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Tuesday, January 7, 2014

How to calculate radioactive "activity"

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 To determine how radioactive something is:
 A* =
0.693 mNA
A* = activity of sample (in disintegrations per time)
m = mass of sample
A = atomic weight of radionuclide (easy: it's the number of the nuclide, like 89 for strontium-89)
T1/2 = half-life of radionuclide (in any unit of time)
Ft = conversion from one unit of time to the desired unit of time
NA = constant called Avogadro's number, or 6.022 x 10 23 atoms/mole
Does this formula really work?
Let's try it on radium.  We know that 1 curie represents the radioactivity of one gram of pure Radium-226 and we also know that 1 curie pumps out 3.700 x 1010 disintegrations per second.
So, to prove that 1 gram of radium pumps out 37 trillion disintegrations per second, let's assign the variables for radium.  Lets assign a mass of 1 (gram).  We know radium's half-life is about 1,603 years and its atomic weight is 226.0254 (radium 226).  For Ft we want to convert from years to seconds, or 31,536,000 seconds per year.
 A* =
(0.693) (1 gram) (6.022 x 10 23 atoms/mole)
= 3.65E10, or 3.65 x 1010 disintegrations per second
(1603 years)(226.0254 grams/mole) (525600 minutes/year)(60 seconds/minute)
Why did we come up short?  Any ideas??
 Example.  If 1 microgram of strontium-89 (which has a 50.6 day half-life) was deposited today in 1 meter by 1 meter area of a rice patty near Tokyo, what is its present activity?
 A* =
(0.693) (0.00001 grams) (6.022 x 10 23 atoms/mole)
= 6.43 x 1011 disintegrations per minute
(50.6 days)(89 grams/mole) (1440 minutes/day)
 Example.  If plutonium-238 depositions from Fukushima caused Namie soils to be 4 becquerels per square meter, what is the activity level?
First 4 becquerels (of pu238) is the same as 108 picocuries (of pu238). Second, one gram of pu238 is 17.44 curies, or (17.44 x 1 trillion) picocuries.  So, the 4 becquerels per square meter represents 108/(17.44 x 1 trillion) = 6.3 x 10-12 grams.
 A* =
(0.693) (6.3 x 10-12 grams) (6.022 x 10 23 atoms/mole)
= 238.8 disintegrations per minute
(88 years)(238 grams/mole) (525600 minutes/year)

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