Problem 72 - Chemistry The Molecular Science

Question

The half-life of molybdenum-99 is 67.0 hours. Calculate how much of a \(1.000-\mathrm{mg}\) sample of \({ }^{99} \mathrm{Mo}\) is left after 335 hours. Determine how many half-lives it underwent.

Step-by-Step Solution

Answer

The sample underwent 5 half-lives, leaving 0.03125 mg of molybdenum-99.

Step 1: Understanding Half-life

The half-life of a substance is the time taken for half of the radioactive atoms in a sample to decay. For molybdenum-99, this half-life is given as 67.0 hours.

Step 2: Determine Total Time

We are asked to find the remaining amount of molybdenum-99 after 335 hours of decay.

Step 3: Calculate Number of Half-lives

To find the number of half-lives that have elapsed, divide the total time by the half-life:\[\text{Number of half-lives} = \frac{335 \text{ hours}}{67.0 \text{ hours/half-life}}\] Calculate this to find the number of half-lives.

Step 4: Apply Exponential Decay Formula

The remaining quantity of a radioactive substance is given by:\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{n} \] where \(N(t)\) is the remaining quantity after time \(t\), \(N_0\) is the initial quantity (1.000 mg here), and \(n\) is the number of half-lives calculated in Step 3.

Step 5: Calculate Remaining Quantity

Substitute the number of half-lives from Step 3 and the initial quantity into the exponential decay formula to find the remaining quantity of molybdenum-99.

Key Concepts

Radioactive Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This happens continuously over time as atoms attempt to reach a more stable state. During decay, different types of radiation can be released, such as alpha particles, beta particles, or gamma rays. The substance undergoing this decay changes chemically, resulting in the formation of new elements or isotopes.

Understanding the basics of radioactive decay is crucial for calculating how a material diminishes over time. This process is not uniform but follows specific rules defined by the material's half-life. In each half-life, precisely half of the radioactive atoms present will decay. This predictable pattern helps scientists make calculations about how much of a radioactive sample remains after a given period. For instance, in the case of molybdenum-99, its behavior over time is determined by its half-life, which is intrinsic to its nature.

Molybdenum-99

Molybdenum-99 (Mo-99) is an important radioactive isotope commonly used in medical imaging and diagnostic procedures. It decays into technetium-99m, which is used in millions of medical tests annually. What makes Mo-99 particularly valuable is its ability to emit radiation that can be captured by imaging devices, helping doctors to view internal organs without invasive techniques.

Mo-99 has a half-life of 67 hours, meaning that every 67 hours, half of the substance will decay naturally. This relatively short half-life makes it suitable for medical applications, where quick decay is beneficial because it reduces the patient's exposure to radioactivity. To calculate decay, knowing the half-life is essential so you can determine how much will remain after a given time. This knowledge is applied practically by calculating how long a sample of Mo-99 remains effective for medical purposes.

Exponential Decay Formula

The exponential decay formula is a mathematical tool used to calculate the remaining quantity of a radioactive substance over time. It considers the initial amount and the number of elapsed half-lives to determine the remaining mass. The formula can be expressed as:

• \[N(t) = N_0 \times \left(\frac{1}{2}\right)^{n}\]

where:

• \(N(t)\) is the remaining quantity after time \(t\).
• \(N_0\) is the initial quantity.
• \(n\) is the number of half-lives that have passed over that time period.

The exponential decay formula reflects the gradual reduction of radioactive material. When you use this formula, it simplifies the process of determining how much of a substance like molybdenum-99 remains after a certain number of half-lives.

For example, with an initial sample of 1.000 mg of molybdenum-99 and 5 half-lives (as calculated), you can determine the remaining mass through this formula by placing these values into the equation. This calculation informs decisions in medical and scientific contexts by revealing precisely how much of the isotope is still useful at a given time.