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Changes of ozone with time
1. General equation
I believe that the equation describing the time evolution of the ozone concentration has the following form:
dn = −kn2 + q (1) dt
where, t is the time, k is the reaction rate constant which is independent of concentration, but, probably, depends on the temperature; n is the concentration (number of the molecules per unit volume):
n= M NA (2) μV
μ is the molar weight of ozone, M is the mass of ozone inside the volume, V , in a given moment of time, NA is the Avogadro number
In Eq.(1), q is the number of molecules delivered per unit time into the volume and calculate per unit volume::
q= m NA (3) μV
where m is the mass of ozone delivered into volume, V , per unit time.
2. Closed volume
Consider the case of q = 0 . For such a case, integration of Eq. (1) yields n= n0 (4)
1+n0kt where n0 is the initial concentration. Using (2):
n0 =M0 NA (5) μV
where M0 is the initial mass of ozone. Eq.(4) can be rewritten as
n= n0 (6) 1+ t
τ where we introduced the time scale parameter τ : when t = τ the concentration of molecules (and the mass of ozone, for a given volume) decreases by factor of two, see Eq.(6).
Comparing Eqs.(4) and (6) one obtains:
τ= 1 (7) n0k
Since k is independent of the concentration, we arrive at an important conclusion: the parameter τ depends on the initial concentration and, for a given volume and pressure, on the initial mass,
M0 . τ= μ =k1 (8)
NAM0k M0
3. Stationary regime
Inspecting Eq.(1) one can conclude that the concentration does not change when q = kn2 (9)
When Eq.(9) is satisfied, the rate of concentration changes, dn / dt , becomes zero, see Eq.(1).
Realizing that the concentration is maintained to be constant in time, i.e.; n = n0 Eqs.(7) we obtain from (9) that
q = n0 (10) τ
Using Eqs.(3) and (5) one can rewrite (10) in terms of mass m = M0 (11)
and using
τ
4. Conclusion
Thus, to maintain a given mass of ozone, M0 , within a fixed volume at a given temperature, one
should deliver ozone with the rate m (mass per unit time) given by Eq.(11). The time scale parameter τ on the right hand side of (11) is determined from an independent experiment. This experiment is to observe a decrease in the ozone mass with time in the same closed volume. The parameter τ coincides with the time corresponding to the ozone mass M = M 0 / 2 . It should be
stressed that, for given volume and temperature, the measured value of τ is strictly linked to the initial mass of ozone M0 which should be common for both the experiments.
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