The following question(s) relate to the reaction between water and a complex ion of Co2+; the rate varies with concentration as follows. (The rate has no dependence on [H2O].)

Trial [Co2+ complex], M Initial rate, M min−1
1 1.14 × 10−3 3.2 × 10−7
2 2.28 × 10−3 6.4 × 10−7
3 3.42 × 10−3 9.6 × 10−7
4 1.14 × 10−2 3.2 × 10−6

Refer to Exhibit 11-1. The rate law for the reaction is rate =

Question

The following question(s) relate to the reaction between water and a complex ion of Co2+; the rate varies with concentration as follows. (The rate has no dependence on [H2O].)

Trial	[Co2+ complex], M	Initial rate, M min−1
1	1.14 × 10−3	3.2 × 10−7
2	2.28 × 10−3	6.4 × 10−7
3	3.42 × 10−3	9.6 × 10−7
4	1.14 × 10−2	3.2 × 10−6

Refer to Exhibit 11-1. The rate law for the reaction is rate =

BlackTom AI · Accepted Answer

To determine the rate law, examine how the initial rate changes as the [Co2+ complex] changes, while noting that there is no dependence on [H2O].

- Compare trials with increasing [Co2+ complex] but the same other conditions to see how the rate scales.
  - Trial 1: [Co2+ complex] = 1.14 × 10^-3 M, rate = 3.2 × 10^-7 M min^-1
  - Trial 2: [Co2+ complex] = 2.28 × 10^-3 M (double Trial 1), rate = 6.4 × 10^-7 M min^-1 (also double)
  This indicates the rate doubles when the reactant concentration doubles, consistent with first-order dependence on [Co2+ complex].
  
- Check another data point to reinforce the trend:
  - Trial 3: [Co2+ complex] = 3.42 × 10^-3 M (triple Trial 1), rate = 9.6 × 10^-7 M min^-1 (also triple)
  Again, the rate scales linearly with concentration, reinforcing first-order behavior in this reactant.
  
- Consider Trial 4: [Co2+ complex] = 1.14 × 10^-2 M, which is 10× Trial 1, and the rate is 3.2 × 10^-6 M min^-1, which is 10× Trial 1’s rate. This also supports a first-order relationship with [Co2+ complex].

Now, evaluate possible rate-law forms:
- Zeroth order in Co2+ complex: rate = k would imply the rate is independent of [Co2+ complex], which contradicts the observed proportional increases in rate with concentration.
- First order in Co2+ complex: rate = k [Co2+ complex]. This matches the data: doubling the concentration doubles the rate, tripling the conc triples the rate, and so on.
- Second order in Co2+ complex or higher: rate would scale with the square or higher power of concentration, which would produce a much stronger dependence than seen (e.g., doubling [Co2+ complex] would give a fourfold increase in rate), inconsistent with the linear scaling in the trials.

Regarding [H2O]: the problem states the rate has no dependence on [H2O], so any rate expression containing [H2O] would be incorrect for this reaction.

Putting it together, the simplest rate law that fits all the data is rate = k [Co2+ complex].

Alternative forms that would be incorrect given the data include rate = k (independent of [Co2+ complex]), rate = k [Co2+ complex]^2, or rate = k [Co2+ complex] [H2O] (or any dependence on [H2O]).

In summary, the rate law that aligns with the observed linear relationship between rate and [Co2+ complex], and with no dependence on [H2O], is rate = k[Co2+ complex].

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