Question

How much time has passed if the activity of the sample has dropped to 775 cpm?

Answer 1

The time elapsed can be calculated using the formula
`\( t = (\ln(N_0/N))/(\lambda) \),` where
`\( t \)` is the time elapsed,
`\( N_0 \)` is the initial activity,
`\( N \)` is the final activity, and
`\( \lambda \)` is the decay constant. Without specific values, a precise time cannot be determined.

Step-by-step explanation:

The time elapsed
`(\( t \))` in radioactive decay can be calculated using the formula mentioned above. The natural logarithm
`(\( \ln \))` of the ratio of the initial activity
`(\( N_0 \))` to the final activity
`(\( N \))` is divided by the decay constant
`(\( \lambda \)).` The decay constant is unique to each radioactive isotope and represents the probability of decay per unit time.

To accurately calculate the time elapsed, the specific decay constant of the radioactive isotope in question is needed. For example, if dealing with carbon-14
`(\( ^(14)C \))` , which has a half-life of approximately 5730 years, the decay constant
`(\( \lambda \))` can be determined as
`\( \lambda = \frac{\ln(2)}{\text{half-life}} \).` Substituting
`\( \lambda \)` and the given activity values into the formula will yield the time elapsed.

Understanding radioactive decay and its mathematical expressions is vital in fields such as archaeology, geology, and nuclear medicine. It allows scientists to estimate the age of materials and track the decay of isotopes in various applications.