Problem 19 - Chemistry and Chemical Reactivity

Question

Hydrogen peroxide, \(\mathrm{H}_{2} \mathrm{O}_{2}(\mathrm{aq}),\) decomposes to \(\mathrm{H}_{2} \mathrm{O}(\ell)\) and \(\mathrm{O}_{2}(\mathrm{g})\) in a reaction that is first-order in \(\mathrm{H}_{2} \mathrm{O}_{2}\) and has a rate constant \(k=1.06 \times 10^{-3} \mathrm{min}^{-1}\) at a given temperature. (a) How long will it take for \(15 \%\) of a sample of \(\mathrm{H}_{2} \mathrm{O}_{2}\) to decompose? (b) How long will it take for \(85 \%\) of the sample to decompose?

Step-by-Step Solution

Answer

150.2 min for 15% decomposed; 1718.9 min for 85% decomposed.

Step 1: Understand the Problem

We are given a first-order decomposition reaction for \( \mathrm{H}_{2}\mathrm{O}_{2} \). We need to calculate the time it takes for \( 15\% \) and \( 85\% \) of the \( \mathrm{H}_{2}\mathrm{O}_{2} \) to decompose. The rate constant \( k \) is provided as \( 1.06 \times 10^{-3} \, \mathrm{min}^{-1} \). We will use the first-order kinetics formula to find the times.

Step 2: Use First-Order Kinetics Formula

The formula for first-order reactions is: \[\ln \left( \frac{[A]_t}{[A]_0} \right) = -kt\]Here, \([A]_0\) is the initial concentration, and \([A]_t\) is the concentration at time \( t \). For both (a) and (b), we will use this formula with the appropriate values for \( [A]_t / [A]_0 \).

Step 3: Solve for 15% Decomposition

For 15% decomposition, 85% of \( \mathrm{H}_{2}\mathrm{O}_{2} \) remains. Thus,\[\frac{[A]_t}{[A]_0} = 0.85\]Plug into the equation:\[\ln(0.85) = - (1.06 \times 10^{-3}) t\]Solving for \( t \),\[t = \frac{\ln(0.85)}{-(1.06 \times 10^{-3})} \approx 150.2 \, \text{min}\]

Step 4: Solve for 85% Decomposition

For 85% decomposition, 15% of \( \mathrm{H}_{2}\mathrm{O}_{2} \) remains. Thus,\[\frac{[A]_t}{[A]_0} = 0.15\]Plug into the equation:\[\ln(0.15) = - (1.06 \times 10^{-3}) t\]Solving for \( t \),\[t = \frac{\ln(0.15)}{-(1.06 \times 10^{-3})} \approx 1718.9 \, \text{min}\]

Step 5: Conclusion

Based on the calculations, it takes approximately 150.2 minutes for 15% of \( \mathrm{H}_{2}\mathrm{O}_{2} \) to decompose and 1718.9 minutes for 85% to decompose.

Key Concepts

Understanding Chemical Decomposition

Chemical decomposition is a type of reaction where a compound breaks down into simpler substances. This process often involves breaking bonds within a molecule, leading to the formation of different products. In the case of hydrogen peroxide (\(\mathrm{H}_{2}\mathrm{O}_{2} \)), it decomposes into water (\(\mathrm{H}_{2}\mathrm{O} \)) and oxygen (\(\mathrm{O}_{2} \)).

This decomposition reaction is classified as first-order, which indicates that the rate of the reaction depends on the concentration of one reactant—in this scenario, hydrogen peroxide. The speed at which \(\mathrm{H}_{2}\mathrm{O}_{2} \) decomposes is affected by various factors, including temperature and the presence of a catalyst. However, for first-order reactions, the primary factor influencing the rate is the concentration of the reactant itself.

Role of the Rate Constant

The rate constant (\(k\)) is a crucial parameter in kinetics, representing how fast a reaction proceeds. For first-order reactions, it's expressed with units of \(\mathrm{min}^{-1} \), indicating a time-dependent rate.

In our hydrogen peroxide decomposition example, the rate constant is given as \(1.06 \times 10^{-3} \; \mathrm{min}^{-1} \).

• A higher value of \(k\) suggests a faster reaction rate, while a lower value indicates a slower process.
• The rate constant is influenced by temperature: as temperature increases, the rate constant usually increases as well, accelerating the reaction.

Understanding the magnitude and influence of the rate constant helps in predicting how quickly a reactant will undergo chemical changes over time.

Significance of Concentration

Concentration refers to the amount of a substance present in a given volume of solution. In the context of chemical kinetics, concentration is directly linked to the reaction rate. For first-order reactions, such as the decomposition of \(\mathrm{H}_{2}\mathrm{O}_{2} \), the rate is dependent on the concentration of \(\mathrm{H}_{2}\mathrm{O}_{2} \) itself.

As the reaction progresses, the concentration of hydrogen peroxide decreases as it is converted into water and oxygen. The rate of this change provides insight into how quickly the reaction occurs.

• Initially, when the concentration is high, the reaction rate is also high.
• As decomposition continues, the concentration drops, consequently slowing down the reaction rate.

This relationship between concentration and reaction rate is central to understanding and predicting the behavior of chemical reactions over time.

Basics of Reaction Time Calculation

Calculating reaction time is essential in kinetic studies to determine how long a given reaction will take to reach a certain level of completion. For first-order reactions, we use the formula:\[\ln \left( \frac{[A]_t}{[A]_0} \right) = -kt\]Here, \([A]_0 \) is the initial concentration, \([A]_t \) is the concentration at time \(t\), and \(k\) is the rate constant.

• For example, if you need to find out how long it takes for 15% of a substance to decompose, you'd set \(\frac{[A]_t}{[A]_0} = 0.85 \) in the formula.
• Solving the equation provides the reaction time, which indicates how long it will take for the reactant's concentration to reduce to 85% of its original value for the 15% decomposition case.

Mastering reaction time calculation enables accurate predictions of reaction durations under various conditions.