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Chapter 14: Problem 41

(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? (b) At \(400^{\circ} \mathrm{C}\) the rate constant is \(0.19 \mathrm{~min}^{-1}\). What is the half-life at this temperature?

Short Answer

Expert verified
(a) The rate constant at \(300^{\circ} \mathrm{C}\) is \(0.277\thinspace\text{day}^{-1}\). (b) The half-life at \(400^{\circ} \mathrm{C}\) is approximately 3.65 minutes.

Step by step solution

01

Recall the formula for half-life of a first-order reaction

For a first-order reaction, the half-life (t₁/₂) and the rate constant (k) are related by the following formula: \[ t_{1/2} = \frac{0.693}{k}\]
02

Plug in the given half-life and solve for k

We're given the half-life as 2.5 days. Let's plug this value into the formula and solve for the rate constant (k): \[ 2.5 \thinspace \text{days} = \frac{0.693}{k} \] Now we can solve for k: \[ k = \frac{0.693}{2.5\thinspace \text{days}} \approx 0.277\thinspace\text{day}^{-1} \] So, the rate constant at 300°C is \(0.277\thinspace\text{day}^{-1}\). b) Finding the half-life at 400°C
03

Recall the formula for half-life of a first-order reaction

As previously mentioned, for a first-order reaction, the half-life (t₁/₂) and the rate constant (k) are related by the following formula: \[ t_{1/2} = \frac{0.693}{k} \]
04

Plug in the given rate constant and solve for half-life

We're given the rate constant as \(0.19\thinspace\text{min}^{-1}\) at 400°C. Let's plug this value into the formula and solve for the half-life (t₁/₂): \[ t_{1/2} = \frac{0.693}{0.19\thinspace\text{min}^{-1}} \approx 3.65\thinspace\text{min} \] So, the half-life at 400°C is approximately 3.65 minutes.

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Most popular questions from this chapter

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