assume that a particular radioactive-isotope has a half-life of 100 million years. Also assume that the parent isotope is incorporated into the minerals found in igneous rocks and that the only daughter atoms in the rock formed from the decay of the parent isotope. Measurements show that one mineral in the igneous rock contains 5 million atoms of the parent isotope and 35 million atoms of daughter.
a) what is the age of this mineral in years?
b) what is the age of rock?
a) To determine the age of the mineral, we can use the ratio of parent to daughter atoms and the half-life of the radioactive isotope.
1. Calculate the ratio of parent to daughter atoms:
- Parent atoms = 5 million atoms
- Daughter atoms = 35 million atoms
The ratio of parent to daughter atoms is 5:35, which can be simplified to 1:7.
2. Determine the number of half-lives that have occurred:
- Since the half-life is 100 million years, each half-life reduces the parent atoms by half.
- In this case, starting with 5 million atoms, we need to determine how many times we can halve the number until we reach 1 atom.
Let's calculate the number of half-lives:
- Number of half-lives = log base 2 of (final number / initial number) = log base 2 of (1 / 5) = -2.3219
We round this to the nearest whole number, which gives us -2 half-lives.
3. Calculate the age of the mineral:
- Age = number of half-lives * half-life duration = -2 * 100 million years = -200 million years
The negative sign indicates that this result is an approximation because the mineral or rock sample is assumed to be older than the half-life.
b) The age of the rock can be assumed to be the same as the mineral, assuming the mineral formed at the same time as the rock and experienced no external disturbances or losses of the radioactive isotopes. Therefore, the age of the rock in this example would also be approximately 200 million years.
To determine the age of the mineral and the rock, we can use the concept of radioactive decay and the ratio of parent to daughter isotopes. Here's how you can calculate the ages:
a) Age of the mineral:
The half-life of the radioactive isotope is given as 100 million years, which means that after every 100 million years, half of the parent isotopes will decay into daughter isotopes. In this case, we have 5 million atoms of the parent isotope and 35 million atoms of the daughter isotope.
The ratio of parent to daughter isotopes can be calculated by dividing the number of parent isotopes by the number of daughter isotopes:
Ratio of parent to daughter = (Number of parent isotopes) / (Number of daughter isotopes)
= 5 million / 35 million
= 1/7
Now, we can determine the number of half-lives that have occurred by comparing the ratio of parent to daughter isotopes to the half-life ratio (1/2):
(Number of half-lives) = log(base 1/2) of (Ratio of parent to daughter)
= log(base 1/2) of (1/7)
≈ 2.807
Since each half-life is 100 million years, we can multiply the number of half-lives by the half-life duration to find the age of the mineral:
Age of the mineral = (Number of half-lives) * (Half-life duration)
= 2.807 * 100 million years
≈ 280.7 million years
Therefore, the age of the mineral is approximately 280.7 million years.
b) Age of the rock:
Assuming that the mineral formed when the rock solidified, the age of the rock can be considered the same as the age of the mineral. Hence, the age of the rock would also be approximately 280.7 million years.
Please note that this calculation assumes a closed system, where there has been no addition or removal of parent or daughter isotopes since the rock's formation.