Choosing a Rate Law and Judging a Proposed Mechanism
Students follow the reaction A + B → products at 25.0 °C by absorbance and record initial rates. Run 1 uses [A]0 = 0.100 M and [B]0 = 0.100 M and gives 2.0 × 10^-4 M/s. Run 2 uses [A]0 = 0.200 M with [B]0 = 0.100 M and gives 8.0 × 10^-4 M/s. Run 3 uses [A]0 = 0.100 M with [B]0 = 0.200 M and gives 4.0 × 10^-4 M/s. Run 4 repeats run 1 with 0.050 M KNO3 added and gives 2.1 × 10^-4 M/s. One student proposes a single elementary step, A + B → products. Another proposes a fast pre-equilibrium 2 A ⇌ A2 followed by slow A2 + B → products. A third notes that the rate rises about 5% when ionic strength is increased and asks whether that matters. The instructor asks for a defended rate law, a value of k with units, and a judgement about which mechanism, if either, the data support.
1. Determine the rate law and the rate constant from runs 1 through 3, showing how each order is extracted.
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Compare runs that change only one concentration.
Order in A: doubling [A] from 0.100 to 0.200 M while [B] is fixed changes the rate from 2.0 × 10^-4 to 8.0 × 10^-4 M/s, a factor of 4. Since 2^m = 4, m = 2 - second order in A.
Order in B: doubling [B] from 0.100 to 0.200 M while [A] is fixed changes the rate from 2.0 × 10^-4 to 4.0 × 10^-4 M/s, a factor of 2. Since 2^n = 2, n = 1 - first order in B.
Rate = k[A]^2[B]. Using run 1: k = 2.0 × 10^-4 / (0.100^2 × 0.100) = 2.0 × 10^-4 / 1.00 × 10^-3 = 0.20 M^-2 s^-1.
Two checks. First, run 3: 0.20 × (0.100)^2 × 0.200 = 4.0 × 10^-4 M/s, which matches. Second, units: the rate is M/s and the concentration product is M^3, so k must carry M^-2 s^-1. A value of k reported as 0.20 s^-1 would be dimensionally wrong and is a common slip when the overall order is 3.
The overall order is 3, but that number says nothing about mechanism; it is only the sum of the experimental exponents.
2. Evaluate both proposed mechanisms against the rate law. Then explain what the rate law alone can and cannot establish.
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The single-step proposal fails. An elementary step A + B → products is bimolecular, so its rate law would be k[A][B] - first order in A, not second. A termolecular step 2 A + B → products would produce the right form, but three-body collisions at these concentrations are rare enough that this is a poor default assumption.
The pre-equilibrium proposal predicts the observed form. With 2 A ⇌ A2 fast (K = [A2]/[A]^2) and A2 + B → products slow, rate = k2[A2][B] = k2K[A]^2[B]. That matches the data with k = k2K, and the identical result comes out of the steady-state treatment whenever the reverse of the first step is much faster than the second.
What the data cannot do is prove the mechanism. A matching rate law is a necessary condition, not sufficient; several mechanisms can yield [A]^2[B]. Confirmation requires independent evidence - detecting A2 spectroscopically, showing the reaction slows in a way consistent with K, measuring an activation energy for the composite k, or testing for an isotope effect. The defensible conclusion is that the data eliminate the simple bimolecular proposal and are consistent with the pre-equilibrium proposal, not that the mechanism has been established.
3. The 5% rate change in run 4 is small. Decide how it should be treated and what the implication is for the reported value of k.
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A 5% change sits close to the reproducibility of absorbance-based initial rates, so it should not be used on its own to add a species to the rate law. It is also physically meaningful rather than merely noise: adding KNO3 raises the ionic strength without changing [A] or [B], and for reactions between ions the observed rate constant depends on activity coefficients through the kinetic salt effect. A genuine ionic-strength dependence would show a trend across several salt concentrations, not a single point.
The right move is to repeat run 1 at three or four ionic strengths, using a salt whose ions are unreactive, and see whether the rate changes monotonically. If it does, the rate law is still correct in form but k is a conditional constant holding only at fixed ionic strength.
This is a real limitation on the reported k = 0.20 M^-2 s^-1. Because runs 1 through 3 were done in media of differing ionic strength (0.100 M B versus 0.200 M B changes the total ionic environment), the k values agree so closely mainly because the salt effect is small here. Reporting k without stating the medium overstates its generality.