Which Reaction Mechanism Aligns with the Rate Law?

[mooncrater]

Homework Statement

This is the question:
Consider the reaction: $$2NO_2(g)+O_3 (g)\longrightarrow N_2O_5 (g)+O_2 (g) $$
The reaction of nitrogen dioxide and ozone is first order in $NO_2(g)$ and in $O_3 (g)$. Which of these possible reaction mechanisms is consistent with the rate law?
MECHANISM I :$$ NO_2(g)+O_3 \longrightarrow NO_3 (g)+O_2 (g)$$------(slow)
$$NO_3 (g)+NO_2 (g) \longrightarrow N_2O_5$(g) $$------(fast)
MECHANISM II:$$O_3 (g)\longrightarrow O_2(g)+[O] $$(this one is in equilibrium)(fast)
$$NO_2 (g)+[O] \longrightarrow NO_3 (g) $$-------(slow)
$$NO_3 (g)+NO_2 (g) \longrightarrow N_2O_5 (g) $$-------(fast)
And the options are :
(A) I only
(B) II only
(C) Both I and II
(D) Neither I not II

Homework Equations

The Attempt at a Solution

The rate equation from the original equation:$$rate=k [NO_2][O_3] $$
And from mechanism 1:$$ r_1=k_1 [O_3][NO_2] $$
And from the mechanism 2 :
$$r_2=k_2 [NO_2][O] $$
$$k_{eq}=\frac {[O][O_2]}{[O_3]} $$
So k_eq[O₃]/[O₂]=[O]
putting this in $r_2$
$$r_2=\frac {k _2k_{eq}[O_3][NO_2]}{[O_2]} $$
Which doesn't correspond with the gas law. So only 1 should be correct and the answer should be (A) but the given answer is (C)... but $r_2$ doesn't correspond to $r$. So where am I wrong?

 

[epenguin]

Homework Statement

This is the question:
Consider the reaction: $$2NO_2(g)+O_3 (g)\longrightarrow N_2O_5 (g)+O_2 (g) $$
The reaction of nitrogen dioxide and ozone is first order in $NO_2(g)$ and in $O_3 (g)$. Which of these possible reaction mechanisms is consistent with the rate law?
MECHANISM I :$$ NO_2(g)+O_3 \longrightarrow NO_3 (g)+O_2 (g)$$------(slow)
$$NO_3 (g)+NO_2 (g) \longrightarrow N_2O_5$(g) $$------(fast)
MECHANISM II:$$O_3 (g)\longrightarrow O_2(g)+[O] $$(this one is in equilibrium)(fast)
$$NO_2 (g)+[O] \longrightarrow NO_3 (g) $$-------(slow)
$$NO_3 (g)+NO_2 (g) \longrightarrow N_2O_5 (g) $$-------(fast)
And the options are :
(A) I only
(B) II only
(C) Both I and II
(D) Neither I not II

Homework Equations

The Attempt at a Solution

The rate equation from the original equation:$$rate=k [NO_2][O_3] $$
And from mechanism 1:$$ r_1=k_1 [O_3][NO_2] $$
And from the mechanism 2 :
$$r_2=k_2 [NO_2][O] $$
$$k_{eq}=\frac {[O][O_2]}{[O_3]} $$
So k_eq[O₃]/[O₂]=[O]
putting this in $r_2$
$$r_2=\frac {k _2k_{eq}[O_3][NO_2]}{[O_2]} $$
Which doesn't correspond with the gas law. So only 1 should be correct and the answer should be (A) but the given answer is (C)... but $r_2$ doesn't correspond to $r$. So where am I wrong?

I think it is OK as far as you got, but you have not got to the end. You need to get rid of the [O_2]] in your last equation and express in terms of initial reactants only.

That said it seems to me still incompatible with mechanism 2.

 

[mooncrater]

I think it is OK as far as you got, but you have not got to the end. You need to get rid of the [O_2]] in your last equation and express in terms of initial reactants only.

That said it seems to me still incompatible with mechanism 2.

But we don't have any other equation in which $O_2$ is present. (In $2^{nd}$ mechanism. ) So I think that this $O_2$ won't go.

 

[epenguin]

But we don't have any other equation in which $O_2$ is present. (In $2^{nd}$ mechanism. ) So I think that this $O_2$ won't go.

True you are not given it, ha ha.
I will let you osider just that first equilibrium by itself and see if you can come with it after all.

 

[mooncrater]

True you are not given it, ha ha.
I will let you osider just that first equilibrium by itself and see if you can come with it after all.

Sorry but I didn't understand what you mean to say.

 

[epenguin]

In general say younstart with substance A and it dissociates A ⇔ B + C

Can you say anything about [ B] at equilibrium?

OK, assume the equilibrium is to the left, I.e. [ B] is small compared to [A].

 

[mooncrater]

In general say younstart with substance A and it dissociates A ⇔ B + C

Can you say anything about [ B] at equilibrium?

OK, assume the equilibrium is to the left, I.e. [ B] is small compared to [A].

Concentration of B will increase.

 

[Borek]

I believe epenguin tries to ask you whether there is any dependency between amount of O₂ and O present (at least initially).

 

[mooncrater]

$k_{eq}=\frac {[O][O_2]}{[O_3]}$
Is one such relation quantitatively.
If $[O]$ is increased then $[O_2]$ will decrease (Le chatelier principle) and vice versa. Are you giving me a hint about something else?

 

[Borek]

Yes, about something else.

Try to build an ICE table for the process, as if you were attempting to solve a simple equilibrium problem - what is the equilibrium concentration of O given K and initial [O₃].

 

[epenguin]

Yes, you see you are trying to do something relatively clever and specialized while missing something very elementary and universal. Let me split my question and ask only for the first step.

In general say younstart with substance A and it dissociates A ⇔ B + C
.

what is the ratio of [ B] and [C] at equilibrium? Or better, if they are not consumed in further reactions (which is an approximation you were assuming) what is their ratio at any time?

 

[mooncrater]

Yes, you see you are trying to do something relatively clever and specialized while missing something very elementary and universal. Let me split my question and ask only for the first step.
what is the ratio of [ B] and [C] at equilibrium? Or better, if they are not consumed in further reactions (which is an approximation you were assuming) what is their ratio at any time?

1:1.

 

[mooncrater]

So $[O_2]=[O]$.
How Is it useful on removing that $extra$ $[O_2]$from the denominator? ($extra$ as the numerator corresponds the the correct rate law... because of which we can't change that)

 

[Borek]

So $[O_2]=[O]$.

Can you use this information - together with the initial concentration of O₃ and K_eq - to calculate the concentration of O?

Don't bother about other things for now, we will see how they fit the problem later.

 

[mooncrater]

Can you use this information - together with the initial concentration of O₃ and K_eq - to calculate the concentration of O?

Don't bother about other things for now, we will see how they fit the problem later.

Yup...
$[O]=\sqrt {K_{eq}[O_3]}$

 

[Borek]

Can't you plug that into the rate equation?

And yes, that would mean mechanism 2 is not consistent with the observed rate (or I am missing something as well).

This is a little bit tricky, as - if I am reading these equation correctly - the observed rate equation will be different at the beginning (when there is no substantial amount of oxygen present) and later (when you can't ignore O₂ presence).

 

[mooncrater]

Can't you plug that into the rate equation?

And yes, that would mean mechanism 2 is not consistent with the observed rate (or I am missing something as well).

This is a little bit tricky, as - if I am reading these equation correctly - the observed rate equation will be different at the beginning (when there is no substantial amount of oxygen present) and later (when you can't ignore O₂ presence).

Yes I can do that... But the real problem (as you are saying) that the mechanism 2 doesn't correspond to the rate law. But the answer is different from that. So can we reach the result that the given answer is wrong?

 

[epenguin]

We could do if that is what the equations tell us. But we haven't seen you tell use what the equation in terms of concentrations of starting reactants is yet. (Assume the overall reaction is irreversible.)