Determining the rate of stratospheric ozone destruction I am a sophomore. A friend and I ran across a kinetic chemistry exercise. We solved it in a slightly different way and we can't come to an agreement on who is right. We would appreciate some help:


  Destruction of stratospheric ozone as determined by using the steady-state approximation.
  
  The balance of ozone in the stratosphere is of critical concern
  because this molecule absorbs ultraviolet light that would be harmful
  to life at Earth's surface. The principal production mechanism for
  ozone is recombination of $\ce O$ atoms with $\ce{O_2}$. The principal
  destruction mechanism is that given below. There is increasing concern
  over alternative destruction mechanisms involving molecules introduced
  into the stratosphere by human activity.
  
  Determine the destruction rate of ozone in the following mechanism.
  $$\begin{align}\ce{O3 + M &amp;&lt;=&gt;[k_1][k_{-1}] O2 + O  + M}\\
\\
\ce{O3 + O &amp;-&gt;[k_2] 2 O2}\end{align}$$


Our answers:

We agree on the beginning, so we have :

$$\begin{align}
-\frac{\mathrm d[\ce{O_3}]}{\mathrm dt} &amp;= k_1[\ce{O_3}][\ce{M}] - k_{-1}[\ce{O_2}][\ce O][\ce M] + k_2[\ce{O_3}][\ce O]\\
-\frac{\mathrm d[\ce{M}]}{\mathrm dt} &amp;= k_1[\ce{O_3}][\ce M] - k_{-1}[\ce{O_2}][\ce O][\ce M]
\\
-\frac{\mathrm d[\ce O]}{\mathrm dt} &amp;= -k_1[\ce{O_3}][\ce M] + k_{-1}[\ce{O_2}][\ce O][\ce M] +k_2[\ce{O_3}][\ce O]
\end{align}$$

Now here comes the disagreement, we need to use the steady-state approximation to say that one of the above speed is null:

My friend argues that $-\frac{\mathrm d[\ce O]}{\mathrm dt}$ should be approximated to null because it is a transient compound.

Even if in most other exercise, it would be correct, I disagree with him because in the proposed mechanism, $\ce{O_3}$ and $\ce O$ play a symmetrical part. So if we approximate $-\frac{\mathrm d[\ce O]}{\mathrm dt}$ to $0$ we should also approximate $-\frac{\mathrm d[\ce{O_3}]}{\mathrm dt}$ to $0$. Since we want to calculate this specific value, we don't want to approximate it.
In my demonstration, I approximate $-\frac{\mathrm d[\ce{M}]}{\mathrm dt}$ to $0$, arguing that the compound $\ce M$ is on both side of the equilibrium equation and therefore it's concentration will remain constant.

We both manage to find a result:
My friend’s calculation:

$$
- \frac{\mathrm d[\ce O]}{\mathrm dt} = -k_1[\ce{O_3}][\ce M] + k_{-1}[\ce{O_2}][\ce O][\ce M] + k_2[\ce{O_3}][\ce O] = 0
$$
$$
k_1[\ce{O_3}][\ce M] - k_{-1}[\ce{O_2}][\ce O][\ce M] = k_2[\ce{O_3}][\ce O]
$$
We substitute the left member in the speed of ozone:
$$
-\frac{\mathrm d[\ce{O_3}]}{\mathrm dt} = 2\times k_2[\ce{O_3}][\ce O]
$$
Because $[\ce O]$ is transient, it is difficult to mesure experimentally, so we expresse it in function of $[\ce{O_3}], [\ce{O_2}]$ and $[\ce M]$ :
$$
k_1[\ce{O_3}][\ce M] = [\ce O] \times ( k_2[\ce{O_3}] + k_{-1}[\ce{O_2}][\ce M] )
$$
$$
[\ce O] = \frac{k_1[\ce{O_3}][\ce M]}{k_2[\ce{O_3}] + k_{-1}[\ce{O_2}][\ce M]}
$$
And finally we have my friend’s result:
$$
-\frac{\mathrm d[\ce{O_3}]}{\mathrm dt} = \frac{2k_1k_2[\ce{O_3}]^2[\ce M]}{k_2[\ce{O_3}] + k_{-1}[\ce{O_2}][\ce M]}
$$

Now my calculations:
$$
-\frac{\mathrm d[\ce M]}{\mathrm dt} = k_1[\ce{O_3}][\ce M] - k_{-1}[\ce{O_2}][\ce O][\ce M] = 0
$$
Which allow us to simplify the destruction rate of ozone:
$$
-\frac{\mathrm d[\ce{O_3}]}{\mathrm dt} = k_2[\ce{O_3}][\ce O]
$$
We already disagree by a factor 2 on the rate. But if we replace $[\ce O]$ like before, we obtain a much simpler result:
$$\begin{align}
k_1[\ce{O_3}][\ce M] &amp;= k_{-1}[\ce{O_2}][\ce{O}][\ce M]
\\
[\ce{O}] &amp;= \frac{k_1[\ce{O_3}][\ce M]}{k_{-1}[\ce{O_2}][\ce M]}
\\
[\ce O] &amp;= \frac{k_1[\ce{O_3}]}{k_{-1}[\ce{O_2}]}
\end{align}$$
Finally I have:
$$
-\frac{\mathrm d[\ce{O_3}]}{\mathrm dt} = \frac{k_1k_2[\ce{O_3}]^2}{k_{-1}[\ce{O_2}]}
$$

Which one of us is wrong and why?
Dioxygen is split in the stratosphere by UV light into oxygen atoms. This is the main source of free oxygen atoms. Their production rate depends on the intensity of the UV light.

In steady-state we can assume that the concentration of oxygen atoms is constant.

$$\frac{d[\ce{O}]}{dt} = 0  \tag{1}$$

For the above destruction mechanism we can formulate the rate laws

$$\begin{align}
\frac{d[\ce{O3}]}{dt} &amp;= -k_1 [\ce{O3}] [\ce{M}] + k_{-1} [\ce{O}] [\ce{O2}] [\ce{M}] - k_2 [\ce{O}] [\ce{O3}] \tag{2}\\
\\
\frac{d[\ce{O}]}{dt} &amp;= k_1 [\ce{O3}] [\ce{M}] - k_{-1} [\ce{O}] [\ce{O2}] [\ce{M}] - k_2 [\ce{O}] [\ce{O3}]  \tag{3}
\end{align}$$

Adding equation $(2)$ and $(3)$ which by $(1)$ is $0$ we get

$$\frac{d[\ce{O3}]}{dt} = -2~k_2 [\ce{O}] [\ce{O3}] \tag{4}$$

With equations $(1)$ and $(3)$ we can express the concentration of oxygen atoms as

$$[\ce{O}] = \frac{k_1 [\ce{O3}] [\ce{M}]}{k_{-1} [\ce{O2}] [\ce{M}] + k_2 [\ce{O3}]} \tag{5}$$

Substituting $(5)$ into equation $(4)$ we get

$$\frac{d[\ce{O3}]}{dt} = \frac{-2~k_1~k_2 [\ce{O3}]^2 [\ce{M}]}{k_{-1} [\ce{O2}] [\ce{M}] + k_2 [\ce{O3}]}  \tag{6}$$

So your friend seems to be right.