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The integrated rate law for first-order reactions is an essential tool in determining the concentration of a reactant at any given time. For a first-order reaction, the rate at which the reaction proceeds depends directly on the concentration of a single reactant. This is represented by the equation:

• \( \text{ln} \frac{[A]_0}{[A]} = kt \)

where:

• \([A]_0\) is the initial concentration of the reactant.
• \([A]\) is the concentration of the reactant at time \(t\).
• \(k\) is the rate constant, which is specific to each reaction.
• \(t\) is the elapsed time.

This equation allows chemists to calculate either the remaining concentration of a reactant or the time taken for a certain amount of reactant to be consumed. It provides a logarithmic relationship between time and concentration, reflecting how the concentration decreases over time.
Understanding the integrated rate law is fundamental to grasping how reactions evolve, especially in scenarios where exact timing and concentration are crucial, such as in industrial processes.

The rate constant, \(k\), is a crucial factor in quantifying the speed of a reaction. For first-order reactions, this constant helps us understand how quickly a reaction progresses at a specific temperature. It remains unchanged as long as the temperature is constant. The units of \(k\) in a first-order reaction are typically \(s^{-1}\), emphasizing that it is a time-based measure. Determining \(k\) involves experimental data:

• Input the initial and remaining concentrations of the reactant.
• Use time taken for these changes into the integrated rate law formula.

For example, consider in our exercise where \([A]_0 = 0.00100 \, M\) and \([A] = 0.00067 \, M\) after \(t = 155 \, s\). By substituting these into the integrated rate law, we find \(k\) as \(0.00259 \, s^{-1}\).
Acknowledging the rate constant's role allows chemists to compare and predict behavior across different reaction conditions, making it a pivotal component in chemical kinetics.

A decomposition reaction involves the breakdown of a single compound into two or more simpler substances. These processes are often endothermic, requiring heat to proceed, as observed with ethyl chloride decomposing into ethylene and hydrogen chloride:

• \( \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}(g) \rightarrow \mathrm{C}_{2} \mathrm{H}_{4}(g) + \mathrm{HCl}(g) \)

This type of reaction is significant in industrial applications and chemical syntheses. The study of decomposition reactions, especially those exhibiting a first-order behavior, helps us predict how quickly a substance will break down under certain conditions. By knowing the rate constant and employing the integrated rate law, chemists can design processes to control the timing and yield of desired products in decomposition. For instance, in our example, by heating ethyl chloride at \(500^\circ C\), we initiate its decomposition, which can be tracked over time to ensure efficient transition to the products.