Radon gas levels in indoor spaces are known to fluctuate considerably, so continuous monitoring is necessary to compute long-term averages. This particular radon detector, which uses continuous air sampling coupled to algorithm-based alpha spectrometry, is designed to do this job and has gained good reviews on Amazon. It is made in Norway by Corentium AS.

My unit has been in continuous operation since October 2015. Although the short-term average figure goes up and down from day to day, and to a lesser extent from week to week (the display shows alternating 1-day and 7-day figures), the long-term average figure is really quite steady.

I was thinking about this the other day, and it occurred to me that since the long-term figure varies so little in a month’s turning, I could use it to estimate the entry rate of radon gas into the enclosed space where the device is located.

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**Formula for computing entry rate**

**1. Units in becquerels per cubic meter (Bq/m ^{3})**

If the device is showing a steady long-term average figure (n) and the enclosed space has a volume of v cubic meters, the entry rate of radon gas is computed as follows:

Entry rate = 8.78nv attomoles per month

For example, if n = 79 and v = 5

Entry rate = 8.78 × 79 × 5 = 3468 attomoles per month

(1 attomole = 10^{-18} moles)

**2. Units in picocuries per litre (pCi/L)**

If the device is showing a steady long-term average figure (n) and the enclosed space has a volume of v cubic meters, the entry rate of radon gas is computed as follows:

Entry rate = 324.74nv attomoles per month

For example, if n = 4.64 and v = 5

Entry rate = 324.74 × 4.64 × 5 = 7534 attomoles per month

(1 attomole = 10^{-18} moles)

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**Explanatory notes**

First I should clarify what I mean by entry rate. Radon is a gas that seeps into enclosed spaces through conduits, joints and cracks; it is also exhaled by diffusion through surfaces. Having infiltrated the space, some of the radon will escape, either through back-diffusion or infiltration into adjacent spaces. Without knowing the rates of ingress and escape, one can conclude that a steady long-term average figure on the detector, which implies a steady concentration of radon in the enclosed space, indicates equilibrium between the rate of radon ingress on the one hand, and the rate of radon escape and decay on the other. In other words at equilibrium

Defining entry rate as the difference between ingress rate and escape rate, we have

The entry rate is thus the rate at which radon atoms enter the enclosed space without escaping.

Given the premise that the concentration of radon gas in the enclosed space is steady, the decay rate can be taken as constant since it is determined solely by the concentration – i.e. the number of radon atoms present in a given volume. So a steady long-term average on the detector means that the entry rate, as here defined, is also constant.

Radon is a very dense gas, almost 8 times as dense as air, and this tempts many to think that radon accumulates at the bottom of an enclosed space. This is not what happens. Like any gas, radon exhibits the phenomenon of diffusion – which is the tendency of a substance to spread uniformly throughout the space available to it. What the density of a gas does affect is the *rate* at which the gas diffuses. But given sufficient time to reach a state of equilibrium, it can be assumed that the concentration of radon gas will be uniform throughout the enclosed space.

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**Assigning a unit of time**

So far I have said nothing concerning the unit of time to be applied in relation to the foregoing rate equation. Now we can address this issue, which constitutes the novelty of the computation scheme.

**— Let the unit of time by which rate is measured be set equal to the half-life of the isotope (Rn-222) of which radon gas is largely composed.**

__Theorem__

Let the entry rate of radon gas into a previously radon-free bounded space be x atoms per unit of time corresponding to the half-life of Rn-222. At the end of the first half-life period, x/2 atoms will have decayed (via α emission) while x/2 atoms remain. At the end of the second half-life period, the first x atoms will have decayed to x/4 while the second x atoms will have decayed to x/2. At the end of the third half-life period, the first x atoms will have decayed to x/8 and the second x atoms to x/4, while the third x atoms will have decayed to x/2 … and so on according to the following scheme

This process forms an absolutely convergent geometric series in which the number of radon atoms remaining in the space after n half-life periods will be

*The conclusion is reached that if the entry rate of radon gas into a previously radon-free bounded space is x radon atoms in the unit of time corresponding to the half-life of Rn-222, the number of radon atoms in this space will over successive half-lives approach a steady-state value of x.*

Assuming diffusion throughout the space, a steady state value of x should be realized in little more than a month since when n = 8 (equivalent to 30.6 days) the series sum is 99.6% of x.

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**Computing entry rate**

Given a steady long-term average figure on the detector, which implies a steady concentration of radon gas throughout the bounded space, the number of radon atoms in this space can be estimated as follows.

*For decay rates measured in becquerels per cubic meter (Bq/m ^{3})*

Let the long-term average figure on the detector, measured in decays per second per cubic meter be n, and let the bounded space be v cubic meters.

In the half-life of Rn-222 (3.8235 days) the number of decays in volume v will be n × v × 330,350

This equals x/2 where x is the steady state population of radon atoms in volume v

Therefore x = n × v × 660,701 radon atoms

By the theorem, x is also the number of radon atoms entering the bounded space in a unit of time equal to the half-life of the isotope (Rn-222) of which radon gas is largely composed – 3.8235 days. The magnitude 8x is therefore the number of radon atoms entering the space in a month (8 x 3.8235 = 30.6). Dividing 8x by the Avogadro number converts the number of radon atoms into moles of radon gas:

Entry rate (moles of radon gas per month) = 8 × n × v × 660,701 / (6.022 x 10^23)

**The entry rate of radon gas ≅ 8.78nv attomoles per month**

(1 attomole = 10^{-18} moles)

*For decay rates measured in picocuries per litre (pCi/L)*

Let the long-term average figure on the detector, measured in picocuries per litre be n, and let the bounded space be v cubic meters.

**The entry rate of radon gas ≅ 324.74nv attomoles per month**

(1 attomole = 10^{-18} moles)

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**Disclaimer**

Please note that the theorem on which the above calculations are based is untested. Until the theorem has been tested and the accuracy of results obtained with it has been determined, the calculation of entry rate as herein defined can only be regarded as a theoretical prediction and should be viewed accordingly.

P Mander May 2017

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