Question

(a) The reaction \(\mathrm{H}_{2} \mathrm{O}_{2}(a q) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)+\frac{1}{2} \mathrm{O}_{2}(g)\), is first order. Near room temperature, the rate constant equals \(7.0 \times 10^{-4} \mathrm{~s}^{-1} .\) Calculate the half-life at this temperature. (b) At \(415^{\circ} \mathrm{C}\), \(\left(\mathrm{CH}_{2}\right)_{2} \mathrm{O}\) decomposes in the gas phase, \(\left(\mathrm{CH}_{2}\right)_{2} \mathrm{O}(g) \longrightarrow \mathrm{CH}_{4}(g)+\mathrm{CO}(g) .\) If the reac- tion is first order with a half-life of \(56.3 \mathrm{~min}\) at this temperature, calculate the rate constant in \(\mathrm{s}^{-1}\).

Short answer

(a) The half-life of the reaction at room temperature is 990 seconds. (b) The rate constant for the reaction at 415°C is approximately \(2.05 \times 10^{-4} s^{-1}\).

Step-by-step solution

Identify the given rate constant

We are given the rate constant as \(k = 7.0 \times 10^{-4} s^{-1}\).

Use the half-life formula

We need to calculate the half-life using the formula: \(t_{1/2} = \frac{0.693}{k}\) Plug in the given value for \(k\): \(t_{1/2} = \frac{0.693}{7.0 \times 10^{-4} s^{-1}}\)

Calculate the half-life

Calculate the half-life: \(t_{1/2} = 990 s \) So the half-life of the reaction is 990 seconds. (b) Calculate the rate constant in \(s^{-1}\)

Identify the given half-life

We are given the half-life as \(t_{1/2} = 56.3 min\). First, we need to convert minutes to seconds: \(56.3 min \times 60 \frac{s}{min} = 3378 s\).

Rearrange the half-life formula

We need to calculate the rate constant from the given half-life. Rearrange the formula to solve for k: \(k = \frac{0.693}{t_{1/2}}\)

Plug in the given half-life

Plug in the value for the half-life \(t_{1/2} = 3378 s\): \(k = \frac{0.693}{3378 s}\)

Calculate the rate constant

Calculate the rate constant: \(k \approx 2.05 \times 10^{-4} s^{-1}\) So the rate constant for the reaction at 415°C is approximately \(2.05 \times 10^{-4} s^{-1}\).

Key concepts

Rate Constant

Understanding the rate constant is crucial in the study of first order reactions. The rate constant, often represented by the symbol \(k\), is a proportionality constant in the rate law equation. It dictates how fast a reaction occurs under certain conditions. For first order reactions, the rate law is typically written as:

• Rate = \(k imes [ ext{A}]\)

The concentration of the reactant \(\text{A}\) decreases over time, and the rate constant \(k\) determines the speed of this decrease.
In the given exercise, we use the rate constant to calculate the half-life of a reaction and vice versa. Remember, unlike 0-order reactions, the rate for a first order reaction depends linearly on reactant concentration, and \(k\) provides insights into the time-dependent behavior of a chemical system. This makes the rate constant a pivotal parameter in reaction kinetics.

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. For first order reactions, the half-life is independent of the initial concentration of the reactant. This unique property simplifies the calculation considerably. The formula used is:

• \(t_{1/2} = \frac{0.693}{k}\)

This equation implies that for first order reactions, the half-life can be directly calculated if the rate constant \(k\) is known.
In part (a) of the exercise, the half-life was calculated using this relationship and the provided rate constant, illustrating how a simple formula can elucidate the decay rate of a reaction. Conversely, if the half-life is known, the rate constant can also be determined.
This interchange adds flexibility to analyzing reaction kinetics, allowing scientists to predict how much time is left for a reaction state to halve itself.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur. It provides valuable insights into how a reaction progresses from reactants to products.
Within the context of this exercise, understanding first order reaction kinetics involves analyzing the exponential decay of reactants over time. For first order reactions:

• The rate of reaction only depends on the concentration of one reactant.
• It follows the equation: Rate = \(-\frac{d[ ext{A}]}{dt} = k[ ext{A}]\)

Studying kinetics can help predict the speed and mechanism of a reaction. In practical applications, such as drug metabolism or industrial processes, knowing the rate of a reaction allows for optimized conditions and improved efficiency.
Reactions can be modeled using different mathematical techniques, but understanding the basic principles, such as determining the rate constant and half-life, is fundamental for analyzing the kinetics of any chemical reaction. This knowledge bridges the gap between theoretical chemistry and real-world applications.