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.

1. The half-life is 1600 years.

2. 8000 years.

Explanation:

1. Determination of the half-life of the element.

The half-life of an element is the time taken for half the element to decay.

Considering the table given above:

Original amount (N₀) of element = 56 g

Next, we shall determine the amount remaining after the 1st half-life of 56 g by dividing it by 2. This is illustrated below:

Original amount (N₀) of element = 56 g

Amount remaining (N) after the 1st half-life =?

N = N₀ / 2

N = 56 / 2

N = 28 g

Thus, 28 g of the element is remaining after the 1st half-life.

Comparing the amount remaining (i.e 28 g) after the 1st half-life with those given in the table (i.e in the question), we can conclude that the half-life of the element is 1600 years since it gives the time for the element to become half its original size (i.e 28 g)

2. Determination of the time taken for 312 g of the sample to decay to 9.75 g.

We'll begin by calculating the number of half-lives (n) taken for 312 g of the sample to decay to 9.75 g

This can be obtained as illustrated below:

Original amount (N₀) = 312 g

Amount remaining (N) = 9.75 g

Number of half-lives (n) =?

N = 1/2ⁿ × N₀

9.75 = 1/2ⁿ × 312

Cross multiply

9.75 × 2ⁿ = 312

Divide both side by 9.75

2ⁿ = 312 / 9.75

2ⁿ = 32

Express 32 in index form with 2 as the base.

2ⁿ = 2⁵

n = 5

Thus, the number of half-lives taken for 312 g of the sample to decay to 9.75 g is 5 half-lives.

Gina, we shall determine the time taken. This can be obtained as follow:

1 half-life = 1600 years

Therefore,

5 half-life = 5 × 1600

5 half-life = 8000 years.

Therefore, it will take 8000 years for 312 g of the sample to decay to 9.75 g.

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