The reaction

2NO3(g) → 2NO2(g) + O2(g)

has the rate law: rate = k[NO3]2. The half-life for the reaction is 53.5 s when the initial concentration of NO3 is 0.845 M. Calculate the time needed for the concentration of NO3 to decrease to one-fourth of its original value.

Question

The reaction

2NO3(g) → 2NO2(g) + O2(g)

has the rate law:  rate = k[NO3]2.  The half-life for the reaction is 53.5 s when the initial concentration of NO3 is 0.845 M. Calculate the time needed for the concentration of NO3 to decrease to one-fourth of its original value.

BlackTom AI · Accepted Answer

We start by identifying the reaction is second-order in NO3 with rate = k[NO3]^2. For a second-order reaction, the half-life t1/2 relates to the initial concentration via t1/2 = 1/(k[A]0).

First, determine the rate constant k from the given data: t1/2 = 53.5 s when [NO3]0 = 0.845 M.
53.5 = 1/(k × 0.845) ⇒ k × 0.845 = 1/53.5 ⇒ k ≈ 1/(53.5 × 0.845) ≈ 0.02216 M−1 s−1.

Next, use the integrated rate law for a second-order reaction with respect to a single reactant: 1/[A]t = 1/[A]0 + kt.
We want [A]t = [A]0/4, so 1/[A]t = 1/([A]0/4) = 4/[A]0. Plugging into the equation:
4/[A]0 = 1/[A]0 + kt ⇒ kt = 3/[A]0.
With [A]0 = 0.845 M, kt = 3/0.845 ≈ 3.549 s−1. Therefore t ≈ 3.549 / k ≈ 3.549 / 0.02216 ≈ 160 s.

Now evaluate the answer options:
- 161 s: This is the closest value to the calculated ≈160 s and aligns with the approximate arithmetic, so it is a plausible correct choice.
- 80.3 s: This would correspond to roughly half of the time calculated for quarter reduction or a misapplication of a first-/zero-order formula; it does not fit the second-order relation used here.
- 53.5 s: This is the given half-life, not the time required to drop to one-quarter of the initial concentration; using it would ignore the additional decay required to reach quarter, so this is incorrect.
- 214 s: This would imply a much longer time than needed for a quarter reduction under the same k and initial concentration; it overestimates the required duration for a second-order decay.
- 107 s: This lies between the half-life and the quarter-reduction time but does not satisfy the exact second-order relationship derived from kt = 3/[A]0; thus it is inconsistent with the calculation.

In summary, only the 161 s option satisfies the derived relationship for a second-order decay with the given initial concentration and time-half data, while the other choices reflect common misconceptions or misapplications of rate laws.

The reaction 2NO3(g) → 2NO2(g) + O2(g) has the rate law: rate = k[NO3]2. The half-life for the reaction is 53.5 s when the initial concentration of NO3 is 0.845 M. Calculate the time needed for the concentration of NO3 to decrease to one-fourth of its original value.单项选择题

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