Sodium-24 and Its Applications: Understanding Radioactive Decay and Half-Life

Sodium-24 is a valuable beta emitter used in medical diagnostics, particularly for exams of vascular circulation. Its half-life of 15.0 hours is crucial in understanding how quickly the substance degrades, and this knowledge aids in accurate medical assessment and treatment planning.

Understanding Radioactive Decay and Half-Life

Radioactive decay is a fundamental concept in nuclear physics, where unstable atomic nuclei spontaneously transform into more stable forms. This transformation often involves the emission of particles such as alpha or beta particles, or gamma rays, followed by the transition to a new atomic nucleus with a lower energy state.

The Importance of Half-Life

The half-life of a radioactive isotope is a measure of the time required for half of the substance to decay into another atomic form. For Sodium-24, the half-life is 15.0 hours. This means that after each 15-hour period, exactly half of the original sample decays, and this process repeats itself with each subsequent half-life.

Calculating Remaining Sodium-24 After 60 Hours

To determine the amount of Sodium-24 remaining after 60 hours, one must perform a calculation based on its half-life.

First, it is important to recognize that 60.0 hours corresponds to 4 half-lives since ( frac{60 text{ hours}}{15 text{ hours/half-life}} 4 ) half-lives.

Using the formula for radioactive decay, which is given by ( left( frac{1}{2} right)^n ) where ( n ) is the number of half-lives, we can determine the fraction of the original sample that remains. For 4 half-lives, this calculation is:

( left( frac{1}{2} right)^4 frac{1}{16} )

Converting this to a percentage, we have:

( frac{1}{16} times 100 6.25% )

Therefore, after 60.0 hours, only 6.25% of an enriched sample of Sodium-24 remains.

Considering Bioelimination

The problem asks whether to consider only the radioactive decay or whether to also consider bioelimination. This is an important distinction because it depends on the context and location of the sample. For instance, if the sample is within a living organism, bioelimination would also play a significant role, leading to a quicker decay of the Sodium-24.

Radioactive Decay Chart

Mass Length of Time 1 0 (beginning of problem) 1/2 15 hours (after one half-life) 1/4 30 hours (after two half-lives) 1/8 45 hours (after three half-lives) 1/16 60 hours (after four half-lives)

From the chart, it is clear that after four half-lives (60 hours), only ( frac{1}{16} ) of the original Sodium-24 remains. This is a direct application of the concept that after each half-life, the amount of radioactive material is halved.

Conclusion

In summary, understanding radioactive decay and half-life is crucial for applications such as the use of Sodium-24 in medical diagnostics. The fraction of Sodium-24 remaining after 60 hours is ( frac{1}{16} ), or 6.25%. This knowledge is essential for accurate medical assessments and treatment planning.

Moreover, the consideration of bioelimination, if applicable, can further refine decay calculations, ensuring more precise results in real-world scenarios.