Problem 73 - Chemistry The Molecular Nature of Matter

Question

If we begin with \(3.00 \mathrm{mg}\) of iodine- \(131\left(t_{1 / 2}=8.07 \mathrm{hr}\right)\), how much remains after six half-life periods?

Step-by-Step Solution

Answer

After six half-life periods, 0.046875 mg of iodine-131 remains.

Step 1: Understanding the Half-life Concept

Half-life is the amount of time it takes for half of a radioactive sample to decay. After each half-life period, the amount of the substance is reduced by half.

Step 2: Calculate the Quantity After Each Half-life

To find the amount of iodine-131 remaining after six half-life periods, use the formula: \( N = N_0 \cdot \left(\frac{1}{2}\right)^n \), where \( N \) is the final quantity, \( N_0 \) is the initial quantity, and \( n \) is the number of half-life periods.

Step 3: Substitute the Given Values into the Formula

Substitute the initial amount \( N_0 = 3.00 \mathrm{mg} \) and the number of half-life periods \( n = 6 \) into the formula to calculate the final amount \( N \): \( N = 3.00 \mathrm{mg} \cdot \left(\frac{1}{2}\right)^6 \).

Step 4: Calculate the Final Amount

Perform the calculation: \( 3.00 \mathrm{mg} \cdot \left(\frac{1}{2}\right)^6 = 3.00 \mathrm{mg} \cdot \frac{1}{64} = 0.046875 \mathrm{mg} \).

Key Concepts

Understanding Half-life

Half-life is a fundamental concept when studying radioactive decay. It is defined as the time required for half of the radioactive isotope samples to decay or transform into a different element or isotope.

For example, if you start with 100 grams of a radioactive isotope and its half-life is 5 years, after 5 years, only 50 grams of the isotope will remain. Another 5 years later, and only 25 grams will be left, and so on. This decrease by half continues until the material becomes stable or is no longer detectable.

• Half-life is constant and unique for each radioactive isotope.
• It is unaffected by external conditions such as temperature or pressure.
• Understanding half-life helps determine the age of archaeological artifacts (carbon dating) or the proper dosages for medical treatments.

This predictable pattern provides invaluable information in various scientific fields, from geology to medicine, where knowing the duration of radioactivity is crucial.

Radioactive Isotopes and Their Use

Radioactive isotopes, also known as radioisotopes, are atoms with an unstable nucleus that release energy in the form of radiation to reach a more stable state. Each isotope has a specific set of properties, including its half-life.

Some commonly known radioactive isotopes include:

• Iodine-131, used in medical diagnostics and treatment, particularly for thyroid conditions.
• Carbon-14, used in dating organic materials in archaeology (radiocarbon dating).
• Uranium-238, used for estimating the age of the earth and in nuclear power generation.

These isotopes have practical applications in healthcare, research, and energy production. It's critical for students and professionals working with radioisotopes to understand their decay patterns, safety handling, and disposal procedures.

The Mathematics of Exponential Decay

Exponential decay is a mathematical concept that describes the process by which a quantity decreases at a rate proportional to its current value. As applied in radioactive decay, this implies that the larger the amount of the substance present, the faster it decays.

The formula to describe this relationship in the context of half-life is: \[ N = N_0 \times \bigg(\frac{1}{2}\bigg)^n \] where:

• \( N \) is the final amount left after \( n \) half-life periods.
• \( N_0 \) is the initial amount of the substance.
• \( n \) is the number of half-lives that have passed.

This formula is a representation of exponential decay and is crucial in determining the amount of a radioactive isotope remaining over time. It illustrates that the decay of radioactive substances is not linear, but rather happens more rapidly at first and slows over time as the substance becomes less and less plentiful.