Analysis of Potassium-40 / Argon-40 determines that 37.5% of the Potassium-40 remains and 62.5% has decayed to Argon-40. What is the age of the rock? A. 1.3 billion years B. 0.65 billion years C. 1.95 billion years D. 0.163 billion years

Option C is correct.

t = 1.95 billion years.

Explanation:

Radioactive decay follows a first order reaction kinetics.

On solving the dynamic equation (the differential equation), this is obtained

C(t) = C₀ e⁻ᵏᵗ

C(t) = amount of radioactive material remaining after time t = 37.5%

C₀ = Initial amount of radioactive material = 100%

t = time that has passed = ?

k = decay constant.

For a first order reaction, the decay constant is related to the half life through the relation

k = (In 2)/T

T = half life = 1.38 billion years

k = (In 2)/1.38

k = 0.5023 per billion years.

C(t) = C₀ e⁻ᵏᵗ

0.375 = e⁻ᵏᵗ

e⁻ᵏᵗ = 0.375

In e⁻ᵏᵗ = In 0.375 = -0.981

-kt = -0.981

t = (0.981/0.5023) = 1.95 billion years.

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