Question

\(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) decomposes in a first-order process with a half life of \(4.88 \times 10^{3} \mathrm{~s}\). If the original concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(0.012 \mathrm{M}\), how many seconds will it take for the \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to reach \(0.0020 \mathrm{M}\) ?

Short answer

The reaction will take \( 1.95 \times 10^{4} \) seconds for the concentration of \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \) to reach \( 0.0020 \) M.

Step-by-step solution

Determine the First-Order Rate Constant

For a first-order reaction, the rate constant (k) can be determined from the half-life using the equation: \( k = \frac{0.693}{t_{1/2}} \) where \( t_{1/2} \) is the half-life. Plugging in the half-life of \( 4.88 \times 10^{3} \) s, we calculate the rate constant.

Use the First-Order Integrated Rate Law

The integrated rate law for a first-order reaction relates the concentration of reactant at any time t to the initial concentration and the rate constant k: \( \ln\left(\frac{[A]_0}{[A]_t}\right) = kt \). We need to solve for t when the initial concentration \( [A]_0 = 0.012 \) M and the final concentration \( [A]_t = 0.0020 \) M.

Solve for Time (t)

Rearrange the first-order integrated rate law to solve for time (t): \( t = \frac{1}{k} \ln\left(\frac{[A]_0}{[A]_t}\right) \). Substitute in the values of \( [A]_0 \), \( [A]_t \), and the rate constant k to calculate the time taken for the concentration of \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \) to reach \( 0.0020 \) M.

Key concepts

Rate Constant Calculation for First-Order Reactions

Understanding the rate constant is crucial when studying the kinetics of a first-order reaction. In simple terms, the rate constant, denoted as 'k', gives us an idea of how quickly the reaction proceeds. For first-order reactions, where the rate of reaction is directly proportional to the concentration of one reactant, calculating the rate constant can be easily done using the relationship with the half-life of the reaction.

As the step-by-step solution shows, the formula to find 'k' from the half-life (\( t_{1/2} \)) is given by: \[ k = \frac{0.693}{t_{1/2}} \.
\] The constant 0.693 is actually the natural logarithm of 2, which comes into play because the half-life is the time it takes for the concentration of the reactant to decrease by half. The smaller the value of the half-life, the larger the rate constant, indicating a faster reaction. Conversely, a longer half-life suggests a slower reaction and a smaller rate constant.

In practice, once you've been given a half-life, all you need is this formula to calculate the rate constant, enabling you to analyze further details about the reaction's kinetics.

Using the Integrated Rate Law for First-Order Reactions

The first-order integrated rate law is a mathematical expression that tells us how the concentration of a reactant decreases over time. For a reaction where a single reactant A breaks down, the integrated rate law is represented as: \[ \ln\left(\frac{[A]_0}{[A]_t}\right) = kt \.
\] In this equation, \([A]_0\) is the initial concentration, \([A]_t\) is the concentration at time 't', and 'k' is the rate constant we previously discussed.

To use this law, you take the natural logarithm of the ratio of the initial concentration to the concentration at a particular time. This value equals the product of the rate constant and time. It essentially shows the exponential decay of the reactant concentration in a first-order reaction.

By rearranging the integrated rate law, one can solve for various unknowns, such as the concentration at a specific time or the time required to reach a certain concentration from the initial value. This versatility makes the integrated rate law a powerful tool in reaction kinetics analysis.

Understanding the Half-Life of a Reaction

The half-life of a reaction, commonly abbreviated as \( t_{1/2} \), is a concept that often intrigues students. It refers to the time required for half of the reactant to be used up or converted in a chemical reaction. In the context of a first-order reaction, the half-life is particularly significant because it is independent of the initial concentration.

For first-order reactions, the half-life can be calculated if the rate constant 'k' is known, using the equation: \[ t_{1/2} = \frac{0.693}{k} \.
\] The unique aspect of first-order reactions is that their half-life remains constant throughout the reaction, regardless of the concentration of the reactant. This distinctive property allows chemists and students to predict how long it will take for a certain amount of reactant to decompose and can be crucial when considering reactions in pharmaceuticals, environmental science, and many other fields.

Moreover, understanding half-life provides insights into the stability of compounds and can help in determining the conditions under which a reaction should be carried out for optimum efficiency. It is a straightforward yet powerful concept that, when grasped, greatly enhances one's comprehension of chemical kinetics.