Problem 70 - Chemistry: An Atoms-Focused Approach

Question

Because the units of concentration in the term \(\ln \left([\mathrm{X}] /[\mathrm{X}]_{0}\right)\) cancel out in the integrated rate law for first- order reactions (Equation 13.16 ), molar concentration can be replaced by any concentration term. With gases, for example, partial pressures may be used. The decomposition of phosphine gas \(\left(\mathrm{PH}_{3}\right)\) at \(600^{\circ} \mathrm{C}\) is first order in \(\mathrm{PH}_{3}\) with \(k=0.023 \mathrm{s}^{-1}\) $$4 \mathrm{PH}_{3}(g) \rightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g)$$ If the initial partial pressure of \(\mathrm{PH}_{3}\) is 375 torr, what percent of \(\mathrm{PH}_{3}\) reacts in \(1 \mathrm{min}\) ?

Step-by-Step Solution

Answer

The percentage of PH3 that reacts in 1 minute is approximately 64.86%.

Step 1: Write the integrated rate law for first-order reactions using partial pressure

The integrated rate law for first-order reactions is given by: $$ \ln \frac{[\mathrm{A}]}{[\mathrm{A}]_{0}} = -kt $$ Since we can replace molar concentration with partial pressures, we can rewrite this as: $$ \ln \frac{P_\mathrm{PH_3}}{P_\mathrm{PH_3(0)}} = -kt $$

Step 2: Insert the given values into the equation

We are given the initial partial pressure \(P_\mathrm{PH_3(0)} = 375 \ \text{torr}\), the rate constant \(k=0.023 \ \mathrm{s}^{-1}\), and the time \(t=1 \ \text{min}\) (we need to convert this to seconds: \(t=60 \ \mathrm{s}\)). Now, we can plug these values into the equation: $$ \ln \frac{P_\mathrm{PH_3}}{375 \ \text{torr}} = -(0.023 \ \mathrm{s}^{-1})(60 \ \mathrm{s}) $$

Step 3: Solve for \(P_\mathrm{PH_3}\)

Now we need to solve the equation for \(P_\mathrm{PH_3}\), which is the partial pressure of PH3 after 1 minute: $$ \ln \frac{P_\mathrm{PH_3}}{375 \ \text{torr}} = -1.380 $$ $$ \frac{P_\mathrm{PH_3}}{375 \ \text{torr}} = e^{-1.380} $$ $$ P_\mathrm{PH_3} = 375 \ \text{torr} \cdot e^{-1.380} $$ $$ P_\mathrm{PH_3} \approx 131.80 \ \text{torr} $$

Step 4: Calculate the percentage of PH3 reacted

To find the percentage of PH3 reacted, we need to determine the difference in the initial and final partial pressures and then divide by the initial partial pressure: $$ \text{Percentage of PH3 reacted} = \frac{375 \ \text{torr} - 131.80 \ \text{torr}}{375 \ \text{torr}} \times 100 $$ $$ \text{Percentage of PH3 reacted} \approx 64.86\% $$ So, approximately 64.86% of PH3 reacts in 1 minute.

Key Concepts

Integrated Rate Law

In the study of chemical kinetics, the integrated rate law is a pivotal formula used to describe the relationship between the concentration of a reactant and time. For first-order reactions, the focus is on how the concentration of a reactant decreases over time due to a specific reaction. This kind of reaction assumes that the rate is directly proportional to the concentration of one reactant.

The integrated rate law for a first-order reaction is expressed as: \[ \ln \frac{[\text{A}]}{[\text{A}]_0} = -kt \]where:

• \( [\text{A}] \) is the concentration of the reactant at time \( t \).
• \( [\text{A}]_0 \) is the initial concentration of the reactant.
• \( k \) is the rate constant.
• \( t \) is the time elapsed.

This equation highlights how the natural log of a reactant's concentration ratio equals the product of its rate constant and time.

A unique feature of the first-order integrated rate law is that the concentration unit cancels out, meaning it can equally apply to gases using their partial pressures instead of concentrations. This flexibility makes it ideal for studying gaseous reactions, enabling scientists to track changes in partial pressures over time.

Partial Pressure

In chemistry, especially when dealing with gases, partial pressure is an essential concept. It refers to the pressure that a gas would exert if it occupied the entire volume alone at the same temperature. Partial pressures are key in predicting how gases will react under various conditions.

For instance, in the decomposition of phosphine gas (PH3), the use of partial pressures instead of concentrations allows for the employment of the integrated rate law to track the reaction's progress. Phosphine decomposes into phosphorus and hydrogen using the first-order reaction kinetics, where the rate depends on the partial pressure of PH3.

• Partial pressures provide a direct method of relating the observed reaction progress with measurable quantities: the pressure of the gas components.
• They offer a tangible and straightforward parameter for calculations, especially in closed systems like gas-filled reaction vessels.

This conversion is crucial because it helps bridge theoretical equations with real-world experimental data, making predictions about reaction behavior more accessible and accurate.

Chemical Kinetics

Chemical kinetics is the field of chemistry that examines the speed or rate of a chemical reaction and the factors affecting these rates. It explores how different conditions, like concentration, temperature, and presence of catalysts, can drastically alter how quickly reactions proceed.

Key points about chemical kinetics include:

• Understanding how and why reactions occur at specific rates.
• Studying the effects of variables such as temperature and concentration on reaction speed.
• Using mathematical relationships, like the integrated rate law, to model and predict reaction behavior over time.

By calculating reaction rates and leveraging equations like the integrated rate law, chemists can predict how a reaction proceeds over time. For example, predicting the percentage of a reactant that reacts within a certain time, as seen with the phosphine decomposition problem. Here, kinetics gives us insights into the amount of phosphine decomposed within 60 seconds by using the rate constant and initial partial pressure. This foundational knowledge is critical for fields like pharmaceuticals, where reaction rates can dictate drug formulation and stability. Overall, understanding chemical kinetics not only aids in academic research but also enhances practical applications across numerous industries.