and environmental science.

Reaction Rate Laws

Reaction rate laws, or rate equations, describe the relationship between the concentrations of reactants and the reaction rate. These laws are empirical, meaning they are determined by experimental data rather than theoretical predictions. The general form of a rate law for a reaction is:

Rate = k[A]^{m}[B]^{n}

The exponents 'm' and 'n' are called the reaction order with respect to reactant A and B, respectively, and the overall order of the reaction is the sum (m+n). For the experiments involving isomers A and B, the rate laws are second-order with respect to each isomer individually, with the rate being directly proportional to the square of the reactant's concentration.

Integrated Rate Laws

Integrated rate laws are equations derived from rate laws that give the concentrations of reactants or products as a function of time. For second-order reactions, the integrated rate law is:

\[ \frac{1}{[A]} = kt + \frac{1}{[A]_0} \]

where '[A]' is the concentration at time 't', '[A]_0' is the initial concentration, and 'k' is the second-order rate constant. The integrated rate law helps determine the concentration of reactants at any given time and can be used to calculate the concentration of isomer A and its dimer A2 in the exercise, as well as the rate constant for the reaction of isomer B.

Reaction Mechanisms

Reaction mechanisms provide a step-by-step description of how a chemical reaction proceeds at the molecular level. It includes details such as the sequence of elementary steps, the formation and breakdown of intermediates, and the determination of rate-determining steps. These reaction mechanisms are essential to explain the observed rate laws, which often involve complex processes. In simple second-order reactions, like the dimerization of isomers A and B, the mechanism may involve the collision and bonding of two reactant molecules to form a product molecule.

Half-Life of Reactions

The half-life of a reaction, denoted as \(t_{1/2}\), is the time required for the concentration of a reactant to reduce to half its initial value. For second-order reactions, the half-life is inversely proportional to the initial concentration of the reactant and the rate constant, given by:

\[ t_{1/2} = \frac{1}{k[A]_0} \]

Half-life is an important concept, especially in pharmacodynamics, where it determines how long a drug remains effective in the body. In the case of isomer A in the exercise, the half-life calculation uses the initial concentration and the known rate constant for A.