Based on their activation energies and energy changes and assuming that all collision factors are the same, rank the following reactions from slowest to fastest. (a) $E_{a}=75 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=-20 \mathrm{~kJ} / \mathrm{mol}$ (b) $E_{a}=100 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=+30 \mathrm{~kJ} / \mathrm{mol}$ (c) $E_{a}=85 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=-50 \mathrm{~kJ} / \mathrm{mol}$

To rank the reactions from slowest to fastest, we can first look at their activation energies. The reaction with the highest activation energy is expected to be the slowest because it requires more energy for the reaction to proceed. Recall that the activation energies for the three reactions are: (a) \(E_{a}=75 \mathrm{~kJ} / \mathrm{mol}\) (b) \(E_{a}=100 \mathrm{~kJ} / \mathrm{mol}\) (c) \(E_{a}=85 \mathrm{~kJ} / \mathrm{mol}\) From these values, we can see that reaction (b) has the highest activation energy, followed by reaction (c) and then reaction (a).

Now let's consider the energy changes for each reaction. The energy change indicates whether the reaction is endothermic (positive \(\Delta E\)) or exothermic (negative \(\Delta E\)). In general, exothermic reactions tend to be faster than endothermic reactions, as they release energy and promote more particle collisions. The energy changes for the three reactions are: (a) \(\Delta E=-20 \mathrm{~kJ} / \mathrm{mol}\) (b) \(\Delta E=+30 \mathrm{~kJ} / \mathrm{mol}\) (c) \(\Delta E=-50 \mathrm{~kJ} / \mathrm{mol}\) From these values, we can see that reaction (a) and (c) are exothermic while reaction (b) is endothermic. Since exothermic reactions are generally faster, we can tentatively rank the reactions as: (c) > (a) > (b).

Finally, we can combine the rankings from the activation energies and energy changes to determine the overall ranking from slowest to fastest. Taking into account the activation energy ranking from Step 1 and the energy change ranking from Step 2, we have: Slowest to Fastest: (b) > (c) > (a) Therefore, the ranking of the reactions from slowest to fastest is: (b), (c), (a).