Radioactivity of a sample (z = 22) decreases 90% after 10 years. What will be the half-life of the sample?a)5 yearsb)2 yearsc)3 yearsd)10 yearsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Half-life is defined as the time it takes for half of the radioactive nuclei in a sample to decay. In this case, we are given that the radioactivity of a sample with atomic number (z) 22 decreases by 90% after 10 years. We need to determine the half-life of the sample.

Let's assume that initially, the sample had N0 radioactive nuclei. After 10 years, the radioactivity of the sample decreases by 90%, which means only 10% of the radioactive nuclei remain.

Therefore, the number of radioactive nuclei remaining after 10 years is given by:
N10 = 0.10 * N0

We can express the ratio of the number of radioactive nuclei remaining after a certain time (Nt) to the initial number of radioactive nuclei (N0) as:
Nt/N0 = (1/2)^(t/T)

Where T is the half-life of the sample and t is the time.

In this case, we can substitute N10 = 0.10 * N0 and t = 10 years into the equation to solve for T:

N10/N0 = (1/2)^(10/T)

0.10 = (1/2)^(10/T)

Taking the logarithm of both sides:

log(0.10) = log((1/2)^(10/T))

log(0.10) = (10/T) * log(1/2)

Using the property log(a^b) = b * log(a), we can rewrite the equation as:

log(0.10) = (10/T) * (-log(2))

Now we can solve for T by rearranging the equation:

T = (10 * log(2)) / log(0.10)

Using a calculator, we find that T ≈ 3.32 years.

Since we are looking for the half-life in years, the closest option is c) 3 years.

Answer

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