Problem 27 - Chemistry: Principles and Reactions

Question

Hydrogen bromide is a highly reactive and corrosive gas used mainly as a catalyst for organic reactions. It is produced by reacting hydrogen and bromine gases together. $$\mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g) \longrightarrow 2 \mathrm{HBr}(g)$$ The rate is followed by measuring the intensity of the orange color of the bromine gas. The following data are obtained: $$\begin{array}{cccc}\hline \text { Expt. } & {\left[\mathrm{H}_{2}\right]} & {\left[\mathrm{Br}_{2}\right]} & \text { Initial Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s}) \\ \hline 1 & 0.100 & 0.100 & 4.74 \times 10^{-3} \\ 2 & 0.100 & 0.200 & 6.71 \times 10^{-3} \\ 3 & 0.250 & 0.200 & 1.68 \times 10^{-2} \\ \hline\end{array}$$(a) What is the order of the reaction with respect to hydrogen, bromine, and overall? (b) Write the rate expression for the reaction. (c) Calculate \(k\) for the reaction. What are the units for \(k ?\) (d) When \(\left[\mathrm{H}_{2}\right]=0.455 \mathrm{M}\) and \(\left[\mathrm{Br}_{2}\right]=0.215 M\), what is the rate of the reaction?

Step-by-Step Solution

Answer

Answer: The order of the reaction with respect to hydrogen is 1 (first order), and the order with respect to bromine is 0.5 (half order). The rate constant k is approximately 0.474 M^(0.5)s^(-1).

Step 1: Determine the order of the reaction with respect to hydrogen

To determine the order of the reaction with respect to hydrogen, compare Experiment 2 and Experiment 3. Notice that the ratio between [H₂] in Experiment 3 to [H₂] in Experiment 2 is 0.25/0.1 = 2.5. The initial rate of Experiment 3 to Experiment 2 is (1.68 × 10⁻²) / (6.71 × 10⁻³) = 2.5. Since the ratio of initial concentrations equals the ratio of initial rates for hydrogen, we can conclude that the order of the reaction with respect to hydrogen (n) is 1 (first order).

Step 2: Determine the order of the reaction with respect to bromine

To determine the order of the reaction with respect to bromine, compare Experiment 1 and Experiment 2. The ratio between the concentrations of [Br₂] in Experiment 2 to [Br₂] in Experiment 1 is 0.2/0.1 = 2. The initial rate of Experiment 2 to Experiment 1 is (6.71 × 10⁻³) / (4.74 × 10⁻³) ≈ 1.4. Since the ratio of initial concentrations does not equal the ratio of initial rates for bromine, we can conclude that the order of the reaction with respect to bromine (m) is not 1. Try the square root: Taking the square root of the ratio of initial concentrations, we get √(0.2/0.1) = 1.4. Now the ratio of the initial rate equals the square root of the ratio of concentrations. We can conclude that the order of the reaction with respect to bromine is 0.5 (half order).

Step 3: Determine the overall order of the reaction

The overall order of the reaction is the sum of the orders of hydrogen and bromine. Thus, the overall order is 1 (for hydrogen) + 0.5 (for bromine) = 1.5.

Step 4: Write the rate expression for the reaction

Using the reaction order information we obtained in steps 1 and 2, we can write the rate expression for the reaction as follows: $$\text{Rate} = k[\mathrm{H}_{2}]^{1}[\mathrm{Br}_{2}]^{0.5}$$

Step 5: Calculate the rate constant (k) and its units

We can use any experiment data to calculate the rate constant (k). Let's use Experiment 1. Rearrange the rate expression to solve for k, then plug in the given concentrations and rate: $$k = \frac{\text{Rate}}{[\mathrm{H}_{2}]^{1}[\mathrm{Br}_{2}]^{0.5}}$$ $$k = \frac{4.74 \times 10^{-3} M \cdot s^{-1}}{0.1^1 M \cdot (0.1)^{0.5} M^{0.5}}$$ $$k ≈ 0.474\, M^{0.5}s^{-1} $$ So the rate constant, k, is approximately 0.474 M^(0.5)s^(-1).

Step 6: Calculate the rate of the reaction for given concentrations

We are given the concentrations [H₂] = 0.455 M and [Br₂] = 0.215 M, and we need to find the rate of the reaction. We will use the rate expression from Step 4 and substitute the values with the calculated k: $$\text{Rate} = (0.474\,M^{0.5}s^{-1})[(0.455\,M)^{1}][(0.215\,M)^{0.5}]$$ $$\text{Rate} ≈ 0.195\, M\cdot s^{-1}$$ The rate of the reaction, when [H₂] = 0.455 M and [Br₂] = 0.215 M, is approximately 0.195 M/s.

Key Concepts

Reaction Order

When we talk about reaction order, we're essentially discussing how the concentration of reactants affects the rate of a chemical reaction. It tells us the power to which the concentration of a reactant is raised in the rate expression. For instance, if the reaction is first order with respect to hydrogen, it means that when the concentration of hydrogen doubles, the rate of reaction also doubles. Here are some key points:

• First Order: If a reactant is first order, the rate of reaction changes linearly with its concentration.
• Zero Order: Rate is constant and independent of the concentration of the reactant.
• Second Order: The rate is proportional to the square of the reactant concentration.
• Fractional Order: It's possible to have non-integer orders, like 0.5 in the case of bromine.

The overall order of a reaction is simply the sum of the orders with respect to each reactant. In this exercise, we found hydrogen to be first-order and bromine to be half-order, giving us an overall order of 1.5.

Rate Expression

A rate expression captures the mathematical relationship between the rate of a reaction and the concentrations of its reactants. It is usually written as:

• Rate = k[A]^m[B]^n

Here, k represents the rate constant, and A and B are reactants with m and n denoting their reaction orders. This expression helps predict how changes in conditions, like reactant concentrations, affect the speed of a reaction.
In this example, the rate expression was determined to be Rate = k[H₂]¹[Br₂]⁰. ⁵. This means the rate is directly proportional to the concentration of hydrogen raised to the power of one and the concentration of bromine raised to the power of half. Understanding the rate expression is key to controlling and optimizing reactions.

Rate Constant

The rate constant, denoted as k, is a crucial part of the rate expression. It links the rate of reaction to the concentrations of reactants, and it varies with temperature. Here’s what to consider about k:

• Units: The units of the rate constant depend on the overall order of the reaction. In our example, with an overall order of 1.5, the units were M^(0.5)s^(-1).
• Determination: To determine k, we rearrange the rate expression using the known initial rates and concentrations from experiments.
• Temperature Dependency: An increase in temperature will generally increase k, thus speeding up the reaction.

The value of k represents how efficiently a reaction proceeds and is a unique characteristic for each reaction.

Organic Reactions

Organic reactions involve organic compounds, typically involving processes like substitution, elimination, or addition reactions. Hydrogen bromide (HBr) plays a vital role in these reactions. Here’s why this matters:

• Catalysis: HBr can act as a catalyst, speeding up the reaction without being consumed in the process.
• Reactivity: Due to its nature, HBr is highly reactive, particularly with unsaturated compounds such as alkenes and alkynes.
• Mechanism: Understanding the mechanism of how HBr participates in a reaction—whether it's an attack on a double bond or another process—is crucial for predicting product formation.

Grasping the role of these chemical reactions is instrumental for fields like pharmaceuticals, material sciences, and more. It's not just about knowing the reactants and products, but about understanding how to drive these reactions efficiently.