Problem 41 - Chemistry The Central Science In Si Units

Question

(a) The gas-phase decomposition of sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right), \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)\) is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), At \(300^{\circ} \mathrm{C}\) the half-life for this process is two and a half days. What is the rate constant at this temperature?

Step-by-Step Solution

Answer

The rate constant \( k \) at 300°C is approximately \( 3.21 \times 10^{-6} \text{ s}^{-1} \).

Step 1: Identify the Half-Life Equation for First Order Reactions

The half-life equation for a first order reaction is given by:\[ t_{1/2} = \frac{0.693}{k} \]where \( t_{1/2} \) is the half-life of the reaction and \( k \) is the rate constant.

Step 2: Convert Half-Life to Consistent Units

We need to convert the half-life from days to seconds for consistency with rate constant units. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, so we have:\[ t_{1/2} = 2.5 \text{ days} = 2.5 \times 24 \times 60 \times 60 \text{ seconds} \]

Step 3: Calculate the Number of Seconds

Calculating the total number of seconds in 2.5 days:\[ t_{1/2} = 2.5 \times 24 \times 60 \times 60 = 216,000 \text{ seconds} \]

Step 4: Solve for Rate Constant k

Rearrange the half-life equation to solve for \( k \):\[ k = \frac{0.693}{t_{1/2}} \]Substitute \( t_{1/2} = 216,000 \text{ seconds} \) into the equation:\[ k = \frac{0.693}{216,000} \]

Step 5: Calculate the Rate Constant

Compute the value of \( k \):\[ k = \frac{0.693}{216,000} \approx 3.21 \times 10^{-6} \text{ s}^{-1} \]

Key Concepts

Rate Constant Calculation

In the realm of chemical kinetics, calculating the rate constant, especially for a first-order reaction, is crucial for understanding how fast a reaction proceeds. The rate constant, often denoted as \( k \), is a proportionality factor in the rate equation. For a first-order reaction, the rate is directly proportional to the concentration of one reactant, making it simpler to determine.

The rate constant \( k \) can be determined using the half-life of a reaction—this is the time required for the concentration of a reactant to reduce to half its initial value. The specific relationship for first-order reactions is:
\[ k = \frac{0.693}{t_{1/2}} \]This formula shows that the rate constant inversely depends on the half-life; thus, a shorter half-life results in a larger rate constant, indicating a faster reaction.

In the given sulfuryl chloride decomposition, knowing the half-life helps us calculate \( k \), which sheds light on the reaction speed at a specified temperature, such as 300 degrees Celsius. Recognizing this relationship helps in predicting the behavior of chemical reactions under different conditions.

Half-Life Equation

The half-life equation is pivotal in kinetic studies and differs across reaction orders. For first-order reactions, it's unique because the half-life, \( t_{1/2} \), is constant and independent of the initial concentration of reactants. This constancy simplifies calculations, allowing predictions about reaction lifespan solely based on time.

The standard form of the equation for first-order reactions is:
\[ t_{1/2} = \frac{0.693}{k} \]This expression is derived from integration of the rate law, and it directly links the half-life to the rate constant \( k \). The value 0.693 is the natural logarithm of two (\( \ln(2) \)), occurring because we are continually halving the quantity of reactant.

This foundational equation is vital for deterministic calculations in both laboratories and industry applications, allowing chemists to foresee how quickly a reactant will diminish halfway under set conditions. In our exercise, converting days into seconds ensures that units remain consistent when calculating the rate constant, reinforcing the scientific accuracy of the process.

Sulfuryl Chloride Decomposition

Sulfuryl chloride, \( \text{SO}_2\text{Cl}_2 \), decomposes into sulfur dioxide \( \text{SO}_2 \) and chlorine gas \( \text{Cl}_2 \), making it a classic study in inorganic chemistry. Understanding its decomposition is essential, not just for theoretical chemistry, but also for industrial applications where control over gas emissions or reactant concentrations is crucial.

The decomposition follows a first-order kinetics pattern, meaning it solely depends on the concentration of \( \text{SO}_2\text{Cl}_2 \). As the temperature increases, so does the rate of decomposition, which is why the half-life and rate constants play such critical roles in measuring this reaction objectively.

Knowing the precise kinetics allows for better management of reaction conditions in chemical manufacturing or material processing, where sulfuryl chloride might serve as an intermediary. By applying the rate constant and half-life formulas, scientists can anticipate how changes in temperature, pressure, or other environmental factors could accelerate or decelerate the rate of decomposition, reaching desired products effectively.