Problem 24 - Chemistry: The Central Science

Question

Consider a hypothetical reaction between \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) that is firstorder in \(\mathrm{A}\), zero order in \(\mathrm{B}\), and second order in C. (a) Write the rate law for the reaction. (b) How does the rate change when \([\mathrm{A}]\) is doubled and the other reactant concentrations are held constant? (c) How does the rate change when [B] is tripled and the other reactant concentrations are held constant? (d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant? (e) By what factor does the rate change when the concentrations of all three reactants are tripled?

Step-by-Step Solution

Answer

The rate law for the given hypothetical reaction is \(Rate = k [\mathrm{A}] [\mathrm{C}]^{2}\). When the concentration of A is doubled, the rate doubles; when the concentration of B is tripled, the rate remains unchanged; when the concentration of C is tripled, the rate increases nine-fold. If the concentrations of all three reactants are tripled, the rate increases by a factor of 27.

Step 1: (a) Write the rate law for the reaction.

To write the rate law for the reaction, we'll use the following general form: \[Rate = k [\mathrm{A}]^{m} [\mathrm{B}]^{n} [\mathrm{C}]^{p}\] Given the order of each reactant: 1st order in A, 0th order in B, and 2nd order in C, we can substitute those into the rate equation: \[Rate = k [\mathrm{A}]^{1} [\mathrm{B}]^{0} [\mathrm{C}]^{2}\] Simplifying the expression: \[Rate = k [\mathrm{A}] [\mathrm{C}]^{2}\]

Step 2: (b) How does the rate change when \([\mathrm{A}]\) is doubled and the other reactant concentrations are held constant?

Let's denote the initial reaction rate as \(Rate_1 = k [\mathrm{A}] [\mathrm{C}]^{2}\). Now, we will double the concentration of A, while keeping concentrations of B and C constant. The new rate, \(Rate_2\), is given by: \[Rate_2 = k [2 \times \mathrm{A}] [\mathrm{C}]^{2} = 2k [\mathrm{A}] [\mathrm{C}]^{2}\] To find out the factor by which the rate changes, divide \(Rate_2\) by \(Rate_1\): \[\frac{Rate_2}{Rate_1} = \frac{2k [\mathrm{A}] [\mathrm{C}]^{2}}{k [\mathrm{A}] [\mathrm{C}]^{2}} = 2\] When the concentration of A is doubled, the reaction rate is doubled.

Step 3: (c) How does the rate change when \([\mathrm{B}]\) is tripled and the other reactant concentrations are held constant?

Since the reaction is zero order in B, changes in the concentration of B do not affect the rate of the reaction.

Step 4: (d) How does the rate change when \([\mathrm{C}]\) is tripled and the other reactant concentrations are held constant?

Let's denote the initial reaction rate as \(Rate_1 = k [\mathrm{A}] [\mathrm{C}]^{2}\). Now, we will triple the concentration of C, while keeping concentrations of A and B constant. The new rate, \(Rate_2\), is given by: \[Rate_2 = k [\mathrm{A}] [(3 \times \mathrm{C})]^{2} = 9k [\mathrm{A}] [\mathrm{C}]^{2}\] To find out the factor by which the rate changes, divide \(Rate_2\) by \(Rate_1\): \[\frac{Rate_2}{Rate_1} = \frac{9k [\mathrm{A}] [\mathrm{C}]^{2}}{k [\mathrm{A}] [\mathrm{C}]^{2}} = 9\] When the concentration of C is tripled, the reaction rate increases nine-fold.

Step 5: (e) By what factor does the rate change when the concentrations of all three reactants are tripled?

Let's denote the initial reaction rate as \(Rate_1 = k [\mathrm{A}] [\mathrm{C}]^{2}\). We are interested in the factor by which the rate changes, but we know that the concentration of B has no effect on the rate, so we only evaluate the impact of tripling A and C while B is tripled: \[Rate_2 = k [3 \times \mathrm{A}] [(3 \times \mathrm{C})]^2\] To find out the factor by which the rate changes, divide \(Rate_2\) by \(Rate_1\): \[\frac{Rate_2}{Rate_1} = \frac{k [3 \times \mathrm{A}] [(3\times\mathrm{C})]^2}{k [\mathrm{A}] [\mathrm{C}]^{2}} = \frac{3 [\mathrm{A}] [(3\times \mathrm{C})^2]}{[\mathrm{A}] [\mathrm{C}]^2} = 3 \times [(3 \times \mathrm{C})^2][\mathrm{C}^{-2}] = 3 \times 9 = 27\] When the concentrations of all three reactants are tripled, the reaction rate increases by a factor of 27.

Key Concepts

Understanding the Rate Law

The rate law is an algebraic equation that quantifies the effect of the concentration of reactants on the rate of a chemical reaction. It's a foundational concept in the study of chemical kinetics and crucial for predicting how a reaction proceeds over time.

To grasp this concept, consider the rate law expression:
\[Rate = k [\mathrm{A}]^{m} [\mathrm{B}]^{n} [\mathrm{C}]^{p}\]
In this equation, \(k\) is the rate constant, which varies with temperature and the presence of a catalyst but is independent of the concentration of reactants. The exponents \(m\), \(n\), and \(p\) are the orders of the reaction with respect to each reactant A, B, and C, respectively. The overall reaction order is the sum of these individual orders.

As such, for the hypothetical reaction given in the exercise, where the reaction is first-order in A, zero-order in B, and second-order in C, the simplified rate law would be:
\[Rate = k [\mathrm{A}] [\mathrm{C}]^{2}\]
This tells us the reaction rate is directly proportional to the concentration of A and the square of the concentration of C, but is independent of the concentration of B.

Determining Reaction Order

Reaction order is significant as it reveals how the rate of reaction responds to changes in reactant concentration. It indicates the power to which the concentration of a reactant is raised in the rate law. In our example, with the orders given, we can explore the effects on reaction rate:

• Doubling \([\mathrm{A}]\) when it is first order will double the reaction rate since the rate is directly proportional to \([\mathrm{A}]\).
• Tripling \([\mathrm{B}]\), which is zero order, has no effect on the rate, indicating that the concentration of B does not contribute to reaction speed.
• Tripling \([\mathrm{C}]\), which is second order, causes the rate to increase by a factor of nine (since \(3^2 = 9\)).

The understanding of reaction order aids in predicting the behavior of chemical reactions under variable conditions, such as concentration changes, which is immensely helpful in industrial and laboratory settings.

The Role of the Rate Constant

The rate constant \(k\) in the context of a rate law is a crucial parameter that provides the speed of a chemical reaction at a given temperature. It incorporates factors such as the activation energy and the frequency of successful collisions between reactant molecules.

In equations, the value of \(k\) determines the proportionality between the concentration terms and the rate of reaction. For a given set of conditions, it's constant, and its units vary depending on the overall reaction order to ensure the rate has units of concentration per unit time.

To illustrate the impact of the rate constant, when all reactant concentrations are tripled in our exercise scenario, the reaction rate increases by a factor of 27 (since \(3\) for A and \(3^2\) for C multiply together). Here, \(k\) doesn't change; rather, the alteration in concentrations causes the rate to accelerate. This makes the rate constant a powerful tool for comparing reactions under different conditions or scaling-up processes from labs to industrial scales.