Understanding Radioactive Decay and Half-Life

Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This process follows exponential decay. A key concept describing the rate of radioactive decay is half-life.

What is Half-Life?

The half-life (\(t_{\frac{1}{2}}\)) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. It's a constant value for a given isotope and does not depend on the initial amount of the substance.

Calculating Substance Left After Decay Time

When we know the half-life of a substance and the total time that has passed, we can calculate the amount of the substance that remains using a specific formula. The formula relates the initial amount, the final amount, the total time, and the half-life.

Given information in this problem:

• Original mass (initial amount), \(N_0\)
• Half-life, \(t_{\frac{1}{2}} = 4\) years
• Total time elapsed, \(t = 12\) years

We want to find the amount of substance left, \(N\), after 12 years.

Formula for Amount Remaining After Radioactive Decay

The amount of substance \(N\) remaining after time \(t\) can be calculated using the formula:

\(N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{\frac{1}{2}}}}\)

Alternatively, we can first calculate the number of half-lives \(n\) that have occurred during time \(t\):

\(n = \frac{t}{t_{\frac{1}{2}}}\)

Then, the amount remaining is given by:

\(N = N_0 \left(\frac{1}{2}\right)^{n}\)

Step-by-Step Calculation of Amount Left

Let's use the step-by-step approach:

Step 1: Calculate the number of half-lives (n)

The number of half-lives is the total time divided by the half-life period.

\(n = \frac{t}{t_{\frac{1}{2}}} = \frac{12 \text{ years}}{4 \text{ years}}\)

\(n = 3\)

So, 3 half-lives have passed in 12 years.

Step 2: Calculate the amount of substance remaining (N)

Using the formula \(N = N_0 \left(\frac{1}{2}\right)^{n}\) with \(n=3\):

\(N = N_0 \left(\frac{1}{2}\right)^{3}\)

\(N = N_0 \times \left(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\right)\)

\(N = N_0 \times \frac{1}{8}\)

\(N = \frac{N_0}{8}\)

Therefore, the amount of substance left after 12 years is \(N_0/8\).

Radioactive Decay Concepts Explained

Radioactive decay is a first-order kinetic process. This means the rate of decay is proportional to the number of radioactive nuclei present. The concept of half-life is a direct consequence of this first-order kinetics.

• After 1 half-life, 1/2 of the original substance remains.
• After 2 half-lives, (1/2) * (1/2) = 1/4 of the original substance remains.
• After 3 half-lives, (1/2) * (1/2) * (1/2) = 1/8 of the original substance remains.

In this problem, since 3 half-lives passed, the amount remaining is \(N_0/8\).

Revision Table: Radioactive Decay Calculations

Additional Information on Radioactive Decay

Radioactive decay is a random process at the level of single atoms, meaning we cannot predict when a specific atom will decay. However, with a large number of atoms, the overall decay rate is predictable and follows the exponential decay law. The decay constant (\(\lambda\)) is another parameter used to describe the decay rate, related to half-life by the equation \(t_{\frac{1}{2}} = \frac{\ln(2)}{\lambda}\). The decay formula can also be written as \(N = N_0 e^{-\lambda t}\), which is equivalent to the formula using half-life.

Radioactive decay is fundamental in various fields, including nuclear physics, nuclear chemistry, geology (radioisotope dating), medicine (nuclear medicine), and energy production (nuclear reactors).