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 3.1% of the radioactive copper(II) acetate remains after 64 hours.

Step 1: Understand Half-Life Formula

The concept of half-life is used to calculate how much of a radioactive substance remains after a certain period of time. The formula used is: \( N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \), where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.

Step 2: Identify Given Values

Identify the values given in the problem: \( T_{1/2} = 12.70 \text{ hours} \) (half-life) and \( t = 64 \text{ hours} \) (time elapsed). These will be used in the half-life formula.

Step 3: Calculate Number of Half-Lives

Calculate the number of half-lives that occur over 64 hours: \( \frac{t}{T_{1/2}} = \frac{64}{12.70} \approx 5.039 \). This tells us that approximately 5.039 half-lives have passed.

Step 4: Calculate Remaining Fraction

Substitute the number of half-lives into the half-life formula to find the remaining fraction of the substance: \( N(t)/N_0 = \left( \frac{1}{2} \right)^{5.039} \approx 0.031 \). This means about 3.1% of the substance remains after 64 hours.

Key Concepts

half-life calculation

Half-life is a crucial concept in understanding radioactive decay. It is the time required for half of a radioactive substance to decay. This means that after one half-life, only half of the original amount remains. To calculate the remaining amount of a substance after a certain time, we use the formula: \( N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \). Here, \( N(t) \) is the quantity that remains, \( N_0 \) is the initial quantity, \( t \) is the time passed, and \( T_{1/2} \) is the half-life of the substance.

• The more the half-lives, the less the remaining quantity.
• This exponential decay means the process never completely ends, but effectively diminishes the substance over time.

Understanding this formula helps in predicting how much of a radioactive isotope like \(^{64} \mathrm{Cu}\) remains after a certain period. With isotopes having different half-lives, it's important to know how to handle these calculations for applications like medical studies.

isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but a different number of neutrons. This means they have the same atomic number but different mass numbers. While most isotopes are stable, others are radioactive and can decay over time. Radioactive isotopes, like \(^{64} \mathrm{Cu}\), are often used in medical research, treatment, and diagnostics. Different isotopes of the same element can exhibit entirely different behaviors.

• Atoms of the same element with varying neutron numbers
• Radioactive isotopes decay at predictable rates, helping in dating ancient materials and studying bodily processes through imaging

Understanding isotopes is important because it allows scientists to harness their properties for various benefits, from archaeological dating to the treatment of diseases.

nuclear chemistry

Nuclear chemistry is the branch of chemistry dealing with radioactivity, nuclear processes, and properties. It includes the study of both natural and artificial isotopes and their reactions. The field focuses on the properties and changes of atomic nuclei, allowing us to understand processes like radioactive decay. Nuclear chemistry plays a critical role in:

• Medical imaging and therapies, using isotopes to diagnose and treat illness
• Nuclear power, providing a significant source of energy
• Weapons development, through the exploration of nuclear reactions

Research in nuclear chemistry helps improve techniques in industrial applications, reduces waste, and finds innovative ways to utilize radioactive materials safely. Understanding the basics of nuclear chemistry is key to exploring the potential and implications of nuclear technologies.