Estimate
of Radon Emanation
This
discussion looks at the question of how a sample containing some amount
of uranium (C, expressed as parts per million) will approach radioactive
equilibrium if some fraction of radon is initially missing. To begin,
we need to define a number of variables and constants.
Concentration of uranium
in the sample (ppm)
Variable t is a sequence of times in units of days with an interval of 0.2
Variable
n is an index ranging
from 0 to 18 with an interval
of 1
Variable f is an array of values representing the initial fraction of radon that is present. The value of f ranges from
0.05 to 0.95 with an interval
of 0.05.
Avogadro's number
Atomic
mass of U238
Number
of atoms of 238U
per gram of sample
Half-life
of 238U (years)
Half-life
of 226Ra (years)
Half-life
of 222Rn (days)
If
we assume radioactive equilibrium between 238U and 226Ra, we can easily calculate the number
of atoms of radium based upon the fact that the activities are equal.
The activity is given by the product of the decay constant (l) and the number of atoms.
First let us define an array of the half-lives (T) expressed in days and
then calculate an array of the decay constants (l).
Number
of atoms of 226Ra per gram
of sample
If
we now assume that the radon is not in equilibrium and that the radon
activity at time t = 0 is some fraction, f, of the radium activity, we
can define the number of atoms of radon as a function of time by the equations
Number
of 226Ra atoms as
a function of time
Rate
of change of the
number of radon atoms
where
the first term is the number of new radon atoms produced by the decay
of radium and the second term represents a loss of radon as the result
of radioactive decay. We can define an array of the activity of the radioactivie
isotopes (uranium, radium, and radon) at time t = 0 as
If
we now assume that the solution to the differential equation can be written
in the general form,
we
can solve for the constants using the known amount of radon at time t
= 0 and the rate of change at time t = 0. The matrix of the constants
is defined below.
We
can now calculate the equivalent uranium concentration as
Because
we cannot determine the concentration of uranium at an instant in time,
we must measure the concentration over a period of time and then calculate
the average concentration for that time period. Define the length of the
time period (d) as a fraction of a day as show below.
Measurement
time in seconds
Measurement
time in days
where
we have used a measurement livetime defined in seconds to calculate time
period in days. We will define a measurement starting time, T1(m), and
an ending time, T2(m) and we will make successive measurements as indicated
by the variable m.
Variable
m is an index ranging
from 0 to 100 with an interval of 1.
Measurement
start time
Measurement
end time
The
average uranium concentration is then defined by the equation
You can
that list the apparent uranium concentrations as a function of time expressed
as ratios to the initial measurement. The values of the ratios are independent
of the uranium concentration and are solely a function of the radon disequilibrium.
The
assume that the initial fraction of radon is given by the value of f as shown
above each of the matrices. The radon emanation fraction is given by 1
- f. Power series equations for each of the measurement intervals were
calculated to enable calculation of the radon emanation using the value
of the ratio. The derived coefficients for the power series are available
in a spreadsheet.
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