The half-life of the hydrogen isotope tritium is about 12 years. After a certain amount of time a fraction \(31 / 32\) of the atoms in the original sample has decayed. The time is most nearly equal to (A) 12 years (B) 24 years (C) 36 years (D) 48 years (E) 60 years

Understanding the concept of half-life is crucial in fields like nuclear physics, chemistry, and environmental science. A half-life is the time required for a quantity to reduce to half its initial value. In radioactive decay, it's the time it takes for half of the radioactive atoms in a sample to disintegrate.

To calculate the half-life when given a decay constant \(\lambda\), we use the formula: \[ T_{1/2} = \frac{\ln{2}}{\lambda} \]. Knowing the half-life helps us understand the longevity and decay rate of radioactive substances.

Step-by-Step Approach to Half-Life Problems

Given the decay constant or the half-life, one can determine the remaining amount of a substance after any period using the right equations. It's a basic yet powerful tool in solving exponential decay problems.

Tritium, or hydrogen-3, is a radioactive isotope with notable use in self-illuminating devices and as a tracer in environmental studies. Its decay is a great example to explore the principles of radioactivity. Tritium decays via beta emission, transforming into helium-3 with a relatively short half-life of about 12 years.

Due to its decay properties, tritium is an excellent candidate for half-life studies in physics and chemistry education.

Decay Representation

The decay of tritium can be represented in equations that express the initial and remaining quantities of the isotope over time, helping us predict the amount present after a specific period.

Radioactive decay is mathematically represented by equations that describe how unstable atoms lose energy by emitting radiation. The general formula for exponential decay of a radioactive substance is: \[ N(t) = N_0 e^{-\lambda t} \], where \(N(t)\) is the number of atoms remaining after time \(t\), \(N_0\) is the initial number of atoms, \(\lambda\) is the decay constant, and \(t\) is the elapsed time.

This equation is vital for understanding how radioactive substances change over time, and is a fundamental piece of knowledge for students studying nuclear physics.

Application in Real-World Scenarios

By applying these equations, scientists and engineers can manage nuclear materials, date archaeological finds, and ensure safety in medical treatments involving radioactive isotopes.

Engaging with problems like the tritium decay example is perfect for AP Physics practice. These problems encompass a variety of concepts including exponential functions, logarithms, and of course, physics. They encourage critical thinking and a deep understanding of physical principles.

The practice of working through such exercises prepares students for the problem-solving nature of AP examinations.

Essential Skills

Through these exercises, students develop skills in equation manipulation, exponential decay understanding, and application of logarithmic functions, all while reinforcing their knowledge of the subject.