Question

A sample of a radioactive substance decayed to 97% of its original amount after a year. (Round your answers to two decimal places.) (a) What is the half-life of the substance

Answer 1

23 years

Explanation:

Step 1: Calculate the rate constant (k) for the radioactive decay

A radioactive substance with initial concentration [A]₀ decays to 97% of its initial amount, that is, [A] = 0.97 [A]₀, after t = 1 year. Considering first-order kinetics, we can calculate the rate constant using the following expression.

ln [A]/[A]₀ = - k.t

k = ln [A]/[A]₀ / -t

k = ln 0.97 [A]₀/[A]₀ / -1 year

k = 0.03 year⁻¹

Step 2: Calculate the half-life of the substance

We will use the following expression.

`t_(1/2)` = ln2/ k = ln 2 / 0.03 year⁻¹ = 23 years