Problem 40 - Chemistry and Chemical Reactivity

Question

Radioactive cobalt-60 is used extensively in nuclear medicine as a \(\gamma\) -ray source. It is made by a neutron capture reaction from cobalt-59, and it is a \(\beta\) emitter; \(\beta\) emission is accompanied by strong \(\gamma\) radiation. The half-life of cobalt-60 is 5.27 years. (a) How long will it take for a cobalt-60 source to decrease to one eighth of its original activity? (b) What fraction of the activity of a cobalt-60 source remains after 1.0 year?

Step-by-Step Solution

Answer

(a) 15.81 years. (b) Approximately 88% remains.

Step 1: Identify the Relationship with Half-life

We're given that the half-life of cobalt-60 is 5.27 years. The formula for a half-life relationship is that if a substance decreases to half its activity in one half-life, over successive half-lives it will continue to decrease by half.

Step 2: Calculate the Time for Activity to Decrease to One Eighth

Since the activity decreases by half every 5.27 years, to find when it decreases to one-eighth, use the fact that \[\frac{1}{2^n} = \frac{1}{8}\].This gives us \(n = 3\) half-lives because \(2^3 = 8\). The time is \(3 \times 5.27 = 15.81\) years.

Step 3: Use Exponential Decay Formula for Fraction Remaining

The remaining activity can be calculated using the decay formula \[N(t) = N_0\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\],where \(t_{1/2}\) is the half-life, \(t\) is the time elapsed, and \(N_0\) is the initial amount. Evaluate this for 1.0 year.

Step 4: Calculate the Fraction After 1 Year

Using \(t = 1.0\) year:\[\text{Fraction of the activity remaining} = \left(\frac{1}{2}\right)^{\frac{1.0}{5.27}}\].Calculate this:\[\text{Remaining fraction} \approx 0.88\,\text{or}\, 88\%\].

Key Concepts

Nuclear Medicine

Nuclear medicine is a fascinating field that uses radioactive elements to diagnose and treat diseases. It leverages the properties of radioactivity to target specific areas in the body. For example, cobalt-60 is commonly used due to its emission of gamma rays.

• These radioactive compounds are introduced into the body and travel to where they're needed.
• The radiation they emit can be detected with special cameras, helping doctors to see how organs or tissues are functioning.
• In therapy, the radiation can kill cancerous cells or shrink tumors.

Understanding how these radioactive substances decay over time is vital, as it affects the dosage and timing of medical treatments. Cobalt-60's properties make it particularly useful, thanks to its predictable decay rate and intense gamma radiation.

Half-life

Half-life is a concept that describes how quickly a radioactive substance decays. It is the time required for half of the radioactive atoms in a sample to decay. For cobalt-60, this key characteristic is 5.27 years.

• This means every 5.27 years, the amount of cobalt-60 left is half what it was at the start of that period.
• The half-life remains constant, meaning the decay happens at a predictable rate, which is useful in planning medical treatments and ensuring safety.

Knowing the half-life of a substance helps in calculating how much of the radioactive material will remain after a certain period, and this can be applied to determine how long a source lasts or when it might need to be replaced.

Gamma Radiation

Gamma radiation is one of the most energetic forms of electromagnetic radiation, produced during radioactive decay. It's highly penetrating, making it both useful and hazardous.

• In nuclear medicine, gamma rays are used to obtain clear images of the inside of the body because they can pass through tissues.
• This allows medical practitioners to detect abnormalities without invasive procedures.
• However, because gamma radiation can damage living tissue, its use must be carefully controlled.

Cobalt-60 emits strong gamma radiation during its decay process, making it effective in both diagnostic imaging and cancer treatment, where its energy is used to target and destroy diseased cells.

Beta Emission

Beta emission occurs when a radioactive nucleus emits a beta particle, either an electron or positron. This process is part of the transformation of an atom from one element to another, often releasing significant energy.

• Beta particles are smaller and less penetrating than gamma rays, but they can still pose a risk to biological tissues if not properly managed.
• They play a crucial role in changing the properties of an atom, as seen with cobalt-60, which transforms from cobalt-59 through neutron capture and beta decay.
• In the medical field, beta emissions can be used to destroy targeted cells, which is useful in certain cancer treatments.

Understanding beta emission helps in safely applying nuclear materials in therapeutic settings, ensuring effective treatment while minimizing potential harm to healthy tissues.