Start with a hypothetical sample of 1,024 atoms of a radioactive isotope. Fill out the following table:
# of half lives:0 Remaining Parent: 1024 Daughter: 0 Simplified P:D Ratio -
1
2
3
4
5
6
7
8
9
10
1024 is 2^10
after each half-life , half of the parent isotope has become the daughter
1 ... 512 ... 512 ... 1
2 ... 256 ... 768 ... 1/3
3 ... 128 ... 896 ... 1/7
etc.
To fill out the table, we need to understand how radioactive decay works. Radioactive decay is a random process in which the unstable nuclei of radioactive isotopes change into more stable isotopes by emitting radiation. The rate at which this decay occurs is measured by the half-life of the isotope, which is the time it takes for half of the original sample to decay.
Let's start with the initial sample of 1,024 atoms of the radioactive isotope:
# of half lives: 0
Remaining Parent: 1024
Daughter: 0
At the start, since no time has passed, all 1,024 atoms are still the parent isotope, and none of them have decayed yet. Therefore, the daughter isotope count is 0.
Now, let's calculate the remaining parent and daughter isotopes after each half-life:
# of half lives: 1
Remaining Parent: 512 (half of the previous count)
Daughter: 512 (the decayed portion of the parent isotope)
# of half lives: 2
Remaining Parent: 256 (half of the previous count)
Daughter: 768 (sum of previous daughter count and half of previous parent count)
# of half lives: 3
Remaining Parent: 128
Daughter: 896
# of half lives: 4
Remaining Parent: 64
Daughter: 960
# of half lives: 5
Remaining Parent: 32
Daughter: 992
# of half lives: 6
Remaining Parent: 16
Daughter: 1008
# of half lives: 7
Remaining Parent: 8
Daughter: 1016
# of half lives: 8
Remaining Parent: 4
Daughter: 1020
# of half lives: 9
Remaining Parent: 2
Daughter: 1022
# of half lives: 10
Remaining Parent: 1
Daughter: 1023
After 10 half-lives, we can see that the parent isotope count decreases exponentially while the daughter isotope count increases. The simplified P:D ratio is as follows:
1:1023, 2:1022, 3:1021, 4:1020, 5:1019, 6:1018, 7:1017, 8:1016, 9:1015, 10:1014
Each ratio represents the remaining parent isotope count followed by a colon and the daughter isotope count.