Question

(a) The gas-phase decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}, \mathrm{SO}_{2} \mathrm{Cl}_{2}(g)\) \(\longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g),\) is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2} .\) At 600 \(\mathrm{K}\) the half-life for this process is \(2.3 \times 10^{5} \mathrm{s}\) . What is the rate constant at this temperature? (b) At 320 "C the rate constant is \(2.2 \times 10^{-5} \mathrm{s}^{-1} .\) What is the half-life at this temperature?

Short answer

(a) The rate constant at 600K is \(3.01 × 10^{-6} s^{-1}\). (b) The half-life at 320°C is \(3.15 × 10^4 s\).

Step-by-step solution

(a) Calculate the rate constant at 600K using the given half-life

As mentioned earlier, we will use the first-order reaction formula \(t_{1/2} = \frac{\ln{2}}{k}\). Rearranging it to solve for k: \(k = \frac{\ln{2}}{t_{1/2}}\) Given half-life, \(t_{1/2} = 2.3 × 10^5 s\) Calculating k, \(k = \frac{\ln{2}}{2.3 × 10^5 s}\) \(k = 3.01 × 10^{-6} s^{-1}\) So, the rate constant at 600K is \(3.01 × 10^{-6} s^{-1}\).

(b) Calculate the half-life at 320°C using the given rate constant

The given rate constant, \(k = 2.2 × 10^{-5} s^{-1}\). Now, we will use the first-order reaction formula \(t_{1/2} = \frac{\ln{2}}{k}\) to calculate half-life. \( t_{1/2} = \frac{\ln{2}}{2.2 × 10^{-5} s^{-1}} \) \( t_{1/2} = 3.15 × 10^4 s \) So, the half-life at 320°C is \(3.15 × 10^4 s\).

Key concepts

Understanding First-Order Reactions

A first-order reaction is a type of chemical reaction where the rate at which it occurs is directly proportional to the concentration of one of the reactants. In simpler terms, if you were to double the amount of this reactant, the reaction rate would also double.

In mathematical terms, a first-order reaction can be described by the equation: \[ \text{Rate} = k[\text{Reactant}] \]where \( k \) is the rate constant, and \( [\text{Reactant}] \) represents the concentration of the reactant. This kind of reaction has distinctive characteristics, such as a constant half-life regardless of the initial concentration, which makes them especially interesting in fields like pharmacokinetics and radioactive decay.

Rate Constant Calculation

The rate constant, symbolized by \( k \), is a crucial parameter in chemical kinetics as it dictates the speed of the chemical reaction. For first-order reactions, it is related to the half-life of the reaction, which is the time required for the concentration of a reactant to drop to half its initial value.

The formula connecting the half-life (\( t_{1/2} \)) and the rate constant for a first-order reaction is:\[ t_{1/2} = \frac{\ln{2}}{k} \]By rearranging this formula, we can solve for the rate constant if we know the half-life:\[ k = \frac{\ln{2}}{t_{1/2}} \]For instance, given a certain half-life, we can calculate the rate constant, which is essential to predict how fast the reaction proceeds under various conditions. Understanding this relationship is foundational for students studying reaction kinetics.

The Concept of Half-Life in Chemical Kinetics

The half-life of a chemical reaction is a term that describes the duration needed for half of the reactant to undergo the reaction. In the context of a first-order reaction, the half-life is independent of the initial concentration of the reactant, making it a valuable tool for scientists to understand the reaction's behavior over time.

The half-life is inversely proportional to the rate constant, as seen in the equation:\[ t_{1/2} = \frac{\ln{2}}{k} \]This means that a greater rate constant results in a shorter half-life, implying a faster reaction. Conversely, a smaller rate constant indicates a slower reaction with a longer half-life. By measuring the half-life, scientists can deduce the rate constant and vice versa, facilitating the prediction of how long it will take for a reaction to reach a specific stage or to complete.