An excited ozone molecule, O3*, in the atmosphere can undergo one of the following reactions,
O3* → O3 (1) fluorescence
O3* → O + O2 (2) decomposition
O3* + M → O3 + M (3)deactivation,
where M is an inert molecule, the rate constant for the fluorescence reaction is k1, the rate constant for the decomposition reaction is k2, and the rate constant for the deactivation reaction is k3. Write a simplified expression for the fraction, X, of ozone molecules undergoing deactivation in terms of the rate constants. (Use the following as necessary: k1, k2, k3, cO for [O3*], and cM for [M].)
To find the fraction, X, of ozone molecules undergoing deactivation, we need to write an expression using the rate constants.
Let's first consider the total rate of reactions involving the excited ozone molecule, O3*.
The total rate of reactions can be described by the rate of fluorescence (k1*[O3*]) plus the rate of decomposition (k2*[O3*]) plus the rate of deactivation (k3*[O3*][M]):
Total rate = k1*[O3*] + k2*[O3*] + k3*[O3*][M]
Now, to find the fraction X of ozone molecules undergoing deactivation, we need to isolate the rate of deactivation from the total rate.
The rate of deactivation can be isolated by dividing k3*[O3*][M] by the total rate:
Rate of deactivation = k3*[O3*][M] / (k1*[O3*] + k2*[O3*] + k3*[O3*][M])
Next, we can simplify this expression by factoring out the common term [O3*]:
Rate of deactivation = [O3*] * (k3*[M]) / ([O3*]*(k1 + k2 + k3*[M]))
Now, we can cancel out the [O3*] term:
Rate of deactivation = k3*[M] / (k1 + k2 + k3*[M])
Finally, we substitute [O3*] with cO and [M] with cM:
Rate of deactivation = k3*cM / (k1 + k2 + k3*cM)
Therefore, the simplified expression for the fraction X of ozone molecules undergoing deactivation is:
X = k3*cM / (k1 + k2 + k3*cM)