Problem 57 - Chemistry and Chemical Reactivity

Question

Formic acid decomposes at \(550^{\circ} \mathrm{C}\) according to the equation $$ \mathrm{HCO}_{2} \mathrm{H}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) $$ The reaction follows first-order kinetics. In an experiment, it is determined that \(75 \%\) of a sample of \(\mathrm{HCO}_{2} \mathrm{H}\) has decomposed in 72 seconds. Determine \(t_{1 / 2}\) for this reaction.

Step-by-Step Solution

Answer

The half-life, \(t_{1/2}\), is approximately 35.9 seconds.

Step 1: Understand First-Order Reaction Kinetics

First-order reactions have the rate law: \( \ln [A]_t = \ln [A]_0 - kt \), where \([A]_t\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, and \(k\) is the rate constant.

Step 2: Define the Problem

In the problem, we know that 75% of formic acid has decomposed in 72 seconds, which means 25% remains. We also know the reaction follows first-order kinetics.

Step 3: Apply the First-Order Kinetics Formula

Using the formula \( \ln ([A]_t/[A]_0) = -kt \), where \([A]_t/[A]_0 = 0.25 \) after 72 seconds, we can solve for \(k\). Substitute \([A]_t/[A]_0 = 0.25\) and \(t = 72\) seconds.

Step 4: Solve for the Rate Constant \(k\)

Using the equation from Step 3: \( \ln(0.25) = -72k \). Therefore, \(k = -\frac{\ln(0.25)}{72} \approx 0.0193 \, \text{s}^{-1} \).

Step 5: Calculate the Half-Life \(t_{1/2}\)

The half-life for a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \). Substitute the value of \(k\) found in Step 4: \( t_{1/2} = \frac{0.693}{0.0193} \approx 35.9 \, \text{s} \).

Key Concepts

Rate Constant

The rate constant, represented by the symbol \(k\), is a crucial parameter in the study of reaction kinetics. It provides valuable information about the speed of a reaction. For first-order reactions, the relationship between the rate constant and concentration change over time can be described by the equation: \[ \ln [A]_t = \ln [A]_0 - kt \]where \([A]_0\) is the initial concentration, \([A]_t\) is the concentration at time \(t\), and \(k\) is the rate constant. An important aspect of first-order reactions is that the rate of reaction is directly proportional to the concentration of the single reactant. This means:

• The rate of reaction remains constant as long as a constant proportion of reactant is converted into products.

In the context of the given exercise, determining the rate constant is key to further calculations. It outlines how quickly the formic acid decomposes under specified conditions, which is at a significant 0.0193 s^-1.

Half-Life Calculation

In chemical kinetics, the half-life of a reaction, denoted as \(t_{1/2}\), is the time required for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, the half-life is distinct in that it remains constant over time. The formula for the half-life in first-order kinetics is: \[ t_{1/2} = \frac{0.693}{k} \]This implies that once \(k\) is determined, calculating \(t_{1/2}\) becomes straightforward. In our problem, substituting the calculated rate constant \(k = 0.0193 \, \text{s}^{-1}\), we find:

• \(t_{1/2} \approx 35.9 \, \text{s}\)

This means that at 550°C, it takes approximately 35.9 seconds for half of the formic acid to decompose. The constancy of the half-life reflects the consistent rate of reaction over time, highlighting a key feature of first-order reactions.

Reaction Kinetics

Reaction kinetics is the study of the rates of chemical processes. It provides insights into how different conditions affect the speed and mechanism of reactions. In our context of first-order kinetics, this type of reaction follows a linear decay when plotted as the natural logarithm of concentration versus time. This reveals several important points:

• First-order kinetics suggests that the reaction rate depends on the concentration of a single reactant.
• The rate does not depend on the concentration of any other substances possibly introduced into the reaction.

Understanding first-order kinetics is crucial when predicting how the concentration of a reactant decreases over time and for planning the conditions required to achieve desired reaction completion. Overall, mastering reaction kinetics helps you anticipate changes in reaction behavior and optimize conditions accordingly, providing a comprehensive view of the dynamic nature of chemical reactions.