What is the age in years of a mineral sample that has a mass ratio of ⁴⁰Ar to ⁴⁰K of 0.330? Potassium-40 decays to argon-40 with a half-life of 1.27 × 10⁹ yr.

Question

Quizplus · Accepted Answer

We can use the decay equation for Potassium-40 to calculate the ratio of remaining Potassium-40 to the initial amount. Let N0 be the initial amount of Potassium-40 and Nf be the remaining amount after time t, then:

Nf = N0 * (1/2)^(t/T1/2)

where T1/2 is the half-life of Potassium-40.

Similarly, we can calculate the ratio of remaining Argon-40 to the initial amount produced by the decay of Potassium-40 using the same equation, since the two isotopes are produced in a 1:1 ratio. Let M0 be the initial amount of Argon-40 and Mf be the remaining amount after time t, then:

Mf = M0 * (1/2)^(t/T1/2)

The ratio of remaining Argon-40 to the initial amount produced by the decay of Potassium-40 is then:

Mf/M0 = (1/2)^(t/T1/2)

We are given the mass ratio of Argon-40 to Potassium-40, which is proportional to the above ratio:

Mf/M0 = (Ar/K) * (40/39.962)

where (Ar/K) is the given mass ratio.

Substituting the given values, we get:

0.330 = (1/2)^(t/1.27e9) * (40/39.962)

Solving for t, we get:

t = 5.23e8 years

Therefore, the age of the mineral sample is 5.23 × 10^8 years, which is option B.

What Is the Age in Years of a Mineral Sample