Problem 34 - Chemistry and Chemical Reactivity

Question

The decomposition of HOF occurs at \(25^{\circ} \mathrm{C}\) $$\mathrm{HOF}(\mathrm{g}) \rightarrow \mathrm{HF}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g})$$ Using the data in the table below, determine the rate law, and then calculate the rate constant. $$\begin{array}{cc}{[\mathrm{HOF}](\mathrm{mol} / \mathrm{L})} & \mathrm{Time}(\mathrm{min}) \\ \hline 0.850 & 0 \\\0.810 & 2.00 \\\0.754 & 5.00 \\\0.526 & 20.0 \\\0.243 & 50.0 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The reaction is first-order with a rate constant of approximately 0.0246 min^{-1}.

Step 1: Determine Reaction Order

To determine the reaction order, we should plot the concentration of HOF against time. We have three common orders for reactions: zero-order, first-order, and second-order. Plot: - Zero-order: [HOF] vs. Time should be linear. - First-order: ln[HOF] vs. Time should be linear. - Second-order: 1/[HOF] vs. Time should be linear. By plotting, we find the first-order plot is linear, indicating a first-order reaction with respect to HOF.

Step 2: Use the First-Order Rate Law

The rate law for a first-order reaction is given by \[ ext{rate} = k[ ext{HOF}] \] Where \(k\) is the rate constant. Since we've determined the reaction is first-order, we'll use this relationship to find the rate constant \(k\).

Step 3: Apply the First-Order Integrated Rate Law

The integrated rate law for a first-order reaction is given by \[ ext{ln}([A]_t) = -kt + ext{ln}([A]_0) \]Substitute the initial concentration \([A]_0 = 0.850 \, \text{mol/L}\) and use the concentration at \(t = 5.00 \, \text{min}\) to solve for \(k\).

Step 4: Calculate the Rate Constant

Using \[ ext{ln}(0.754) = ext{ln}(0.850) - k(5.00) \]Calculate:\[ k = \frac{ ext{ln}(0.850) - ext{ln}(0.754)}{5.00} \approx 0.0246 \, \text{min}^{-1} \]

Step 5: Validate the Calculation and Result

Verify by checking if \(k\) remains consistent when using other data points. Substitute the concentration at \(t = 20.0 \, \text{min}\) and compare with initial concentration. The values should confirm that \(k \approx 0.0246 \, \text{min}^{-1}\).

Key Concepts

Rate Law

In chemical reactions, the rate law expresses the relationship between the reaction rate and the concentrations of reactants. The rate at which a chemical reaction proceeds can often depend on the concentration of the reactants. The rate law provides the mathematical description of this dependency, serving as a crucial concept in reaction kinetics. When you determine a rate law, you identify the order of the reaction with respect to each reactant. This is done experimentally by observing how changes in concentration affect the reaction speed. There are several potential orders a reaction can have, including zero, first, and second order.

• Zero-order: The rate does not change with concentration changes.
• First-order: The rate changes linearly with concentration changes.
• Second-order: The rate changes with the square of the concentration.

Understanding the rate law is essential as it helps chemists predict how a reaction behaves under different conditions, and it aids in determining all necessary parameters for controlling the reaction.

First-Order Reaction

A first-order reaction is a type of reaction where the rate depends linearly on the concentration of one reactant. This means that if you double the concentration of the reactant, the rate of the reaction also doubles. A first-order reaction is characterized by its straightforward mathematical relationship between concentration and time, which is expressed by the equation:\[\text{ln}([A]_t) = -kt + \text{ln}([A]_0)\]Here,

• \([A]_t\) is the concentration at time \( t \).
• \([A]_0\) is the initial concentration.
• \(k\) is the rate constant.

This equation is the integrated rate law for first-order reactions. It tells us that the natural logarithm of the concentration of the reactant decreases linearly with time. When examining data to identify reagents with first-order behavior, plotting \(\text{ln}([A])\) versus time will yield a straight line. The slope of this line is equal to the negative of the rate constant \(k\). This linear relationship makes it easier to analyze and predict the behavior of chemical reactions over time.

Rate Constant Calculation

The rate constant \(k\) is a crucial parameter in the study of reaction kinetics. It indicates the speed of a reaction for a particular set of conditions and is specific to each reaction and temperature. In a first-order reaction, the rate constant can be determined using experimental data and the equation derived from the integrated rate law:\[k = \frac{\text{ln}([A]_0) - \text{ln}([A]_t)}{t}\]This formula allows you to calculate \(k\) by substituting the initial concentration \([A]_0\), the concentration at a particular time \([A]_t\), and the time \(t\). For accurate calculations, you'll typically use several data points to ensure consistency:

• Check if the calculated \(k\) is consistent for all data points.
• Any variation indicates potential errors in data or the assumption about the reaction order.

The value of \(k\) is essential because it informs us about the reaction's velocity and can predict how the reaction progresses under varying conditions. Additionally, performing calculations at different temperatures helps understand the effect of temperature on the reaction rate, using the Arrhenius equation to relate \(k\) to the temperature of the reaction.