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Half-life of Ozone

CHANGES IN OZONE WITH TIME

1. General equation

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.

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 a 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

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

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 a 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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