Problem 31 - Chemistry and Chemical Reactivity

Question

Formic acid decomposes at \(550^{\circ} \mathrm{C}\) according to the equation $$\mathrm{HCO}_{2} \mathrm{H}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ 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 \(t_{1/2}\) for this reaction is approximately 36.09 seconds.

Step 1: Understand the Kinetics

This reaction follows first-order kinetics, which means the rate of reaction depends linearly on the concentration of formic acid. For first-order reactions, the relationship between concentration and time is given by the equation:\[ ln([A]_0/[A]) = kt \]where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant.

Step 2: Determine the Rate Constant, k

We are told that 75% of HCO₂H has decomposed in 72 seconds. This means 25% remains. Using the relationship for a first-order reaction we have:\[ ln([A]_0/[A]) = kt \]Substitute \([A]_0/[A] = 1/0.25 = 4\) and \(t = 72 \, \text{seconds}\) to find \(k\):\[ ln(4) = k \times 72 \]\[ k = \frac{ln(4)}{72} \approx 0.0192 \, \text{seconds}^{-1} \]

Step 3: Calculate the Half-life, t_{1/2}

For first-order reactions, the half-life \(t_{1/2}\) is given by the formula:\[ t_{1/2} = \frac{0.693}{k} \]Substitute \(k = 0.0192 \, \text{seconds}^{-1}\):\[ t_{1/2} = \frac{0.693}{0.0192} \approx 36.09 \, \text{seconds} \]

Key Concepts

Half-life calculation

The half-life of a reaction, denoted as \( t_{1/2} \), is the time it takes for half of the reactant to be consumed in a chemical reaction. It is an essential concept in understanding how quickly a reaction proceeds. For first-order reactions, the half-life is independent of the initial concentration of the reactant. This characteristic reliably informs us about the dynamics of chemical interactions without altering due to concentration changes. First-order reactions simplify the half-life calculation with the formula: \[ t_{1/2} = \frac{0.693}{k} \] where \( k \) is the rate constant. This formula highlights how half-life is a constant value, unchanged by varying amounts of starting material, making it extremely useful for predicting reaction schedules and durations. By knowing \( k \), you can straightforwardly determine how long it will take for half the reactant to decompose, offering a clear picture of the reaction's pace.

Rate constant determination

Determining the rate constant, \( k \), is fundamental for analyzing reaction kinetics, as it gives insight into the speed of a chemical reaction. The rate constant is unique to each reaction and varies with conditions such as temperature. For first-order reactions, the equation used is: \[ ln([A]_0/[A]) = kt \] In this equation: - \([A]_0\) is the initial concentration of the reactant. - \([A]\) is the concentration after time \( t \). By rearranging and solving for \( k \), researchers can quantify how a reaction progresses over time. For example, in a scenario where \( 75\% \) of a substance has decomposed, only \( 25\% \) remains. This remaining concentration can substitute in the equation to find \( k \). Using this process helps to illustrate the reaction rate, providing valuable insight into the efficiency of the reaction.

Chemical decomposition

Chemical decomposition is a process where a single compound breaks down into two or more simpler substances. This occurs through chemical reactions and often involves energy to disrupt chemical bonds. Decomposition reactions are categorized by their mechanism, conditions, and the substances they generate. In the context of the reaction \( \text{HCO}_2\text{H} \rightarrow \text{CO}_2 + \text{H}_2 \), decomposition results in the formation of carbon dioxide and hydrogen gas. These reactions are essential in both biological and industrial processes, underlining the role of energy changes and the resultant new product formation. Understanding the decomposition provides insights into reaction mechanisms and helps clarify conditions required for such transformations. It allows scientists to predict product formations and understand the energy dynamics involved.

Reaction kinetics

Reaction kinetics explores the rates of chemical processes and the steps that control these rates. It examines how different conditions like concentration, temperature, and catalysts affect the speed of a reaction. First-order kinetics specifically examines cases where the rate depends on the concentration of a single reactant. The rate laws derived from reaction kinetics provide mathematical relationships that describe how reactants transform into products over time. For first-order reactions, this relationship is a simple logarithmic decay, expressed as: - \[ ln([A]_0/[A]) = kt \] This formula allows us to calculate the time required for a given amount of reactant to decompose. Therefore, understanding reaction kinetics is crucial for predicting how fast reactions occur, optimizing conditions for industrial chemical processes, and enhancing productivity by controlling and applying heat or pressure efficiently. It lays the foundation for further studies on complex reaction networks.