Problem 64 - Chemistry and Chemical Reactivity

Question

The decomposition of phosphine, \(\mathrm{PH}_{3},\) proceeds according to the equation $$\mathrm{PH}_{3}(\mathrm{g}) \rightarrow 1 / 4 \mathrm{P}_{4}(\mathrm{g})+3 / 2 \mathrm{H}_{2}(\mathrm{g})$$ It is found that the reaction has the following rate equation: Rate \(=k\left[\mathrm{PH}_{3}\right] .\) The half-life of \(\mathrm{PH}_{3}\) is 37.9 seconds at \(120^{\circ} \mathrm{C}.\) (a) How much time is required for three fourths of the \(\mathrm{PH}_{3}\) to decompose? (b) What fraction of the original sample of \(\mathrm{PH}_{3}\) remains after 1.00 minute?

Step-by-Step Solution

Answer

(a) 75.8 seconds, (b) \(\frac{1}{3}\) remains after 1 minute.

Step 1: Understand the Reaction Order

The rate equation given is of the form \( \text{Rate} = k[\text{PH}_3] \), which indicates a first-order reaction since the rate depends linearly on the concentration of \( \text{PH}_3 \). For first-order reactions, the rate constant \( k \) and the half-life \( t_{1/2} \) are related as \( t_{1/2} = \frac{0.693}{k} \).

Step 2: Use Half-Life to Find Rate Constant

Given the half-life \( t_{1/2} = 37.9 \) seconds, we calculate \( k \):\[ k = \frac{0.693}{37.9} \approx 0.0183 \text{ s}^{-1} \]

Step 3: Calculate Time for Three Fourths to Decompose

The integrated rate law for a first-order reaction is\[ \ln \left( \frac{[\text{PH}_3]_0}{[\text{PH}_3]} \right) = kt \]If three fourths of \( \text{PH}_3 \) is decomposed, \( [\text{PH}_3] = \frac{1}{4}[\text{PH}_3]_0 \). Substitute into the equation:\[ \ln \left( \frac{[\text{PH}_3]_0}{\frac{1}{4}[\text{PH}_3]_0} \right) = (0.0183 \text{ s}^{-1}) t \]\[ \ln(4) = 0.0183t \]\[ t = \frac{\ln(4)}{0.0183} \approx 75.8 \text{ seconds} \]

Step 4: Calculate Fraction Remaining after 1 Minute

To find the fraction of \( \text{PH}_3 \) remaining after 1 minute (60 seconds), we use:\[ \ln \left( \frac{[\text{PH}_3]_0}{[\text{PH}_3]} \right) = (0.0183 \text{ s}^{-1})\times 60 \]\[ \ln \left( \frac{[\text{PH}_3]_0}{[\text{PH}_3]} \right) = 1.098 \]\( e^{1.098} \approx \frac{[\text{PH}_3]_0}{[\text{PH}_3]} = 3 \ [\text{PH}_3] = \frac{[\text{PH}_3]_0}{3} \)Therefore, \( \frac{1}{3} \) of the original sample remains.

Key Concepts

First-Order Reaction

In chemical kinetics, a first-order reaction is one where the rate of reaction is directly proportional to the concentration of a single reactant. This means that the decay or transformation of the reactant follows an exponential pattern. First-order reactions are common in chemical processes, including radioactive decay and enzyme-catalyzed reactions.
The general rate equation for a first-order reaction is expressed as:

• Rate = k[A]

Here, 'Rate' signifies the speed at which the reaction progresses, 'k' is the reaction rate constant, and '[A]' is the concentration of the reactant 'A'.
First-order reactions are characterized by their simplicity as they do not involve interactions between multiple reactants, making their study and predictions easier than complex reactions.

Rate Equation

The rate equation is crucial in understanding how a chemical reaction progresses over time. It's an equation that links the rate of reaction to the concentrations of the reactants, each raised to some power, called the order of reaction. For a first-order reaction, the rate equation takes the form:

• Rate = k[A]

In this equation, *k* is the rate constant and is unique for every distinct reaction. It represents the speed at which a reaction occurs, showing how fast the concentration of a reactant decreases. For first-order reactions, the rate is directly proportional to the concentration of the reactant. This means a decrease in reactant concentration leads to a directly related decrease in reaction rate.
The ability of the rate equation to predict how quickly reactants convert into products is fundamental in fields like pharmacology for drug degradation and environmental science for pollutant decay.

Half-Life Calculation

Half-life is a concept often associated with radioactive decay, but it's equally important in the study of chemical reactions, particularly first-order reactions. The half-life (\(t_{1/2}\)) of a reaction is the time required for half of the reactant to be consumed or transformed.
In first-order reactions, the half-life is constant and does not depend on the initial concentration of the reactant. It's calculated with the formula:

• \(t_{1/2} = \frac{0.693}{k}\)

where *k* is the reaction rate constant. This constancy comes from the exponential nature of first-order decay, where the time to degrade a fixed percentage of reactants remains unchanged irrespective of their starting amounts.
Understanding half-life helps in predicting how long a substance will remain active or visible in a given environment, which is critical in areas like medicine absorption and nuclear waste management.

Reaction Rate Constant

The reaction rate constant, denoted as *k*, is a numerical value that helps us understand the speed of a chemical reaction. It is important for quantifying how quick or slow a reaction proceeds. In the context of a first-order reaction, the rate constant can be determined if you know the half-life of the reaction.
For a first-order reaction, the rate constant is obtained using the half-life formula:

• \(k = \frac{0.693}{t_{1/2}}\)

This constant is specific to each reaction and influences the rate at which reactant concentrations decrease. The larger the value of *k*, the faster the reaction proceeds, indicating a shorter duration for the reactant to be converted into product.
Knowing *k* provides insight into reaction dynamics and is pivotal for designing and controlling chemical processes, like reactors used in industry or labs for synthesizing materials.