Defining Radioactivity
In this section, you will learn the definition of these terms
- activity
- becquerel (unit)
- specific activity
- curie (unit)
and you will use information from previous sections about
- decay constant
Now that we have defined the idea of a decay constant, and described how it can be related to the half-life of a radionuclide, we can more rigorously define the idea of radioactivity, or more simply, activity.
Activity is the rate at which radioactive disintegrations are taking place. It is related to both the number of radioactive nuclei present in a sample, and to the decay constant of those radionuclei. For a pure sample consisting of only one type of radionuclide, it is very easy to calculate activity. We use the letter A to represent activity.
The activity of a pure sample is simply the decay constant of that material multiplied by the number of radionuclei present.
For example, if we have 100,000 nuclei of bismuth-213, which has a half-life of 45.5 minutes (2730 seconds), we can quickly calculate the decay constant of Bi-213 to be (ln 2/2730 sec = 0.000254/sec). The activity of this sample is then 100,000*(0.000254/sec) = 25.4/sec.
This naturally leads us into a discussion of what the “units” of activity should be. We calculated the activity of this sample to be 25.4/sec. What does that mean? What is taking place? Well, an activity of 25.4/sec means that every second, 25.4 nuclei of bismuth-213 are decaying into something else. Of course, if 25.4 are decaying every second, then our sample of 100,000 isn’t 100,000 very long. One second later it is 99,975, and then one second later it is 99,949, and so on.
The units of activity are disintegrations per second, and this is given a specific name in the SI system of units. It’s called a becquerel, in honor of Henri Becquerel who was a pioneer in the discovery of radioactivity, and is has the abbreviation Bq. So a sample of bismuth-213 with 100,000 atoms has an activity of 25.4 becquerels, or 25.4 Bq. But just for a moment. A second later the activity will be lower due to radioactive decay. In fact, we can apply the same equation that we used when we introduced the decay constant to define the decrease in activity over time. Just as the number of radionuclei present decreases over time according to the decay constant.
We can multiply both sides of the equation by the decay constant to redefine the equation in terms of activity instead of number.
Now A0 represents the initial activity of the sample, and A represents the activity of a sample after a given amount of time.
A becquerel is a very small unit of activity. It is so small in fact that a becquerel of radioactivity is almost undetectable. Even a billion becquerels (a giga-becquerel, or GBq) of activity is a modest value of radioactivity.
Specific Activity
Now that we’ve defined activity and the decay constant, we can move on to the idea of specific activity, which is going to be a very useful concept in figuring out how much radioactivity is in a given amount of some material. We’ve been talking about the number of radionuclei present in a sample when in reality, even a gram of a material can have a trillion trillion atoms or more.
It would be nice to have an idea how much activity of given amount of material has, for instance, if we were holding a gram of natural thorium, how much activity would it have?
Let’s assume we were holding a gram of pure thorium-232. It has an atomic mass of 232, which means that 232 grams of thorium would have one mole of number. A mole is defined as 0.6022 trillion trillion (1024) of something. So one gram of thorium would have 1/232 of a mole of thorium, and thus there would be (0.6022 x 1024)*(1/232) = 2.60 x 1021 atoms per gram. So every gram of thorium has 2.6 billion trillion atoms in it. To figure out the activity, we would multiply this number by the decay constant of thorium-232. Thorium-232 has a VERY long half-life, 14 billion years, and so to find the decay constant first we need to figure out how many seconds are in 14 billion years. (60 seconds/minute)*(60 minutes/hour)*(24 hours/day)*(365.24 days/year)*(14 billion years) = 4.4 x 1017 seconds. The decay constant of thorium-232 is then (ln 2)/(4.4 x 1017 seconds) = 1.57 x 10-18/second. The activity of one gram of thorium is then (2.60 x 1021/gram)*(1.57 x 10-18/second) = 4080/second/gram = 4080 Bq/gram.
This is the specific activity of pure thorium-232: 4080 Bq/gram, and it is rather low. That’s because thorium-232 has such an exceptionally long half-life.
Here is a basic principle:
The longer the half-life, the LESS radioactive a material is, per unit mass.
The shorter the half-life, the MORE radioactive a material is, per unit mass.
Now let’s consider the specific activity of another substance, like bismuth-213 for instance. A mole of bismuth-213 has a mass of 213 grams, so one gram of bismuth-213 has 1/213th of a mole of substance. That means that one gram of bismuth-213 has (0.6022 x 1024)*(1/213) = 2.83 x 1021 atoms per gram. Bismuth-213 has a half-life of 45.5 minutes, so it has a decay constant of (ln 2)/(45.5 minutes*60 seconds/minute) = 2.54 x 10-4/sec. The specific activity of Bi-213 is then (2.83 x 1021/gram)*(2.54 x 10-4/sec) = 718 x 1015/second/gram, or 718 x 1015 Bq/gram.
Thus, bismuth-213 is almost a million billion times more radioactive than thorium-232. Why? Because its half-life is so much smaller (45.5 minutes versus 14 billion years), or examined another way, because its decay constant is so much greater (2.54 x 10-4/sec versus 1.57 x 10-18/sec).
From these descriptions of the specific activity of a sample we can nearly deduce the algebraic formula we would use to calculate specific activity.
In this equation, SA is the specific activity, in becquerels per gram, of some substance that we would like to study. NA is Avogadro’s number (0.6022 x 1024) and M is the atomic mass of the radionuclide in question. M was 232 for thorium-232 and 213 for bismuth-213, and would be whatever the atomic mass of the substance in question was. If you know the decay constant, you can use it directly, as in the middle expression, or if you only know the half-life (in seconds) you can use the expression on the right.
Specific activity is a tremendously useful quantity to know, because it allows us to move directly between concepts of mass and activity. Once we know how many grams of a substance we might have, we can calculate the activity of that substance. Conversely, if we have a measured activity, we can know how many grams we are dealing with. For substances with short half-lives, like the bismuth-213 we used in the example, we will almost certainly be given a description of the amount in terms of activity rather than in terms of mass. A doctor might administer a 0.5 GBq treatment of Bi-213 to a patient, never once stopping to think that this is a mass of only 70 trillionths of a gram.
There is another unit of activity that is commonly used, especially in older documents. It is the curie, and it was originally defined as the activity in one gram of radium-226. The curie has since been redefined as 37 billion becquerels of activity, or 37 GBq. To convert activity values from becquerels to curies, simply divide the value in becquerels by 37 billion. To convert activity in curies to becquerels, multiply by 37 billion. The curie has the abbreviation Ci, and is often expressed in smaller units like millicuries (mCi) and microcuries (?Ci). Unlike a becquerel, a curie is a significant amount of activity.
Here is a table of specific activities of different radionuclides:
Example problem 1: The average adult human body contains 250 g of normal potassium, of which 0.012% is the radioactive beta emitter potassium-40 (half-life 1.3 x 109 yr). What is the activity of potassium-40 in the body?
The mass of 40K in the body is (250)*(0.00012) = 0.03g. From the table, the specific activity of potassium-40 is 254,000 Bq/gm. Therefore, the activity of potassium-40 in the body is (254000 Bq/gm)*(0.03gm) = 7600 Bq.
Example problem 2: A sample of radioactive iodine-131 (half-life of 8.02 d) has an activity of 37 GBq. What is its mass?
From the table, the specific activity of iodine-131 is 4.6 x 1015 Bq/gm. An activity of 37 GBq corresponds to a mass of (37 x 109 Bq)/(4.6 x 1015 Bq/gm) = 8.0 x 10-6 gm or 8 micrograms of iodine-131.
I wish I’d seen your website about ten years ago when I was working as a radiochemist. I had to figure this same thing ie. specific activity of natural uranium to make a comparison between two analytical methods used in the lab. I came up with essentially the same result, though I was using the units of picocuries rather than becquerels(though you show how easily those are converted). Your site would have saved me some time.
Mike