Question

Potassium-40 decays to argon-40 with a half-life of 1.27 × 109 yr. The age of a mineral sample that has a mass ratio of 40ar to 40k of 0.812 is ________ yr.

Answer 1

Answer: The age of the mineral sample is $1.09\times 10^9yrs$

Explanation:

The equation used to calculate rate constant from given half life for first order kinetics:

$t_{1/2}=\frac{0.693}{k}$

where,

$t_{1/2}$ = half life of the reaction = $1.27\times 10^9yrs$

Putting values in above equation, we get:

$k=\frac{0.693}{1.27\times 10^9yrs}=5.46\times 10^{-10}yrs^{-1}$

We are given:

Mass ratio of K-40 to Ar-40 = 0.812 : 1.00

Rate law expression for first order kinetics is given by the equation:

$k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}$

where,

k = rate constant = $5.46\times 10^{-10}yr^{-1}$

t = time taken for decay process = ? yr

$[A_o]$ = initial amount of the sample = [1.00 + 0.812] = 1.812 grams

[A] = amount left after decay process = 1.00 grams

Putting values in above equation, we get:

$5.46\times 10^{-10}=\frac{2.303}{t}\log\frac{1.812}{1}\\\\t=1.09\times 10^9yrs$

Hence, the age of the mineral sample is $1.09\times 10^9yrs$