Question

For each of the questions, four choices have been provided. Select the correct alternative. For a particular reaction, \(A+B \rightarrow C\) was studied at \(25^{\circ} \mathrm{C}\). The following results are obtained: $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} {[\mathrm{A}]} \\ \text { mole } / \mathrm{I} \end{array} & \begin{array}{c} {[\mathrm{B}]} \\ \text { moles } / \mathrm{I} \end{array} & \begin{array}{c} \text { Rate } \\ \left(\mathrm{mole} \mathrm{I}^{-1} \mathrm{~s}^{-1}\right) \end{array} \\ \hline 9 \times 10^{-5} & 1.5 \times 10^{-2} & 0.06 \\ \hline 9 \times 10^{-5} & 3 \times 10^{-3} & 0.012 \\ \hline 3 \times 10^{-5} & 3 \times 10^{-3} & 0.004 \\ \hline 6 \times 10^{-5} & \mathrm{x} & 0.024 \\ \hline \end{array} $$ Then the value of \(\mathrm{x}\) is ________ (a) \(6 \times 10^{-3} \mathrm{~m} \ell^{-1}\) (b) \(3 \times 10^{-3} \mathrm{~m} \ell^{-1}\) (c) \(4.5 \times 10^{-3} \mathrm{~m} \ell^{-1}\) (d) \(9 \times 10^{-3} \mathrm{~m} \ell^{-1}\)

Short answer

Answer: The concentration x is \(9 \times 10^{-3}\ \mathrm{m\ell^{-1}}\).

Step-by-step solution

Determine the order of the reaction with respect to A

We start by comparing the first and second row of the data. The concentration of A is the same in both cases, while the concentration of B is 5 times smaller in the second row. Observe how the rate changes: $$\frac{0.012}{0.06} = \frac{1}{5}$$ This implies that the reaction is first-order with respect to B.

Determine the order of the reaction with respect to B

Now, we will compare the second and third row of the data. The concentration of B is the same in both cases, while the concentration of A is 3 times smaller in the third row. Observe how the rate changes: $$\frac{0.004}{0.012} = \frac{1}{3}$$ This implies that the reaction is first-order with respect to A.

Apply the rate law to the reaction

Since the reaction is first-order with respect to both A and B, the rate law is: $$\text{Rate} = k[\mathrm{A}][\mathrm{B}]$$ Where k is the rate constant.

Calculate the rate constant (k) using the first row of data

Use the rate, concentration of A, and concentration of B from the first row of data to calculate k: $$k = \frac{\text{Rate}}{[\mathrm{A}][\mathrm{B}]} = \frac{0.06}{(9 \times 10^{-5})(1.5 \times 10^{-2})}$$ $$k = 0.44444$$

Find the concentration x using the last row of data

Now, we will use the rate and concentration of A from the last row of the data, and the rate constant k, to solve for x (concentration of B): $$0.024 = 0.44444(6 \times 10^{-5})[\mathrm{x}]$$ $$[\mathrm{x}] = \frac{0.024}{(0.44444)(6 \times 10^{-5})}$$ $$[\mathrm{x}] = 9 \times 10^{-3}\ \mathrm{m\ell^{-1}}$$ The correct answer is (d) \(9 \times 10^{-3}\ \mathrm{m\ell^{-1}}\).

Key concepts

Reaction Order

Understanding reaction order is essential in determining how the concentration of reactants affects the reaction rate. In reaction kinetics, the reaction order is derived from the exponents of the concentration terms in the rate law expression. For a given reaction, the order can be with respect to individual reactants or the overall reaction. It dictates how the rate of reaction changes when the concentration of a reactant is altered.

In the given problem, by examining how changes in concentrations of reactants affect the rate, we establish that the reaction is first-order with respect to both reactant A and reactant B. This implies that doubling the concentration of A or B separately would double the reaction rate. Each reactant has a power of 1 in the rate law, indicating a direct dependency.

Rate Law

The rate law is a mathematical expression that describes the rate of a chemical reaction as a function of the concentration of its reactants. It provides a relationship between the rate of reaction and the concentration of reactants involved.

For the reaction given in the exercise, the rate law is expressed as \(\text{Rate} = k[A][B]\). This law shows that the rate of reaction depends on the concentration of A and B, and it multiplies both concentrations by the rate constant, \(k\). Knowing the rate law enables us to predict how the reaction rate will change with varying reactant concentrations. Here, both A and B are first-order, confirmed by analyzing the data in the exercise.

Rate Constant

The rate constant \(k\) is a crucial factor in the rate law that affects the speed of a chemical reaction under given conditions. Unlike reaction order, it does not change with concentrations but varies with temperature and the presence of catalysts.

In this exercise, the rate constant was calculated using the provided data from the first row. By inserting known values, \(k\) was derived using the formula \(k = \frac{\text{Rate}}{[A][B]}\). The calculated value was approximately 0.44444 indicating how fast the reaction proceeds per unit concentration of A and B at the given temperature of 25°C. Remember, that the rate constant gives insight into the intrinsic rate of a chemical reaction.

Concentration Dependency

Concentration dependency refers to how the rate of reaction changes in response to varying concentrations of reactants. For many reactions, as in the case here, the concentration dependency is directly tied to the reaction order.

Analyze the correlation using the rate law \(\text{Rate} = k[A][B]\), as the concentrations of A and B influence the rate. In our example, if either concentration is increased, the reaction rate increases proportionally due to the first-order nature of both reactants. By calculating the unknown concentration in the final step, knowing the concentration dependency becomes essential for predicting how the system responds to fluctuations in reactant levels. Understanding this concept highlights how dynamics of kinetic studies impact chemical processes in practice.