Which of the following linear plots do you expect for a reaction \(A \longrightarrow\) products if the kinetics are (a) zero order, (b) first order, or (c) second order? [Section 14.4\(]\)

First, let's review the mathematical expressions for each type of kinetics. 1. Zero-order kinetics: The reaction rate is constant. The rate equation is: \[r = k[A]^0 = k\] 2. First-order kinetics: The reaction rate is proportional to the concentration of the single reactant A. The rate equation is: \[r = k[A]\] 3. Second-order kinetics: The reaction rate is proportional to the square of the concentration of the single reactant A. The rate equation is: \[r = k[A]^2\] Now that we have the rate equations for each type of kinetics, we can proceed to determine their linear plots.

We will now find the linear plots for each type of kinetics by integrating the corresponding rate equations and looking for linear relationships. 1. Zero-order kinetics: For a zero-order reaction, after integrating the rate equation, we get: \[[A] = -kt + [A]_0\] This equation represents a linear plot; the concentration [A] decreases linearly with time t. Thus, a graph of [A] vs. t gives a straight line with a negative slope equal to -k. 2. First-order kinetics: For a first-order reaction, after integrating the rate equation, we get: \[ln[A] = -kt + ln[A]_0\] In this case, a linear plot can be achieved by plotting the natural logarithm of the concentration ln[A] vs. time t. This plot gives a straight line with a negative slope equal to -k. 3. Second-order kinetics: For a second-order reaction, after integrating the rate equation, we get: \[\frac{1}{[A]} = kt + \frac{1}{[A]_0}\] Here, a linear plot can be obtained by plotting the reciprocal of the concentration 1/[A] vs. time t. This plot results in a straight line with a positive slope equal to k. In summary, for reactions with (a) zero-order, (b) first-order, and (c) second-order kinetics, the expected linear plots are: a) [A] vs. t, with a straight line and a negative slope equal to -k b) ln[A] vs. t, with a straight line and a negative slope equal to -k c) 1/[A] vs. t, with a straight line and a positive slope equal to k