Problem 29 - Chemistry and Chemical Reactivity

Question

The radioactive isotope \(^{64} \mathrm{Cu}\) is used in the form of copper(II) acetate to study Wilson's disease. The isotope has a half-life of 12.70 hours. What fraction of radioactive copper(II) acetate remains after 64 hours?

Step-by-Step Solution

Answer

Approximately 3.13% of radioactive copper(II) acetate remains after 64 hours.

Step 1: Understand the Half-Life Definition

The half-life of a radioactive substance is the time required for half of the substance to decay. For the isotope \(^{64} \mathrm{Cu}\), the half-life is given as 12.70 hours.

Step 2: Calculate the Number of Half-Lives

To find the total number of half-lives that have passed in 64 hours, divide 64 hours by the half-life of 12.70 hours:\[\text{Number of half-lives} = \frac{64}{12.70} \approx 5.04\]

Step 3: Use the Exponential Decay Formula

The fraction remaining after \( n \) half-lives can be calculated using the formula:\[\left(\frac{1}{2}\right)^n\]Where \( n = 5.04 \).

Step 4: Calculate the Remaining Fraction

Substitute \( n = 5.04 \) into the decay formula:\[\left(\frac{1}{2}\right)^{5.04} \approx 0.0313\]This result represents the fraction of the radioactive copper(II) acetate that remains after 64 hours.

Key Concepts

half-life

In radioactive decay, the concept of half-life is essential to understanding how substances change over time. The half-life is the period it takes for half of a sample of a radioactive substance to decay and lose its radioactivity. This means that after one half-life, only half of the original substance remains radioactive.
When calculating how much of a substance remains after a certain time, we use the idea of half-lives to break the total time into smaller intervals. In the context of the copper-64 exercise, the half-life was given as 12.70 hours. If we wish to know the remaining fraction after 64 hours, we first determine how many half-lives fit into this period. By dividing 64 by 12.70, we find approximately 5.04 half-lives have passed.
This number helps us understand that the radioactive isotope has significantly decayed over this time frame. Using the formula \( \left(\frac{1}{2}\right)^n \) where \( n \) is the number of half-lives, we quantify the fraction remaining. In our example, \( n = 5.04 \), thus resulting in a fraction of about 0.0313 as remaining from the original sample.

Wilson's disease

Wilson's disease is a rare genetic disorder that causes excessive copper accumulation in organs like the liver and brain. Our bodies typically regulate copper absorption and excretion, but for individuals with Wilson's disease, this regulation is impaired, leading to copper buildup. Over time, the excess copper causes damage to tissues and organs.
To study the disease and its effects, researchers often use tracers such as the radioactive isotope copper-64. These tracers help scientists observe how copper moves and where it accumulates in the body. This information is crucial in diagnosing the disease and determining the appropriate treatment measures.
By understanding how copper-64 behaves in the body, doctors can make more informed decisions about managing the disease, such as assessing how much copper is deposited in specific organs. Early detection and treatment of Wilson's disease are key to preventing long-term complications.

copper-64

Copper-64, or \(^{64}\text{Cu}\), is a radioactive isotope utilized in medical research and diagnostics because of its unique properties. Unlike stable copper, copper-64 emits radiation, which can be detected and tracked by special imaging equipment. This feature makes it especially valuable in studying diseases that involve copper metabolism, like Wilson's disease.
One of the primary attributes of copper-64 is its half-life of 12.70 hours. This relatively short half-life is advantageous because it limits the duration of radiation exposure to patients. Despite its short-lived nature, copper-64 provides enough time for thorough analysis and imaging in medical procedures.
Copper-64 is typically incorporated into compounds like copper(II) acetate, allowing it to be absorbed and utilized in biological studies. By observing copper-64's path through the body, researchers gain valuable insights into how copper metabolism is altered in certain conditions, thereby aiding in the treatment and study of related disorders.