The reaction of compound \(A\) to give compounds \(C\) and \(D\) was found to be second-order in \(A\). The rate constant for the reaction was determined to be \(2.42 \mathrm{L} \mathrm{mol}^{-1} \mathrm{s}^{-1} .\) If the initial concentration is \(0.500 \mathrm{mol} / \mathrm{L},\) what is the value of \(\mathrm{t}_{1 / 2} ?\)

The second-order integrated rate law is pivotal in the realm of chemical kinetics, particularly when dealing with reactions where the rate is dependent on the concentration of one reactant raised to the second power, or two reactants each raised to the first power. In essence, the law provides a relationship between the concentration of the reactants and time.

For a second-order reaction with one reactant, the rate law can be written as \(\frac{d[A]}{dt} = -k[A]^2\), where \(k\) is the rate constant, \(t\) represents time, and \(A\) the concentration of the reactant. Integrating this equation over time gives us the second-order integrated rate law: \(\frac{1}{[A]} - \frac{1}{[A]_0} = kt\), where \( [A]_0 \) is the initial concentration of the reactant. Applying this law is key to solving problems related to the rate of a reaction, as it allows us to track how concentration changes over time. Thus, understanding its derivation and application can markedly enhance a student's ability to solve complex kinematic problems.

In the context of our particular exercise, recognizing that the reaction is second-order allows us to use a simplified formula specifically for calculating the half-life of such reactions, which is directly proportional to the inverse of the product of the rate constant and initial concentration.

The rate constant, symbolized as \(k\), is a quintessential parameter in the kinetics of chemical reactions. It is a measure of the speed at which a reaction proceeds and is specific to the reaction at a given temperature. Mathematically, for a second-order reaction, it has the units of \(\mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}\), which reflects the concentration and time dimensions involved.

In practice, \(k\) is determined experimentally and provides crucial insight into reaction dynamics. The higher the value of \(k\), the faster the reaction takes place. Being aware of \(k\)'s magnitude enables chemists to anticipate and control the rate of chemical processes, which is fundamental in both industrial applications and academic research.

For our exercise, the given rate constant of \(2.42 \mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}\) indicates a moderately fast reaction. This value is essential for computing the half-life of the reactant, serving as a pivotal element in the equation provided for second-order reactions.

The initial concentration, denoted by \( [A]_0 \), of a reactant in a chemical reaction is another foundational component in kinetics. It signifies the starting concentration of a reactant before any reaction has occurred. In the context of second-order kinetics, the initial concentration is especially significant as the half-life of the reaction is inversely proportional to \( [A]_0 \). This highlights an intriguing aspect of second-order reactions: the half-life varies with the initial concentration, unlike first-order reactions where the half-life is constant.

In analyzing our exercise, the initial concentration of \(0.500 \mathrm{mol}\,\mathrm{L}^{-1}\) signifies the quantity of reactant present at the outset. This value is directly used in the half-life equation to solve for \( t_{1/2} \). A deep understanding of how initial concentration affects the reaction time course is imperative for students and professionals when manipulating or predicting reaction outcomes.

Overall, each part of the equation, including the rate constant and initial concentration, plays a vital role in determining the half-life for a second-order reaction, giving us a holistic view of the system’s kinetics.