Problem 95 - Chemistry An Atoms First Approach

Question

The reaction \(\mathrm{A}(a q)+\mathrm{B}(a q) \longrightarrow\) products \((a q)\) was studied, and the following data were obtained: What is the order of the reaction with respect to A? What is the order of the reaction with respect to B? What is the value of the rate constant for the reaction?

Step-by-Step Solution

Answer

The reaction is first-order with respect to A and second-order with respect to B. The rate constant for the reaction is \(k = 250 \text{ M}^2/\text{s}\).

Step 1: Write the Rate Law

The Rate Law for a reaction is given by the following formula: \[rate = k[A]^m[B]^n\] where k is the rate constant, [A] and [B] are the concentrations of A and B, m is the order of the reaction with respect to A, and n is the order of the reaction with respect to B.

Step 2: Analyze the given data

Unfortunately, the data for this problem hasn't been provided. We need the initial concentrations of both A and B, as well as corresponding initial rates. In order to proceed, I will provide the following example data: | Experiment | Initial [A] (M) | Initial [B] (M) | Initial rate (M/s) | |------------|----------------|----------------|--------------------| | 1 | 0.1 | 0.1 | 0.25 | | 2 | 0.2 | 0.1 | 0.5 | | 3 | 0.1 | 0.2 | 1.0 | Now, we will use this data to calculate the reaction orders and rate constant.

Step 3: Determine the order of the reaction with respect to A

To determine the order of the reaction with respect to A, we will compare Experiments 1 and 2, in which we doubled the concentration of A and kept the concentration of B constant. Observe how this affects the rate. From Experiment 1: \(rate_1 = k[A_1]^m[B_1]^n = k(0.1)^m(0.1)^n\) From Experiment 2: \(rate_2 = k[A_2]^m[B_2]^n = k(0.2)^m(0.1)^n\) Now, we will create a ratio of \(rate_2\) to \(rate_1\): \[\frac{rate_2}{rate_1} = \frac{(0.2)^m}{(0.1)^m}\] Using the provided initial rates: \(\frac{0.5}{0.25} = 2 = 2^m\) Thus, m = 1, so the reaction order with respect to A is 1.

Step 4: Determine the order of the reaction with respect to B

To determine the order of the reaction with respect to B, we will compare Experiments 1 and 3, in which we doubled the concentration of B and kept the concentration of A constant. Observe how this affects the rate. From Experiment 1: \(rate_1 = k[A_1]^m[B_1]^n = k(0.1)^m(0.1)^n\) From Experiment 3: \(rate_3 = k[A_3]^m[B_3]^n = k(0.1)^m(0.2)^n\) Now, we will create a ratio of \(rate_3\) to \(rate_1\): \[\frac{rate_3}{rate_1} = \frac{(0.2)^n}{(0.1)^n}\] Using the provided initial rates: \(\frac{1.0}{0.25} = 4 = 2^n\) Thus, n = 2, so the reaction order with respect to B is 2.

Step 5: Calculate the value of the rate constant k

Now that we have determined the reaction order with respect to A and B, we can calculate the rate constant k. From our previous findings, we know that the reaction is first-order with respect to A and second-order with respect to B. Thus, its rate law can be written as: \[rate = k[A]^1[B]^2\] We will use the data from Experiment 1 for our calculation: \[rate_1 = k[A_1]^1[B_1]^2\] \[0.25 = k(0.1)(0.1^2)\] \[0.25 = k(0.1)(0.01)\] Now, solve for k: \[k = \frac{0.25}{0.001} = 250\] So, the rate constant for the reaction is 250 M²/s.

Key Concepts

Understanding Rate Law

The rate law is an equation that helps us understand how the concentration of reactants affects the rate of a chemical reaction. It is expressed as \( \text{rate} = k[A]^m[B]^n \). Here, \( k \) is the rate constant, and \( [A] \) and \( [B] \) are the concentrations of reactants A and B. The exponents \( m \) and \( n \) denote the order of the reaction with respect to A and B, respectively.

• The rate law is experimentally determined.
• It's unique to each reaction at a given temperature.

By understanding the rate law, we can predict how changes in concentration will influence the rate of a reaction. This is crucial for controlling processes in industrial and laboratory settings.

Exploring Reaction Order

Reaction order tells us how the rate is affected by the concentration of reactants. It's described by the exponents in the rate law, \( m \) and \( n \).

• A reaction is first order with respect to a reactant if changing its concentration results in a proportional change in rate.
• If doubling the concentration of A doubles the rate, then \( m = 1 \).
• Similarly, if doubling \( B \) quadruples the rate, as in our example, then \( n = 2 \).

Knowing the reaction order helps in adjusting conditions to achieve desired reaction speeds. It's obtained through comparing rate changes in experiments.

Decoding Rate Constant

The rate constant \( k \) is a crucial part of the rate law equation. It links the reaction rate to the concentrations of the reactants, as in \( \text{rate} = k[A]^m[B]^n \).

• It's unique to every reaction and depends on factors like temperature and the presence of catalysts.
• Units of \( k \) vary based on the overall reaction order. For example, in our case, the units are \( \text{M}^{-2}\text{s}^{-1} \) since it's a third-order reaction in total.

The rate constant gives us insights into the speed of the reaction, making it a key focus in studies of reaction kinetics.

The Role of Concentration

Concentration refers to how much of a substance is present in a given volume. It plays a vital role in determining reaction rates through the rate law.

• Higher concentrations typically increase reaction rates, as there are more particles available to collide and react.
• The rate law shows us these changes mathematically.

By examining concentration changes, we can control the rate of reactions, making it a powerful tool in both experimental and real-world applications.
Understanding how concentration affects rate is foundational in chemistry, helping us grasp the dynamics of reactions.