pls help me :))

Question

Anwita is on a geology field trip and finds a rock that contains 53 % of the original potassium-40, while remaining amount of potassium-40 had decayed into argon-40. The half-life of potassium-40 is 1.25 × 10⁹ years. Assume the only decay product is argon-40. How old is the rock in billions of years (10⁹ years)?

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Answer 1

Hello Gavin!

Knowns:

N = 53% (Leftover parent product in percentage)

D₀ = 0% (Daughter product at the start, assumed here to be 0 to get true age)

D = 47% (Percentage of parent product that has decayed - Current daughter product)

ℑ = 1.25x10⁹ [years]

Formulas:

λ = ln(2) / ℑ [1/years] = 5.54x10^-10 [1/years]

D = D₀ + N(e^λt-1) Formula for parent/daughter decay

Analysis:

e^λt = (D/N) + 1

λt = ln[ (D/N) + 1 ]

Therefore:

t = ln[ (D/N) + 1 ] / λ

Plugging in:

t = ln[ (47% / 53%) + 1 ] / ( 5.54x10^-10 [1/years] )

~ 1.15x10⁹ [years] or 1.15 billion years

Note: This is just shy of 1 half-life. However, notice that 47% of the daughter product exists, which is also just shy of 1 half-life. This should help mentally check that you have done the right process. Hope this helps!