# Consistency with the rate law.

1. May 4, 2015

### mooncrater

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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 keq[O3]/[O2]=[O]
putting this in $r_2$
$$r_2=\frac {k _2k_{eq}[O_3][NO_2]}{[O_2]}$$
Which doesnt 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?

2. May 5, 2015

### epenguin

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 [O2]] in your last equation and express in terms of initial reactants only.

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

3. May 5, 2015

### mooncrater

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.

4. May 5, 2015

### epenguin

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.

5. May 5, 2015

### mooncrater

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

6. May 5, 2015

### 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].

Last edited: May 5, 2015
7. May 6, 2015

### mooncrater

Concentration of B will increase.

8. May 6, 2015

### Staff: Mentor

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

9. May 6, 2015

### 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?

10. May 6, 2015

### Staff: Mentor

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].

11. May 6, 2015

### 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.
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?

Last edited: May 6, 2015
12. May 8, 2015

### mooncrater

1:1.

13. May 8, 2015

### 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)

14. May 8, 2015

### Staff: Mentor

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

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

15. May 8, 2015

### mooncrater

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

16. May 8, 2015

### Staff: Mentor

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).

17. May 8, 2015

### mooncrater

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?

18. May 8, 2015

### 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.)