PS Test 2, question 30

Hi Alex, could you help explain question 30 from the physical sciences
section test 2. I found that question to be really challenging and I am
not quite understanding the explanation.
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This question reads:  “If Step 2 above were the rate-determining step of Reaction 1, which of the following equations would correctly define the rate?”

Remember that the rate-limiting step can be related to the rate by simply taking each of the reactants from that step and raising them to their stoichiometric coefficients.  For this question, that means that rate = k₂[B][D].  There’s a problem here, though.  When we write rate laws, all of the terms must be reactants of the overall reaction, meaning that we’ll have to have our rate dependent on A and D (based on the reaction above the mechanism).  So having [B] in our answer is a problem.  How do we fix this?

The key is then looking at Step 1, where [B] is manufactured.  To find out how to rewrite our concentration of B, let’s use the fact that it tells us this reaction is “slow.”  In reaction kinetics, “slow” means that the reaction is at equilibrium.  If it’s at equilibrium, then the forward and reverse reactions of Step 1 have the same rate.  Mathematically, that means:

rate_f = rate_r

k₁[A] = k_-1[B][C]

The “-1” on the rate constant is indicating that it is the rate constant for the reverse reaction of Step 1.  This is a common convention in kinetics.  Thus, to find [B], we can rearrange:

[B] = k₁[A]/k_-1[C]

Plugging into our rate law from before, we get:

rate = k₂[B][D] = k₂(k₁[A]/k_-1[C])[D] or k₁k₂[A][D]/k_-1[C].  This matches with answer choice C.  We could get picky and point out that C is in our rate law, and it’s not a reactant of the overall reaction.  That’s true, but you’ll see that we actually cannot get any further with [C] than this step.  Ideally, we’d know more information and be able to remove [C] from our rate law, but we can’t do that in this particular scenario.