Problem 41 - Chemistry: Principles and Reactions

Question

The decomposition of phosphine, \(\mathrm{PH}_{3}\), to \(\mathrm{P}_{4}(g)\) and \(\mathrm{H}_{2}(g)\) is firstorder. Its rate constant at a certain temperature is \(1.1 \mathrm{~min}^{-1}\). (a) What is its half-life in seconds? (b) What percentage of phosphine is decomposed after \(1.25 \mathrm{~min}\) ? (c) How long will it take to decompose one fifth of the phosphine?

Step-by-Step Solution

Answer

Answer: The half-life of the decomposition is approximately 0.63 seconds, the percentage of phosphine decomposed after 1.25 minutes is 67.8%, and the time required to decompose one-fifth of the phosphine is approximately 12.18 seconds.

Step 1: (a) Calculate the half-life

For a first-order reaction, the half-life can be calculated using the following formula: t_(1/2) = ln(2)/k, where t_(1/2) is half-life and k is the rate constant. Given, k = 1.1 min^(-1). To convert k to seconds, just multiply the value by 60: k = 1.1 * 60 = 66 s^(-1) Using the formula: t_(1/2) = ln(2)/66 ≈ 0.0105 minutes or 0.0105 * 60 = 0.63 seconds.

Step 2: (b) Percentage of phosphine decomposed after 1.25 minutes

To find the remaining concentration of phosphine after 1.25 minutes, we'll use the first-order reaction equation: [PH3]t = [PH3]_0 * e^(-kt) Given, t = 1.25 minutes, k = 1.1 min^(-1). Let the initial concentration [PH3]_0 be 100%. Then: [PH3]t = 100 * e^(-1.1 * 1.25) ≈ 32.2% Now we can calculate the percentage of phosphine decomposed by subtracting the remaining concentration from the initial concentration: Percentage decomposed = 100 - 32.2 = 67.8%

Step 3: (c) Time to decompose one-fifth of the phosphine

We need to determine the time it takes for 20% (1/5th) of the phosphine to decompose. Let the initial concentration [PH3]_0 be 100%. After decomposing 20%, the remaining concentration becomes 80%. Using the first-order reaction equation: [PH3]_0 * e^(-kt) = 80 Let's divide both sides by [PH3]_0: e^(-kt) = 0.8 Now, we need to solve for t: t = - (ln(0.8) / k) = - (-0.2231 / 1.1) ≈ 0.203 minutes or approximately 12.18 seconds.

Key Concepts

Chemical Kinetics

Chemical kinetics is the study of the speed or rate at which chemical reactions proceed, and the factors that affect these rates. The speed, or rate, is a measure of how quickly reactants are transformed into products over time. In chemical kinetics, we analyze the reaction mechanisms, which are the step-by-step pathways by which reactions occur, to understand better both the speed of reactions and how they happen at a molecular level.

Key elements that influence reaction rates include the concentration of reactants, temperature, catalysts, and the physical state of the reactants. Exploring these components allows scientists to predict how changes in these factors can alter the speed of a reaction. Additionally, understanding kinetic principles is crucial for applications in industries such as pharmaceutical production, environmental engineering, and materials science where controlling reaction rates is often essential.

Reaction Rate Constant

The reaction rate constant, symbolized by 'k', is a quantitative measure of the speed of a chemical reaction. For a given reaction at a constant temperature, 'k' is a fixed value that expresses the relationship between the reactant concentrations and the rate at which they form products. In first-order reactions, such as the decomposition of phosphine, this constant can be used in conjunction with the natural logarithm of 2 (\( ln(2) \)), to find the half-life of the reaction, which represents the time required for half the reactants to be converted into products.

The value of the reaction rate constant is determined experimentally, and it varies with temperature. The Arrhenius equation describes how the rate constant changes with temperature, indicating that as the temperature increases, so too does the rate constant, hence speeding up the reaction. This rate constant is the backbone of many calculations in chemical kinetics because it provides the connecting factor between the concentration of reactants and the reaction rate.

Half-Life of a Reaction

The half-life of a reaction, often denoted as \( t_{1/2} \), is an important concept in chemical kinetics that reflects the time required for half of the reactant to be transformed into product(s) in a chemical reaction. For first-order reactions, the half-life is particularly significant because it remains constant regardless of the initial concentration of the reactant. By contrast, in reactions of other orders, half-life can vary with the initial concentration of reactants.

To calculate the half-life of a first-order reaction, you can use the formula \( t_{1/2} = \frac{ln(2)}{k} \), where 'k' is the reaction rate constant. This formula shows us that the half-life is inversely proportional to the reaction rate constant; a higher 'k' results in a shorter half-life, and vice versa. Understanding half-life is not just important in chemistry but also in fields such as pharmacology and nuclear science, where it helps inform decisions on dosing and safety.