Problem 58 - Chemistry: An Atoms-Focused Approach

Question

Two reactions in which there is a single reactant have nearly the same magnitude rate constant. One is first order; the other is second order. a. If the initial concentrations of the reactants are both \(1.0 \mathrm{mM},\) which reaction will proceed at the higher rate? b. If the initial concentrations of the reactants are both 2.0 \(M,\) which reaction will proceed at the higher rate?

Step-by-Step Solution

Answer

Based on the analysis and solution, when the initial concentration of reactants is 1.0 mM, both the first-order and second-order reactions have the same rate. However, when the initial concentration increases to 2.0 M, the second-order reaction has a higher rate compared to the first-order reaction.

Step 1: Write general rate laws for first-order and second-order reactions

In this step, we will write the general rate law equations for a first-order and a second-order reaction. For a first-order reaction: \(Rate = k[A]\) For a second-order reaction: \(Rate = k[A]^2\) Now, we will start applying these equations for the given initial concentrations. #a.

Step 2: Calculate rates for both reactions with initial concentration 1.0 mM

Now we will plug the given initial concentrations into the equations of step 1. For the first-order reaction: \(Rate_1 = k[1.0 \mathrm{mM}]\) \(Rate_1 = (1.0 \mathrm{mM})k\) For the second-order reaction: \(Rate_2 = k[1.0 \mathrm{mM}]^2\) \(Rate_2 = (1.0 \mathrm{mM})^2k = (1.0^2 \mathrm{mM^2})k\) Since the rates have nearly the same magnitude rate constant \((k)\), we can compare these two rates directly.

Step 3: Compare the rates for initial concentration 1.0 mM

Comparing the two rates: \(Rate_1 = 1.0k\) \(Rate_2 = 1.0k\) In this case, both reactions have the same rate when initial concentration of the reactants is 1.0 mM. #b.

Step 4: Calculate rates for both reactions with initial concentration 2.0 M

Now we will plug the given initial concentrations into the equations of step 1 for this part of the problem. For the first-order reaction: \(Rate_1 = k[2.0 \mathrm{M}]\) \(Rate_1 = (2.0 \mathrm{M})k\) For the second-order reaction: \(Rate_2 = k[2.0 \mathrm{M}]^2\) \(Rate_2 = (2.0 \mathrm{M})^2k = (4.0 \mathrm{M^2})k\)

Step 5: Compare the rates for initial concentration 2.0 M

Comparing the two rates: \(Rate_1 = 2.0k\) \(Rate_2 = 4.0k\) In this case, the second-order reaction has a higher rate when the initial concentration of the reactants is 2.0 M. #Conclusion# As a result, we have found that when the initial concentration of reactants is 1.0 mM, both reactions have the same rate. However, when the initial concentration is 2.0 M, the second-order reaction has a higher rate than the first-order reaction.