Answer:
8,000 years.
Clarification:
- Radioactive isotopes are known to decay following first-order kinetics.
- The half-life is defined as the duration required for a reactant's concentration to halve.
- When a reactant starts with an initial concentration of [A₀], at the half-life it will reach a concentration of ([A₀]/2).
- Furthermore, for first-order decay, the half-life does not depend on the starting concentration.
Part 1: What is the half-life of the element? Explain how you determined this.
- The half-life of this element equals 1,600 years.
This means the reactant reduces from 56.0 g to its half (28.0 g) in 1,600 years.
Thus, the half-life for this sample is 1,600 years.
Part 2: How long would it take for 312 g of the sample to decay down to 9.75 grams? Show your work or explain your answer.
- Using the equations for first-order reactions:
k = ln(2)/(t1/2) = 0.693/(t1/2).
Where k is the reaction's rate constant.
t1/2 represents the half-life of the reaction.
∴ k =0.693/(t1/2) = 0.693/(1,600 years) = 4.33 x 10⁻⁴ year⁻¹.
- Utilizing the integral formula for first-order reaction:
kt = ln([A₀]/[A]),
with k being the reaction's rate constant (k = 4.33 x 10⁻⁴ year⁻¹).
t is the duration of the reaction (t =??? year).
[A₀] indicates the initial concentration of the sample ([A₀] = 312.0 g).
[A] shows the concentration left after decay ([A] = 9.75 g).
∴ t = (1/k) ln([A₀]/[A]) = (1/4.33 x 10⁻⁴ year⁻¹) ln(312.0 g/9.75 g) = 8,000 years.
