Part 1: What is the half-life of the element? Explain how you determined this.
Part 2: How long would it take 308 g of the sample to decay to 4.8125 grams? Show your work or explain your answer.

1. The half-life is 22 years.

2. 132 years.

Explanation:

1. Determination of the the half-life.

The half-life of a substance is the time taken for half the substance to decay.

Thus, we can obtain the half-life of the element by doing the following:

Original amount = 45 g

Half of the original amount = 45 /2 = 22.5 g

Compare this amount (i.e 22.5 g) with those given in the table above.

Thus, we can see that it will take 22 years for the original amount (i.e 45 g) to decay half its original amount (i.e 22.5 g).

Therefore, the half-life of the element is 22 years.

2. Determination of the time taken for 308 g of the sample to decay to 4.8125 grams.

We'll begin calculating the number of half-lives required for 308 g of the sample to decay to 4.8125 grams. This can be obtained as follow:

Original amount (N₀) = 308 g

Amount remaining (N) = 4.8125 g

Number of half-lives (n) =?

N = 1/2ⁿ × N₀

4.8125 = 1/2ⁿ × 308

Cross multiply

4.8125 × 2ⁿ = 308

Divide both side by 4.8125

2ⁿ = 308 / 4.8125

2ⁿ = 64

Express 64 in index for using 2 as the base

2ⁿ = 2⁶

n = 6

Thus, it will take 6 half-lives for 308 g of the sample to decay to 4.8125 grams.

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

1 half-life = 22 years.

Therefore,

6 half-lives = 6 × 22 years

6 half-lives = 132 years

Thus, it will take 132 years for 308 g of the sample to decay to 4.8125 g