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 \(320^{\circ} \mathrm{C}\) 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 600 K is approximately \(4.35 \times 10^{-6} s^{-1}\). (b) The half-life at 320 degrees Celsius (or 593.15 K) is approximately \(4.55 \times 10^4 s\).

Step-by-step solution

Write down the given half-life at 600 K

The given half-life, \(t_{1/2}\), at 600 K is \(2.3 \times 10^5 s\).

Use the first-order half-life equation to find the rate constant "k"

Now, we will use the formula for the half-life of a first-order reaction to find the rate constant: \(k = \dfrac{1}{t_{1/2}}\) Substitute the given half-life: \(k = \dfrac{1}{(2.3 \times 10^5 s)}\)

Calculate the rate constant "k"

Now, we can calculate the rate constant: \(k \approx 4.35 \times 10^{-6} s^{-1}\) The rate constant at 600 K is approximately \(4.35 \times 10^{-6} s^{-1}\). #b) Find the half-life at 320 degrees Celsius#

Convert the temperature to Kelvin

To find the half-life at 320 degrees Celsius, we first need to convert this temperature to Kelvin: \(320^{\circ}C + 273.15 = 593.15 K\) Now, we know that at 593.15 K, the rate constant, \(k\), is \(2.2 \times 10^{-5} s^{-1}\).

Use the first-order half-life equation to find the half-life at 593.15 K

Use the formula for the half-life of a first-order reaction: \(t_{1/2} = \dfrac{1}{k}\) Substitute the given rate constant: \(t_{1/2} = \dfrac{1}{(2.2 \times 10^{-5} s^{-1})}\)

Calculate the half-life at 593.15 K

Now, we can calculate the half-life: \(t_{1/2} \approx 4.55 \times 10^4 s\) The half-life at 320 degrees Celsius (or 593.15 K) is approximately \(4.55 \times 10^4 s\).

Key concepts

First-Order Reaction

Understanding a first-order reaction in chemical kinetics is crucial for students studying the speed of chemical processes. This type of reaction is characterized by a rate that is directly proportional to the concentration of one reactant. In simpler terms, as the reactant concentration decreases, the rate of reaction decreases at a proportional rate.

One unique aspect of a first-order reaction is the consistent time it takes for the concentration of the reactant to reduce by half, no matter what quantity you start with. This is why half-life—an important term discussed later—is a key concept in first-order kinetics. Knowing just the reaction's half-life or rate constant allows you to determine the other through the relationship \( k = \frac{1}{t_{1/2}} \).

This is immediately valuable for exercises where students might only be given one of these pieces of information and need to determine the other, as demonstrated in our textbook example. The example also points to the idea that the same reaction will have different rate constants and half-lives at different temperatures, hinting at the effect of temperature on chemical reactions.

Rate Constant

The rate constant, often symbolized as \( k \), is a coefficient that measures the speed of a chemical reaction. For first-order reactions, the rate constant provides insight into how quickly a reactant is transformed into products per unit time. It is crucial because, once determined, it can be used to predict the behavior of a reaction over time.

To calculate the rate constant from a known half-life, we use the formula \( k = \frac{1}{t_{1/2}} \). The units of \( k \) for first-order reactions are always \( s^{-1} \) as it reflects the frequency of reaction events per second. For example, as we saw in the solution, with a half-life of \(2.3 \times 10^5 s\), the rate constant is calculated to be approximately \(4.35 \times 10^{-6} s^{-1}\).

Understanding the rate constant's role is paramount, not just for solving textbook problems, but for real-world applications where predicting reaction times can be crucial, such as in pharmaceuticals, food preservation, and many industries.

Half-Life

Half-life, denoted by \( t_{1/2} \), is a term widely used across various disciplines, including physics and chemistry. In the context of chemical kinetics and first-order reactions, half-life represents the time required for the concentration of a reactant to reach half of its initial value. It is a constant value for first-order reactions, meaning it remains the same at any concentration level of the reactant.

For instance, if the initial concentration of a substance is halved in 100 seconds, after another 100 seconds, the remaining concentration will once again halve. As a result, half-life is an extremely useful value when we need to understand timing for processes like drug clearance from the body, radioactive decay, or any chemical degradation. Referring back to our textbook example, when the half-life was known \(2.3 \times 10^5 s\), we used it to find the rate constant. Conversely, knowing the rate constant would allow us to calculate the half-life, essential for predicting how long a reaction will take to reach a certain stage.

Temperature Conversion

Temperature conversion is a routinely used process in the study of chemical kinetics, as reactions are highly sensitive to temperature variations. It's common to describe temperatures in degrees Celsius (°C) in daily life, but scientific calculations require temperature in Kelvin (K).

The Kelvin scale is an absolute temperature scale, starting at absolute zero, the theoretical lowest possible temperature. To convert from Celsius to Kelvin, you add \(273.15\) to the Celsius temperature. This conversion is crucial because rate constants and reaction rates are usually given or calculated at a temperature in Kelvin.

For instance, in the example provided, we converted \(320°C\) to Kelvin by adding \(273.15\), yielding \(593.15 K\). This step is necessary to accurately calculate reaction rates or half-lives at different temperatures and to use these values in formulas that are temperature-dependent. This seemingly simple act of conversion underlies the reproducibility of experiments and consistency in interpreting results, no matter where or by whom they are conducted.