Question

A rock sample from the moon includes a mineral that contains small amounts of the radioactive isotope Potassium-40 and its daughter element Argon-40 (half-life of 1.3 billion years). This mineral would not form with any Argon-40. Consider a crystal with 7 atoms of Argon-40 for every 1 atom of Potassium-40. How many atoms of Potassium-40 were present when the crystal formed for each atom of Potassium-40 that exists today

Answer 1

For a crystal with a 7:1 ratio of Argon-40 to Potassium-40, initially, there were 8 atoms of Potassium-40 for every atom of Potassium-40 that remains today, as this indicates that three half-lives have passed.

Step-by-step explanation:

The calculation of the number of atoms of Potassium-40 present when the crystal formed, based on the current ratio of Argon-40 to Potassium-40, involves understanding radioactive decay and the concept of half-lives. In the scenario provided, we have 7 atoms of Argon-40 for every 1 atom of Potassium-40, indicating that multiple half-lives have passed since the crystal initially formed without any Argon-40.

Because the half-life of Potassium-40 is 1.25 billion years, and we know that after one half-life, there would be an equal number of parent and daughter atoms, the presence of a 7:1 ratio of Argon-40 to Potassium-40 suggests that three half-lives have occurred. For each half-life that passes, the number of parent atoms reduces by half, therefore:

• After 1 half-life: 2 atoms of K-40 for each atom of Ar-40
• After 2 half-lives: 4 atoms of K-40 for each atom of Ar-40
• After 3 half-lives: 8 atoms of K-40 for each atom of Ar-40

Thus, initially, for each atom of Potassium-40 that remains today, there were 8 atoms of Potassium-40 when the crystal formed.

Answer 2

There were originally 8 atoms of Potassium-40.

Step-by-step explanation:

The half-life of a radioactive material is the time taken for half the original material to decay or the time required for a quantity of the radioactive substance to reduce to half of its initial value.

If the original material formed without any Argon-40, it means that the atoms originally present were Potassium-40 atoms.

Presently, there are 7 Argon-40 atoms for every 1 of Potassium-40, we can deduce the number of half-lifes the Potassium-40 has undergone as follows :

After one half-life, (1/2) there will be one Potassium-40 atom for every Argon-40 atom.

After a second half life, 1/2 × 1/2 = 1/4: there will be one Potassium-40 atom for every three atoms of Argon-40.

After a third half-life, 1/4 × 1/2 = 1/8: there will be one Potassium-40 atom for every 7 atoms of Argon-40.

Since there are 1/8 atoms of Potassium-40 presently, there were originally 8 atoms of Potassium-40.