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The following data are given for the reaction of $\ce{NO}$ and $\ce{O2}$:

$$ \ce{2NO + O2 -> 2NO2} $$

The the reaction is second order in $\ce{[NO]}$ and first order in $\ce{[O2]}$, and the rate of disappearance of $\ce{NO}$ is $2.5 \times 10^{-5}~\mathrm{mol\over L\,s}$ at the instant when $\ce{[NO] = [O2]} = 0.01~\mathrm{mol\over L}$.

The question asks me to calculate the rate constant.

I've thought of two ways of approaching the calculation—which of these solutions is correct?

1) Take the rate of the reaction as one-half the rate of disappearance of $\ce{NO}$:

$$ \begin{align} R &= {1\over 2} * 2.5 \times 10^{-5} = k \ce{[NO]^2[O2]}=k(0.01)^3 \\ k &= 12.5~\mathrm{L^2\over mol^2\,s} \end{align} $$

2) Take the rate of the reaction as equal to the rate of disappearance of $\ce{NO}$:

$$ \begin{align} R &= 2.5 \times 10^{-5} = k \ce{[NO]^2[O2]} = k(0.01)^3 \\ k &= 25 ~\mathrm{L^2\over mol^2\,s} \end{align} $$

pentavalentcarbon
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Adnan AL-Amleh
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1 Answers1

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Either definition is acceptable, as long as you clearly identify relationship between the calculated rate value $R$ and the rates-of-change for each of the species in the system.

In case (1), you've effectively calculated the rate constant defined as:

$$ R_1 = k_1\ce{[NO]^2[O2]} = -{\mathrm d \ce{[O2]}\over \mathrm dt} = -{1\over 2}{\mathrm d \ce{[NO]}\over \mathrm dt} = {1\over 2}{\mathrm d \ce{[NO2]}\over \mathrm dt} $$

In case (2), you've instead calculated the rate constant defined as:

$$ R_2 = k_2 \ce{[NO^2][O2]} = -2{\mathrm d \ce{[O2]}\over \mathrm dt} = -{\mathrm d \ce{[NO]}\over \mathrm dt} = {\mathrm d \ce{[NO2]}\over \mathrm dt} $$

Both cases accurately describe the kinetics, but you have to be careful to identify which definition of the rate you've used in calculating $k$.

hBy2Py
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  • Which value of the rate constant (k1 = 12.5 or k2 = 25) used in calculating the rate at the instant when [NO] = 0.015 mol/L and [O2] = 0.005 mol/L.with explaning? – Adnan AL-Amleh Jan 15 '17 at 05:18
  • @AdnanAL-Amleh There's really not a lot more I can say. The correct answer depends on which of the two rate relationships the person who wrote the problem had in mind. If you don't have any insight into their thinking, then there's no way to know which one they want. – hBy2Py Jan 15 '17 at 13:12
  • the problem adapted and edited from : Kotz J.C.,TriekelTreichel P.M., and Weaver G.C., Chemistry and Chemical Reactivity .6th Edition. The Answers : K= 12.5 M^-2 S^-1 and THE Rate at the instant when [NO] = 0.015 mol/L and [O2] = 0.005 mol/L=-1.4 ×10^-5 M/sec – Adnan AL-Amleh 20 hours ago – Adnan AL-Amleh Jan 16 '17 at 01:39
  • @AdnanAL-Amleh Well, that would've been useful information to include in the question! If they give 12.5, then clearly they set their rate up as in my $R_1$. – hBy2Py Jan 16 '17 at 02:10
  • I am Ambiguous with these answer , because the Rate of disappearance NO is experimentally determined with no consideration of coefficient factor in the balance eq.(the factors in eq. useful in calculate the ratios of rates-of-change for each of the species in the reaction. ) and rate formula obtained experimentally , so , I think with the second Definition in your R2. I want support with feed Back – Adnan AL-Amleh Jan 16 '17 at 02:31
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    @AdnanAL-Amleh Reported rate constants *are* often ambiguous. You're absolutely right that there is indeed only one actual, physical $\mathrm d \ce{[NO]}\over \mathrm dt$ observed in this system. But, there are an infinite number of rate expressions you can write, all related by multiplicative constants, that describe the kinetics equally well. Maybe someone else can help give you more insight, but I don't know if I can assist much beyond what's contained in my answer above. – hBy2Py Jan 16 '17 at 03:27
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    The IUPAC Gold Book recommends using the definition as in R1 above and this is also what most, if not all textbooks, use. – porphyrin Jan 18 '17 at 13:06
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    @porphyrin I also prefer the $R_1$ definition. Many journal articles do not follow this recommendation, however, and it can take quite a bit of work to back out the proper form of the rate expression. – hBy2Py Jan 18 '17 at 14:13
  • There is no simple correlation between the stoichiometry of the reaction and the rate law, the value of K in the rate law must be determined experimentally , – Adnan AL-Amleh Jan 19 '17 at 08:19
  • when we refer to what given in the question ,I think the rate of the reaction = experimentally determined value of the rate of disappearance of NO , so the value of rate constant =K2= 25 – Adnan AL-Amleh Jan 19 '17 at 08:19
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    @AdnanAL-Amleh "There is no simple correlation between the stoichiometry of the reaction and the rate law" I don't know what you mean by this; to my mind, the stoichiometry of the reaction *defines* the rate law. – hBy2Py Jan 19 '17 at 12:00
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    @AdnanAL-Amleh "I think the rate of the reaction = experimentally determined value of the rate of disappearance of NO." Like I said, this is a *valid choice* of the rate expression. But, *it is definitely not the only valid choice*. – hBy2Py Jan 19 '17 at 12:01