Answer:
8,000 years.
Explanation:
- It is known that the decay of a radioactive isotope isotope obeys first order kinetics.
- Half-life time is the time needed for the reactants to be in its half concentration.
- If reactant has initial concentration [A₀], after half-life time its concentration will be ([A₀]/2).
- Also, it is clear that in first order decay the half-life time is independent of the initial concentration.
Part 1: What is the half-life of the element? Explain how you determined this.
- The half-life of the element is 1,600 years.
Half-life time is the time needed for the reactants to be in its half concentration.
The sample stats with 56.0 g and reaches its half concentration (28.0 g) after 1,600 years.
<em>So, the half-life of the sample is 1,600 years.</em>
<em></em>
Part 2: How long would it take 312 g of the sample to decay to 9.75 grams? Show your work or explain your answer.
- For, first order reactions:
<em>k = ln(2)/(t1/2) = 0.693/(t1/2).</em>
Where, k is the rate constant of the reaction.
t1/2 is the half-life of the reaction.
∴ k =0.693/(t1/2) = 0.693/(1,600 years) = 4.33 x 10⁻⁴ year⁻¹.
- Also, we have the integral law of first order reaction:
<em>kt = ln([A₀]/[A]),</em>
where, k is the rate constant of the reaction (k = 4.33 x 10⁻⁴ year⁻¹).
t is the time of the reaction (t = ??? year).
[A₀] is the initial concentration of the sample ([A₀] = 312.0 g).
[A] is the remaining concentration of the sample ([A] = 9.75 g).
<em>∴ t = (1/k) ln([A₀]/[A])</em> = (1/4.33 x 10⁻⁴ year⁻¹) ln(312.0 g/9.75 g) = <em>8,000 years</em>.
