Question

If \(2400 \mu g\) of hydrogen- 3 decay to \(600 \mu g\) after 24.8 years, what is the half-life of this radionuclide that is used as a chemical tracer?

Short answer

The half-life of hydrogen-3 is 12.4 years.

Step-by-step solution

Understand the Problem

We need to find the half-life of hydrogen-3 given that its initial mass is \(2400 \mu g\) and it decays to \(600 \mu g\) in \(24.8\) years.

Use Decay Formula

The decay of a substance can be modeled using the formula:\[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the time elapsed, and \(T_{1/2}\) is the half-life we are trying to find.

Set Up the Equation

Given:- \(N_0 = 2400 \mu g\)- \(N(t) = 600 \mu g\)- \(t = 24.8\) yearsPlug these values into the decay formula:\[ 600 = 2400 \left(\frac{1}{2}\right)^{\frac{24.8}{T_{1/2}}} \]

Solve for the Half-Life

First, divide both sides by \(2400\):\[ \frac{600}{2400} = \left(\frac{1}{2}\right)^{\frac{24.8}{T_{1/2}}} \]\[ 0.25 = \left(\frac{1}{2}\right)^{\frac{24.8}{T_{1/2}}} \]Next, solve for \(\frac{24.8}{T_{1/2}}\):\[ \log_{0.5}(0.25) = \frac{24.8}{T_{1/2}} \]Convert to base 10 logs:\[ \frac{24.8}{T_{1/2}} = \frac{\log(0.25)}{\log(0.5)} \]\[ \frac{24.8}{T_{1/2}} = 2 \]Finally, solve for \(T_{1/2}\):\[ T_{1/2} = \frac{24.8}{2} = 12.4 \text{ years} \]

Verify the Solution

Let's verify: using the found half-life of \(12.4\) years, after two half-lives (\(24.8\) years), the substance should reduce from \(2400 \mu g\) to \(600 \mu g\), which matches the problem's conditions.

Key concepts

Radioactive Decay

Radioactive decay is a natural process where an unstable atomic nucleus loses energy by emitting radiation. This process occurs in unstable isotopes, which are atoms with excess nuclear energy.
Here's what happens during decay:

• The nucleus releases particles or electromagnetic waves to reach a more stable state.
• It may convert into a different element with a stable nucleus.
• The rate of decay is characterized by a term known as the "half-life." This is the time it takes for half of the radioactive atoms in a sample to decay.

Understanding half-life helps in calculating how quickly a radioactive substance will decay. The decay follows an exponential pattern, meaning that with time, the substance reduces continuously by half.
This concept is crucial in fields like archaeology (carbon dating), medicine (radioactive tracers), and energy (nuclear power). It also has applications in environmental science, particularly in tracking isotopes in nature.

Hydrogen-3

Hydrogen-3, also known as tritium, is one of the isotopes of hydrogen. It is radioactive and has one proton and two neutrons in its nucleus, giving it a greater mass than the more common hydrogen-1.

• Tritium is naturally occurring but can also be artificially produced in nuclear reactors.
• Its radioactive decay is beta decay, where it emits a beta particle and some energy, resulting in a stable helium-3 nucleus.
• The half-life of hydrogen-3 is approximately 12.4 years, which makes it quite useful for long-term studies.

Because tritium is a soft beta emitter, it poses less of a radiation hazard compared to other radioactive materials. It is often used in self-luminous devices such as signs and watch dials, where it provides a steady source of light without the need for power.
In scientific research, its relatively short half-life and harmless radiation make it a popular choice as a chemical tracer.

Chemical Tracer

A chemical tracer is a substance with a detectable property used to track the presence and concentration of fluids in environmental science, medicine, and various research fields. Tracers can be chemical (like dyes) or radioactive isotopes (like tritium or hydrogen-3).
Using a tracer involves injecting it into a system to monitor how substances move or distribute themselves:

• In medicine, they help in imaging and diagnosing diseases by highlighting organs and systems under study.
• In environmental science, tracers can track water movement, for instance, in hydrology studies.
• Tritium, or hydrogen-3, is often favored due to its manageable half-life and low health risk when used correctly.

Due to their traceability, chemical tracers can reveal otherwise difficult-to-detect patterns. This is valuable for improving our understanding of complex systems, confirming theories, or validating simulations. Tracers elevate scientific exploration by providing insight that was perhaps challenging to obtain through direct observation alone.