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Chapter 21: Problem 37

Cobalt-so is a strong gamma emitter that has a half-life of 5.26 yr. The cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75\(\%\) of the original sample. If an original sample was purchased in June \(2016,\) when will it be necessary to replace the cobalt-60?

Short Answer

Expert verified
The Cobalt-60 in the radiotherapy unit must be replaced when its radioactivity falls to 75% of the original sample. Knowing the half-life of Cobalt-60 is 5.26 years and using the decay formula, we find that the time required for the radioactivity to decrease to 75% is approximately 2.082 years. Adding this time to the purchase date (June 2016), we conclude that it will be necessary to replace the Cobalt-60 in July 2018.

Step by step solution

01

List down given information

We know the following information: 1. Half-life of Cobalt-60 (T1/2) is 5.26 years 2. Radioactivity falls to 75% of the original sample 3. Purchase date: June 2016.
02

Calculate the decay constant (k)

The decay constant (k) can be calculated using the half-life formula: \(T_{1/2} = \frac{ln(2)}{k}\) We can solve for k as follows: \(k = \frac{ln(2)}{T_{1/2}}\) Plugging in the given half-life: \(k = \frac{ln(2)}{5.26}\) \(k \approx 0.1319\,yr^{-1}\)
03

Use the decay formula to find the time

The decay formula is given by: \(N(t) = N_0e^{-kt}\) Here, N(t) is the radioactivity at time t, N0 is the initial radioactivity, k is the decay constant, and t is the time elapsed. We need to find the time t when the radioactivity falls to 75% of its initial radioactivity. Hence, \(N(t) = 0.75N_0\). Plugging in the values, we get: \(0.75N_0 = N_0e^{-0.1319t}\) Now, we can solve for t: \(\frac{0.75}{1} = e^{-0.1319t}\)
04

Solve for t (time)

Taking the natural logarithm of both sides, we get: \(ln(0.75) = -0.1319t\) Now, divide by -0.1319 to find the value of t: \(t = \frac{ln(0.75)}{-0.1319}\) \(t \approx 2.082\,years\)
05

Calculate the replacement date

The initial sample was purchased in June 2016, so we need to add the time t (2.082 years) to this date to find the replacement date. June 2016 + 2.082 years ≈ June 2016 + 2 years and 1 month ≈ July 2018 Hence, it will be necessary to replace the Cobalt-60 in the radiotherapy unit in July 2018.

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