Radioactive Decay

1. Physical half-life (T_p)
1. This is the physical decay of a radionuclide
2. It is defined as the time in which it takes number of atoms to disintegrate to ½ the original amount
3. The rate of decay remains constant and cannot be altered, even with the application of temperature, pressure, or chemical interaction
4. As a radionuclide decays over time fewer of the original atoms remain. This remaining fraction of atoms, that decay per unit time, is known as the decay constant (λ)
1. The larger the λ per unit time the shorter the half-life (T_½)
2. This has an inverse relationship
3. While the literature refers to λ as being “constant” in reality it is a statistical value, which means that there maybe a slight variation in the decay time
4. Refer to the chart below an note that the y-axis represents the number of radioatoms and the x-axis represents the number of T_½

5. The decay formula can be expressed in the following formula.


1. A_t = number of atoms at a certain point in time
2. A_o = number of atoms originally present
3. e = base of the natural logarithm, 2.718
4. λ decay constant
5. t = time
6. Expanding “– (λt)” the formula can be re-written. Note 0.693 is the natural log of 2


Let’s apply the decay formula to see how this works!

Question: You have 100 mCi of ^99mTc. How much activity remains in 2 hours?


These entries are used on scientific calculator (TI-30 IIS)

First calculate e^-λt

1. Enter 100
2. Hit the 2nd or shift key
3. Select the ln button (e)
4. Select the +/- button
5. Type the natural log of 2 which is " 0.693"
6. Now enter the time (t), "2" (hours)
7. Then choice the T1/2 "6.02" (hours)
8. (This generates a decay factor, e^-λt which is 0.794 )
9. Enter parenthesis - ")"
10. Equal sign "="
11. A_t = "79.4" mCi

Another approach to decay is the formula below which allows you to decay into the future. If you want to decay into the past make the exponent negative


1. Either way you calculate this your answer will still be same
2. It is up to you to decide which method you wish to apply when calculating decay

2. Here are several decay questions for you to work on. (¹²³I T1/2 = 13 hours and ¹³¹I T1/2 = 8 days)
1. Determine the amount of activity remaining in a vial a ¹²³I if the initial capsule contained 221 μCi at 12N. How much remains at 8:30am the following morning. Answer
2. ^99mTc has 1.0 mCi of activity. Decay this every half-hour, up to 6 hours . Put this information on a separate piece of paper and place that data on a chart, showing time elapsed and the amount of activity remaining. Answer
3. If you had 100 mCi of ¹³¹I today (12N), how much did you have two weeks ago (at 12N) Answer

3. Yet another application of the decay formula - when working with ^99mTc apply its 1 hour decay factor, 0.891
   1. t = time
   2. Time is s ratio of the amount of minutes into the future divided by 60 minutes
   3. Assume you have 100 mCi at 7am, how much activity remains at 8:41? Answer
4. Biological half-life (T_b)
   1. This is the time it takes the body to eliminate ½ of the radioactive material
   2. It has nothing to do with T_p, however, it is related to T_p (more on that in a moment)
   3. Methods in which the body eliminates a radioactive compound would include: urine, feces, perspiration, and/or exhalation
   4. When related to the T_p the biological effects of a radioactive compound is limited if the T_b is very short (even when the T_p is very long)


5. Effective half-life (T_e)
   1. When combining the T_p and T_b the result is T_e
   2. Will allow you to calculate amount of radiation absorbed by the body or even a specific organ
   3. If T_p and T_b are known then the following formula can be used to calculate T_e

 


6. If T_p and T_e are known then this formula can be used to calculate T_b, the biological half-life

7. Calculate the following radioactive compounds and discuss the effects of T_e, T_b, and T_p
1. ^99mTcMDP
2. ¹²³I
3. ^99mTc sulfur colloid