Problem 23 - 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 \(k=1.06 \times 10^{-3} \mathrm{min}^{-1}\) (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

(a) 145 mins for 15% decomposition. (b) 1800 mins for 85% decomposition.

Step 1: Understand First Order Kinetics

For a first order reaction, the rate of reaction depends linearly on the concentration of a single reactant. The integrated rate law for a first order reaction is given by:\[ ext{ln} rac{[A]_t}{[A]_0} = -kt\]Where \([A]_t\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, and \(k\) is the rate constant. We will use this to find the time \(t\) for decomposition.

Step 2: Calculate Time for 15% Decomposition

For the hydrogen peroxide to decompose by 15%, 85% of it remains. So, \(\frac{[A]_t}{[A]_0} = 0.85\). Substitute the known values into the integrated rate law:\[ ext{ln} (0.85) = -1.06 imes 10^{-3} imes t\]Solve for \(t\):\[t = \frac{ ext{ln} (0.85)}{-1.06 imes 10^{-3}}\]Calculate \(t\) using a calculator to get the time in minutes.

Step 3: Calculate Time for 85% Decomposition

For the hydrogen peroxide to decompose by 85%, only 15% of it remains. So, \(\frac{[A]_t}{[A]_0} = 0.15\). Substitute the known values into the same integrated rate law:\[ ext{ln} (0.15) = -1.06 imes 10^{-3} imes t\]Solve for \(t\):\[t = \frac{ ext{ln} (0.15)}{-1.06 imes 10^{-3}}\]Calculate \(t\) using a calculator to find the time in minutes.

Key Concepts

Integrated Rate Law

In chemistry, the integrated rate law is a key formula used to understand how the concentration of a reactant changes over time. For first order reactions, like the decomposition of hydrogen peroxide (H2O2), this law takes a special form that shows how the rate is directly proportional to the remaining concentration of the reactant. The equation is:\[\ln \left(\frac{[A]_t}{[A]_0}\right) = -kt\]Here:

• \([A]_t\) is the concentration of H2O2 at time \(t\).
• \([A]_0\) is the initial concentration of H2O2.
• \(k\) is the reaction rate constant.
• \(t\) is the time in minutes that we are solving for.

This equation allows us to solve for the time it takes for a certain percentage of the reactant to decompose. By rearranging it, you can easily calculate how long it takes for the concentration to reach a certain level.

Reaction Rate Constant

The reaction rate constant, denoted as \(k\), is a crucial factor in determining the speed of a chemical reaction. For a first order reaction like the decomposition of hydrogen peroxide, \(k\) has units of \(\text{min}^{-1}\), meaning it shows the number of reactions occurring per unit of time.In our problem, the reaction rate constant is given as \(1.06 \times 10^{-3} \text{ min}^{-1}\). This value tells us how quickly the H2O2 decomposes in this specific set of conditions. A larger absolute value of \(k\) implies a faster reaction. Understanding the rate constant is important because it helps predict how long processes like decomposition will take under various conditions.When applying the integrated rate law, the reaction rate constant is pivotal as it directly influences the time calculation for reaching a specified reactant concentration.

Decomposition Reaction

A decomposition reaction involves a single compound breaking down into two or more products. In the case of hydrogen peroxide, it undergoes a decomposition reaction:\[\text{H}_2\text{O}_2(\text{aq}) \rightarrow \text{H}_2\text{O}(\ell) + \text{O}_2(\text{g})\]This reaction starts with a single reactant, H2O2, which breaks down into water and oxygen gas. Decomposition reactions often require energy input, yet in hydrogen peroxide, the reaction can proceed slowly on its own due to its intrinsic instability. Such reactions are significant in everyday life and industrial processes, as they can help in applications ranging from cleaning to rocket propulsion. Understanding the specific decomposition of H2O2 unveils important insights into the broader chemical world.

Hydrogen Peroxide

Hydrogen peroxide, H2O2, is a chemical compound commonly used as a disinfectant and bleaching agent. It is known for being a colorless liquid that has strong oxidizing properties. This nature makes it very reactive, often playing crucial roles in various chemical pathways and reactions. In its decomposition, hydrogen peroxide acts as both a reducing and oxidizing agent, facilitating its breakdown into water and oxygen. The decomposition reaction can be catalyzed by various factors, such as impurities, light, or the presence of certain enzymes. Hydrogen peroxide is environmentally friendly because it decomposes into non-toxic end products. Yet, it must be handled with care due to its reactive nature, especially at higher concentrations or temperatures. Understanding its properties equips us to use it safely and effectively in numerous contexts.