The table shows the amount of radioactive element remaining in a sample over a period of time.

radioactive decay rate

amount of radioactive sample

(grams) time

(years)

56.0 0

47.1 400

39.6 800

33.3 1,200

28 1,600

part 1: what is the half-life of the element? explain how you determined this.

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.

8,000 year.

Explanation:

Part 1: What is the half-life of the element? Explain how you determined this.

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.

So, the half-life of the sample is 1,600 years.

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.

k = ln(2)/(t1/2) = 0.693/(t1/2).

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.

kt = ln([A₀]/[A]),

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).

∴ t = (1/k) ln([A₀]/[A]) = (1/4.33 x 10⁻⁴ year⁻¹) ln(312.0 g/9.75 g) = 8,000 year.

Half life of the sample= 1600 years

Part 2: 8000 years

Explanation:

The half life of a radioactive substance is the time taken for the substance to decay to half its initial amount.

Since 56.0g of radioactive material was present at t= 0 years. Half of 56.0g is 28.0g hence the time taken to have 28.0g of the sample left is the half life of the material.

If the original sample contained 312g of radioactive material

First half life. 156g left

Second half life. 78 g left

Third half life. 39gleft

Fourth half life. 19.5 g left

Fifth half life. 9.75 g

If each half life takes 1600 years then five half life will take 5×1600= 8000 years

i'm assuming that the correct answer is indeed the first one/ "a"..

it is the most commonly accepted one