(a) For a generic second-order reaction \(\mathrm{A} \longrightarrow \mathrm{B},\) what quantity, when graphed versus time, will yield a straight line? (b) What is the slope of the straight line from part (a)? (c) How do the half-lives of first-order and second-order reactions differ?

We start by writing the rate equation for a second-order reaction: \(rate = k \times [A]^2\) Where \(k\) is the rate constant, and \([A]\) represents the concentration of reactant A. We are asked to find a quantity that will yield a straight line when graphed against time. To do this, we can convert the rate equation into an integrated rate law. For a second-order reaction, the integrated rate law is: \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\) Where \([A]_t\) is the concentration of A at a specific time, \([A]_0\) is the initial concentration of A, and \(t\) is time. Here, if we plot \(\frac{1}{[A]_t}\) against time (\(t\)), we will get a straight line.

To find the slope of the straight line from part (a), we can rewrite the integrated rate law in the form of a linear equation: \(\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}\) This equation is in the form of \(y = mx + c\), where \(y = \frac{1}{[A]_t}\), \(x = t\), \(m = k\), and \(c = \frac{1}{[A]_0}\). The slope of the straight line, or the coefficient of \(t\), is the rate constant, \(k\).

The half-life, \(t_{1/2}\), is the amount of time it takes for the concentration of the reactant to reduce to half its initial value. The half-life of a reaction depends on the order of the reaction. For a first-order reaction, the half-life is given by: \(t_{1/2}^{(1)} = \frac{0.693}{k}\) Here, we can see that the half-life is constant and does not depend on the initial concentration. For a second-order reaction, the half-life is given by: \(t_{1/2}^{(2)} = \frac{1}{k[A]_0}\) In this case, the half-life depends on the initial concentration of the reactant. As the initial concentration increases, the half-life becomes smaller. In summary, the half-life of a first-order reaction remains constant, independent of the reactant concentration, while the half-life of a second-order reaction depends on the initial concentration of the reactant.