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# Half-life of Ozone

## Typical Gaseous Ozone Half-life time as a Function of Temperature

Half-life time Temp
∼ 3 months -50 ºC
∼ 18 days -35 ºC
∼ 8 days -25 ºC
∼ 3 days 20 ºC
∼ 1.5 hours days 120 ºC
∼ 1.5 seconds 250 ºC

## Typical Dissolved in Water Ozone (PH=7) Half-life time as a Function of Temperature

Half-life time Temp
∼ 30 minutes 15 ºC
∼ 20 minutes  20 ºC
∼ 15 minutes 25 ºC
∼ 12 minutes 30 ºC
∼ 8 minutes 35 ºC

## 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)* NA                 (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) NA       (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= (no/1+no k t)              (4)

where is the initial concentration. Using (2):

no = (Mo/ μV)* NA         (5)

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

n= no/ (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 / no 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 / (NA Mo k) = k1 / Mo         (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/ τ          (10)

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

m = Mo / 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.

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