Question

Cobalt-60 is a strong gamma emitter that has a half-life of \(5.26 \mathrm{yr}\). The cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to \(75 \%\) of the original sample. (a) If an original sample was purchased in June 2006, when will it be necessary to replace the cobalt-60? (b) How can you store cobalt-60 so that it is safe to handle?

Short answer

To find the replacement date, use the radioactive decay formula \(N(t) = N_0 (1/2)^{\frac{t}{T}}\), where \(T\) is the half-life (5.26 years) and \(N(t)\) is 75% of the original amount. Solve for \(t\): \(t ≈ 5.26\frac{\ln(0.75)}{\ln(1/2)} ≈ 2.17\) years. Since the initial purchase was in June 2006, the cobalt-60 needs to be replaced in June 2008. To safely store cobalt-60, use a shielded container made of lead or thick concrete, securely stored and labeled in a designated radioactive storage area. Regularly monitor radiation levels for safety.

Step-by-step solution

Identify the knowns and unknowns

We are given that the half-life of cobalt-60 is 5.26 years, and we want to know when its radioactivity falls to 75% (0.75) of the original amount. Let t be the time elapsed until the cobalt-60 needs to be replaced.

Set up the decay equation

Using the formula for radioactive decay, we have: N(t) = N₀ (1/2)^(t/T), Substituting the knowns into the equation, we get: 0.75 N₀ = N₀ (1/2)^(t/5.26), where N₀ is the original amount of cobalt-60.

Solve for t

To find the time (t) to replace the cobalt-60, we can divide both sides of the equation by N₀ and solve for t: 0.75 = (1/2)^(t/5.26). Now, take the natural logarithm of both sides: ln(0.75) = ln((1/2)^(t/5.26)). Use the property of logarithms to simplify the equation: ln(0.75) = (t/5.26) ln(1/2). Finally, isolate t to find the time elapsed: t = 5.26(ln(0.75) / ln(1/2)).

Calculate the time to replace the cobalt-60

Now, simply plug in the values into the equation and solve for t: t ≈ 5.26(-0.28768 / -0.69315) ≈ 2.17. So, it takes approximately 2.17 years for the radioactivity level to fall to 75% of the original sample.

Find the replacement date

The original sample was purchased in June 2006. Therefore, to find the replacement date, add the time elapsed (2.17 years) to the original purchase date: June 2006 + 2.17 years ≈ June 2008. So, it will be necessary to replace the cobalt-60 in June 2008.

Describe how to safely store cobalt-60

To safely store cobalt-60, it is important to place it in a shielded container made of materials that block gamma radiation, such as lead or thick concrete. The container should also be securely stored in a designated radioactive storage area, away from other materials and personnel, and properly labeled with appropriate warning signs. Regular monitoring of radiation levels in and around the storage area is essential to ensure the safety of personnel working with or near the cobalt-60.