Problem 38 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

The rate law is first-order: \( r = k[\mathrm{HOF}] \), with \( k = 0.046 \) min\(^{-1}\).

Step 1: Understand the Reaction

The given reaction is \( 2 \mathrm{HOF}(\mathrm{g}) \rightarrow 2 \mathrm{HF}(\mathrm{g}) + \mathrm{O}_2(\mathrm{g}) \), indicating a decomposition reaction. We need to determine the rate law, which depends on the concentration of \( \mathrm{HOF} \) and possibly its order.

Step 2: Plot Concentration Data

Before deciding on the rate law, plot the concentration of \( \mathrm{HOF} \) against time to visualize the reaction order. The values are: \[ [\mathrm{HOF}] (\mathrm{mol/L}): 0.850, 0.810, 0.754, 0.526, 0.243 \] with corresponding time: \[ t (\mathrm{min}): 0, 2.00, 5.00, 20.0, 50.0 \]. A linear decrease suggests zero order, exponential suggests first order, and a quadratic curve suggests second order.

Step 3: Determine Reaction Order

Calculate the natural logarithm of concentrations and their inverses: \( \ln([\mathrm{HOF}]) \) and \( 1/[\mathrm{HOF}] \) for each time point. Comparing the plotted lines of \( [\mathrm{HOF}] \), \( \ln([\mathrm{HOF}]) \), and \( 1/[\mathrm{HOF}] \) against time will indicate the order of reaction. First-order reactions are linear in \( \ln([\mathrm{HOF}]) \).

Step 4: Rate Law Identification

Based on the analysis, if a plot of \( \ln([\mathrm{HOF}]) \) vs. time is linear, the reaction is first order. The rate law for a first-order reaction is \( r = k[\mathrm{HOF}] \), where \( r \) is the rate, \( k \) is the rate constant, and \( [\mathrm{HOF}] \) is the concentration.

Step 5: Calculate the Rate Constant

Using the linear relationship of \( \ln([\mathrm{HOF}]) \) vs. time, apply the following first-order kinetics equation: \[ \ln([\mathrm{HOF}]_t) = \ln([\mathrm{HOF}]_0) - kt \] where \( [\mathrm{HOF}]_0 \) is the initial concentration. Calculate \( k \) using two time points, such as \( t = 0 \) with \( [\mathrm{HOF}] = 0.850 \) and \( t = 50 \) with \( [\mathrm{HOF}] = 0.243 \). Rewriting, \( k = \frac{1}{t} (\ln([\mathrm{HOF}]_0) - \ln([\mathrm{HOF}]_t)) = \frac{1}{50} (\ln(0.850) - \ln(0.243)) \).

Step 6: Solve for k

Compute \( k \) as follows: \[ k = \frac{1}{50} (\ln(0.850) - \ln(0.243)) \approx 0.046 \] min\(^{-1}\).

Key Concepts

Reaction Order

In kinetics, understanding the reaction order is crucial because it tells us how the rate is affected by the concentration of reactants. Each reaction can have its own unique order, which might be zero, first, second, or even fractional. To determine the order, we look at how concentration changes over time.
For instance:

• A zero-order reaction experiences a constant rate. Thus, the concentration decreases linearly over time regardless of how much reactant is left.
• A first-order reaction shows an exponential decrease in concentration. The rate changes proportionally with the concentration of the reactant.
• A second-order reaction reveals a rate that's proportional to the square of the reactant's concentration, often seen as a curve when plotted.

In the example of HOF decomposition, plotting the natural logarithm of concentration versus time helps us confirm the reaction is first order. A linear graph for this plot straight away signals that the reaction is first order. This insight allows us to create a rate law that accurately reflects how the HOF decomposes.

Rate Constant

The rate constant, denoted by \( k \), is an essential part of a reaction's rate law. It links the rate of reaction to the concentrations of reactants raised to their respective powers as dictated by the reaction order. Its units vary depending on the reaction order:

• For a zero-order reaction, \( k \) has units of concentration/time (e.g., mol/L/s).
• For a first-order reaction, \( k \) has units of 1/time (e.g., s\(^{-1}\) or min\(^{-1}\)).
• For a second-order reaction, \( k \) has units of 1/(concentration \( \times \) time) (e.g., L/mol/s).

In analyzing the decomposition of HOF, once we've confirmed that it's a first-order reaction, we find \( k \) using a straightforward formula derived from the integrated rate law: \[ k = \frac{1}{t} (\ln([\mathrm{HOF}]_0) - \ln([\mathrm{HOF}]_t)) \]This calculation gives a consistent rate constant, which helps in predicting how fast or slow the reaction progresses under the same conditions.

Kinetics

Kinetics studies the rate at which chemical reactions occur, allowing chemists to understand and predict the behavior of reactions over time. By examining the rate law, which incorporates both the reaction order and rate constant, you gain insight into how different factors influence reaction speed.
Essentially, kinetics bridges the gap between theoretical chemical equations and real reactions observed experimentally. It answers crucial questions:

• How does temperature affect the rate?
• What role do catalysts play in speeding up a reaction?
• How do changes in concentration influence the rate?

In the HOF decomposition reaction, kinetics shows that as the concentration of HOF decreases, the rate of formation of HF and \( \mathrm{O_2} \) slows down. This decrease follows a predictable pattern described by the first-order rate law previously discussed. Understanding this pattern lets scientists plan how to control the reactions practically, whether in industrial settings or laboratory research.

Decomposition Reaction

A decomposition reaction occurs when a single chemical compound breaks down into two or more simpler substances. This type of reaction often requires energy input, such as heat, light, or electricity, to proceed.
Common characteristics of decomposition reactions include:

• The initial compound splitting into individual elements or smaller compounds.
• Reactions being endothermic—absorbing energy to proceed.
• The involvement of gases as an output, more obvious by a release of visible gases or bubbles.

In the case of HOF decomposition, written as \( 2 \mathrm{HOF}(\mathrm{g}) \rightarrow 2 \mathrm{HF}(\mathrm{g}) + \mathrm{O}_2(\mathrm{g}) \), we see hydrogen fluoride and oxygen gas as the products. Understanding decomposition mechanisms helps predict not only what products are formed but also the conditions necessary for the reaction to occur efficiently.