**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>.