Question

Cobalt-62 \(\left(\begin{array}{c}62 \\ 27\end{array}\right.\) o \()\) is a radionuclide with a half-life of 1.5 minutes. What fraction of an initial sample of \(\mathrm{Co}-62\) will remain after \(6 \mathrm{~min} ?\)

Short answer

After 6 minutes, \(\frac{1}{16}\) of the initial Cobalt-62 sample will remain.

Step-by-step solution

Determine the number of half-lives elapsed

To find the number of half-lives, we simply need to divide the given time (6 minutes) by the half-life of Cobalt-62, which is 1.5 minutes: Number of half-lives = \(\frac{6 \ \text{minutes}}{1.5 \ \text{minutes}}\)

Calculate the number of half-lives

Divide the total time by the half-life time: Number of half-lives = \(\frac{6}{1.5} = 4\)

Determine the remaining fraction

After one half-life, half of the initial sample would remain. After each subsequent half-life, the remaining amount will be halved again. In this case, there are four half-lives, so the remaining fraction of the sample can be calculated as: Remaining fraction = \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\) So, after 6 minutes, 1/16 of the initial Cobalt-62 sample will remain.

Key concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This decay occurs because elements like Cobalt-62 contain excess energy that needs to be released for the atom to become stable. During this process, particles or electromagnetic waves are emitted, leading to the transformation of the element into another substance.
For example, a radioactive atom might lose a particle in its nucleus, turning it into a completely different atom. This change is unpredictable for a single atom, but it becomes statistically reliable over a large number of atoms.
When studying radioactive decay, scientists focus on the rate of decay, commonly measured by the "half-life," which is the time it takes for half of the radioactive atoms in a sample to decay.

Cobalt-62

Cobalt-62 is an isotope of the element cobalt. Isotopes are variants of a particular chemical element that have the same number of protons but a different number of neutrons. This affects their stability.
Cobalt-62 specifically is a radionuclide, meaning it is radioactive and decays over time. It possesses a half-life of 1.5 minutes, which means that every 1.5 minutes, half of a given sample of Cobalt-62 will have decayed into another substance.

• Half-life allows scientists to predict how long it will take for the sample to decrease to a specific fraction of its original amount.
• Knowing the properties of Cobalt-62 is crucial in fields like medicine and research, where it is used for imaging and tracing techniques.

Understanding this property helps in calculating how much of the substance remains after a certain period by applying the half-life concept repeatedly.

Exponential Decay

Exponential decay describes the process in which quantities decrease at a rate proportional to their current value. In radioactive decay, this means that as time progresses, the quantity of the material decreases exponentially.
When we say that exponential decay occurs, we're talking about a process where the amount of radioactive material halves at consistent intervals (the half-life).
This concept is crucial for calculating how much of a radioactive isotope remains over time.
For Cobalt-62, which has a 1.5-minute half-life, after each 1.5 minutes, the remaining amount of Cobalt-62 gets reduced by half. Using the formula:\[\text{Remaining fraction} = \left(\frac{1}{2}\right)^n\] Where \(n\) is the number of half-lives that have occurred, you can determine the remaining quantity. After each half-life cycle, the fraction reduces further—1/2 becomes 1/4, then 1/8, etc., showcasing exponential decay in action.