Question

If the half-life of a radioactive atom is 3,500 years. How old is the substance if the parent-daughter ratio is 1:127?

Answer 1

2.447 × 10⁴ years

Step-by-step explanation:

Step 1: Given data

• Half-life of the radioactive atom (t1/2): 3,500 years

• Parent-daughter ratio ([A]/[A]₀): 1:127 (1/127)

Step 2: Calculate the rate constant

Radioactive decay follows first-order kinetics. We can calculate the rate constant (k) using the following equation.

k = ln2 / (t1/2) = ln2 / 3,500 y = 1.980 × 10⁻⁴ y⁻¹

Step 3: Calculate the time elapsed (t)

For first-order kinetics, we will use the following expression.

ln ([A]/[A]₀) = -k × t

t = ln ([A]/[A]₀)/ (-k)

t = ln (1/127) / (1.980 × 10⁻⁴ y⁻¹) = 2.447 × 10⁴ y