Half-life of Ozone

CHANGES IN OZONE OVER TIME

1. General equation

The equation that describes the temporal evolution of the ozone concentration has the following form:

dn/dt = −kn² + q             (1)

Where t is the time, k is the reaction rate constant which is independent of concentration, but, probably, depends on the temperature; is the concentration (number of the molecules per unit volume):

n= (M /μV)* N_A                 (2)

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), is the number of molecules delivered per unit time into the volume and calculated per unit volume:

q= (m/μV) N_A       (3)

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= (n_o/1+n_o k t)              (4)

where is the initial concentration. Using (2):

n_o = (M_o/ μV)* N_A         (5)

where M0 is the initial mass of ozone. Eq.(4) can be rewritten as

n= n_o/ (1 + t / τ) (6)

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 / n_o k         (7)

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 / (N_A M_o k) = k1 / M_o         (8)

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.; and using Eqs. (7) we obtain from (9) that

q = n_o  / τ          (10)

Using Eqs. (3) and (5) one can rewrite (10) in terms of mass

m = M_o / τ        (11)

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 (mass per unit time) given by Eq.(11). The time scale parameter m 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. It should be stressed that, for a given volume and temperature, the measured value of 0 M M= / 2 τ is strictly linked to the initial mass of ozone M0 which should be common for both experiments.