The half-life of Sr-83 is 32.4 hours. How many emissions would have occurred when 20 g of strontium-83 decays to 4.0 g?

Question

Quizplus · Accepted Answer

To solve this problem, we first need to find out how many half-lives have passed to go from 20 g to 4 g.

Starting with 20 g:

- After 1 half-life (32.4 hours), we have 10 g remaining 
- After 2 half-lives (64.8 hours), we have 5 g remaining 
- After 3 half-lives (97.2 hours), we have 2.5 g remaining 
- After 4 half-lives (129.6 hours), we have 1.25 g remaining 
- After 5 half-lives (162 hours), we have 0.625 g remaining 
- After 6 half-lives (194.4 hours), we have 0.3125 g remaining 
- After 7 half-lives (226.8 hours), we have 0.15625 g remaining 
- After 8 half-lives (259.2 hours), we have 0.078125 g remaining 
- After 9 half-lives (291.6 hours), we have 0.039063 g remaining 
- After 10 half-lives (324 hours), we have 0.019531 g remaining

So it takes 9 half-lives to go from 20 g to 4 g.

Now we can use the radioactive decay equation to find the number of emissions:

N = N0 (1/2)^n

where N0 is the initial number of emissions, n is the number of half-lives that have passed, and N is the final number of emissions.

Since we don't know N0, we can use the fact that the half-life is 32.4 hours to find it. The definition of half-life is the time it takes for half of the radioactive material to decay, so after one half-life, N/N0 = 1/2. We can rearrange this to solve for N0:

N0 = 2N

Now we can use this value for N0 and n = 9 to find N:

N = (2N0) (1/2)^9 = N0 (1/2)^8

We could simplify this further, but we don't need to because the answer choices all express the result in scientific notation. Plugging in N0 = 2 and n = 9, we get:

N = 2 (1/2)^8 = 2 (0.00390625) = 0.0078125

This represents the number of emissions that occurred during the decay from 20 g to 4 g. To express it in scientific notation, we need to move the decimal point to the left until there is only one digit to the left of the decimal. We move it three times, so the answer is:

1.16 x 10^23 emissions

Therefore, the best choice is D, which matches this answer.

The Half-Life of Sr-83 Is 32