Problem 39 - Chemistry and Chemical Reactivity

Question

Copper(II) acetate containing \(^{64} \mathrm{Cu}\) is used to study brain tumors. This isotope has a half-life of 12.7 hours. If you begin with \(25.0 \mu \mathrm{g}\) of \(^{64} \mathrm{Cu},\) what mass remains after 63.5 hours?

Step-by-Step Solution

Answer

About 0.781 µg of \(^{64} \mathrm{Cu}\) remains after 63.5 hours.

Step 1: Understand the Concept of Half-Life

Half-life is the time required for half of a radioactive substance to decay. For Copper(II) acetate-containing \(^{64} \mathrm{Cu}\), the half-life is given as 12.7 hours. This means that every 12.7 hours, the amount of \(^{64} \mathrm{Cu}\) remaining is half of what it was at the beginning of that time period.

Step 2: Determine the Number of Half-Life Periods

To find out how many half-life periods have passed, divide the total time elapsed by the length of one half-life. Here, the total time is 63.5 hours. So, the number of half-lives is \( \frac{63.5}{12.7} = 5 \).

Step 3: Apply the Half-Life Formula

The amount of substance remaining after a certain number of half-lives can be calculated with the formula: \( \text{remaining mass} = \text{initial mass} \times \left(\frac{1}{2}\right)^{n} \), where \(n\) is the number of half-lives. Here, the initial mass is 25.0 \(\mu \mathrm{g}\) and \(n = 5\).

Step 4: Calculate the Remaining Mass of \(^{64} \mathrm{Cu}\)

Using the formula from the previous step, compute:\[\text{remaining mass} = 25.0 \times \left(\frac{1}{2}\right)^{5} = 25.0 \times \frac{1}{32} = 0.78125 \mu \mathrm{g}\]This calculation shows that after 63.5 hours, the remaining mass of \(^{64} \mathrm{Cu}\) is approximately 0.781 \(\mu \mathrm{g}\).

Key Concepts

Understanding Half-Life

Half-life is a key concept when studying radioactive decay. It describes the period it takes for half of a radioactive substance to be transformed into a different element through the emission of radiation. This transformation is spontaneous and varies in time for different isotopes. Every isotope, or variant, of an element has its own distinct half-life.

For instance, the half-life of \(^{64} \mathrm{Cu}\) is 12.7 hours. This means that if you begin with a certain amount of \(^{64} \mathrm{Cu}\), such as 25 micrograms, only half of it would remain after one half-life period. Knowing how many half-life periods have passed allows us to predict how much of an isotope, like \(^{64} \mathrm{Cu}\) in this case, will remain.

The general formula used to calculate the remaining mass of a substance after a certain number of half-lives is:

• \( \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{n} \)
• Here, \(n\) is the number of half-lives, calculated by dividing the total time by the half-life duration.

The Role of Radioisotopes

Radioisotopes are isotopes of an element that exhibit radioactivity. They have unstable atomic nuclei and emit energy and particles as they decay into more stable forms. This radioactive property makes them extremely useful in various applications, including medical diagnostics and treatments.

In medical research, \(^{64} \mathrm{Cu}\) is a significant radioisotope because it can be used to study conditions such as brain tumors. By tracking the decay and movement of \(^{64} \mathrm{Cu}\) injected into the body, researchers can gather valuable insights about the tumor’s dynamics. Since the half-life of this radioisotope is relatively short (12.7 hours), it quickly provides information while minimizing the radiation exposure to patients.

Radioisotopes have a wide range of applications beyond medicine too:

• In industry, they are used in radiography to inspect welding seams and metal joints for integrity.
• In archaeology, they help in carbon-dating to determine the age of ancient artifacts.

Nuclear Chemistry Essentials

Nuclear chemistry is a branch of chemistry focused on the reactions and properties of atomic nuclei. Unlike conventional chemistry, which deals with the interactions between electrons, nuclear chemistry concerns itself with the processes that involve changes in the nucleus of an atom.

Radioactive decay—a central topic in nuclear chemistry—involves the transformation of an unstable isotope into a stable one. During this transformation, particles such as alpha particles, beta particles, or gamma rays are emitted. Each type of emission has its unique characteristics and effects.

Nuclear chemistry is crucial in understanding various natural phenomena and applications, such as:

• The formation of elements in stars.
• The principles behind nuclear power generation.
• The development of radiopharmaceuticals for medical purposes.

A practical understanding of nuclear chemistry allows scientists to harness these reactions safely and effectively, providing benefits in fields ranging from energy production to medicine.