Problem 45 - Chemistry: Principles and Reactions

Question

Copper-64 is one of the metals used to study brain activity. Its decay constant is \(0.0546 \mathrm{~h}^{-1}\). If a solution containing \(5.00 \mathrm{mg}\) of \(\mathrm{Cu}-64\) is used, how many milligrams of Cu-64 remain after eight hours?

Step-by-Step Solution

Answer

Answer: Approximately 2.70 mg of Copper-64 remains after 8 hours.

Step 1: Write down the radioactive decay formula

Our first step is to write down the radioactive decay formula, which is given by: \(N_t = N_0 e^{-\lambda t}\)

Step 2: Use the given values

Now we will substitute the given values into the formula: \(N_t = 5.00 \mathrm{mg} \cdot e^{-0.0546 \mathrm{~h}^{-1} \cdot 8 \mathrm{~h}}\)

Step 3: Calculate the remaining amount of Copper-64

We will now calculate \(N_t\): \(N_t \approx 5.00 \mathrm{mg} \cdot e^{-0.0546 \times 8} \approx 5.00 \mathrm{mg} \cdot e^{-0.4368} \approx 2.701 \mathrm{mg}\)

Step 4: Round the result

We will round the result to two decimal places: \(N_t \approx 2.70 \mathrm{mg}\) After eight hours, approximately 2.70 milligrams of Copper-64 remain.

Key Concepts

Copper-64

Copper-64 ( \( -> sometime used as Copper-64 \)) is a radioactive isotope of copper. It is used in a variety of scientific and medical applications; a common application is to study brain activity. Its radioactive nature allows scientists and doctors to trace how substances move within the brain. This utility comes from the isotope's ability to emit positrons, which are used in a scanning technique known as positron emission tomography (PET).
Copper-64 has a moderate half-life and decay rate that are useful for short-term studies and experiments. It's not too short to quickly lose its utility nor too long to overstay the time needed for observation.
The behaviour of Copper-64 and its applications make it an excellent tool for imaging and diagnostic purposes without causing harm from excessive radioactivity.

Half-life

The concept of half-life is really important when dealing with radioactive substances like Copper-64. The half-life is the time it takes for half of a sample of radioactive material to decay. It provides an intuitive way to understand the decay process.
For Copper-64, the half-life is approximately 12.7 hours. This means that if you start with a given amount of Copper-64, half of it will have decayed after 12.7 hours. This allows scientists to predict how long a sample remains useful for experiments or medical scanning.

• A longer half-life means the substance remains radioactive for a longer period.
• A shorter half-life indicates the material will decay more quickly.

Decay constant

The decay constant ( \( \lambda \)) is a number that helps us understand the rate of decay of a radioactive substance. It is a constant used in the exponential decay formula. For Copper-64, the decay constant is \( 0.0546 \, \text{h}^{-1} \).

• The decay constant is inversely related to the half-life. A high decay constant means a short half-life.
• It describes the probability per unit time that a given radioactive nucleus will decay.

In essence, the decay constant provides a measure of how quickly atoms of Copper-64 are disintegrating over time. This constant is crucial for calculating the remaining quantity of a radioactive substance over time using the decay formula.

Exponential decay

Exponential decay describes how the quantity of a radioactive substance decreases over time. It's called "exponential" because the size of the decrease depends on the current amount, leading to the quantity shrinking rapidly at first, and then more slowly over time.
The mathematical formula we use for exponential decay is:\[ N_t = N_0 e^{-\lambda t} \]Here, \( N_0 \) is the initial amount of substance, \( N_t \) is the amount remaining after time \( t \), and \( \lambda \) is the decay constant. This equation helps us calculate how much of a substance is left over time.

• Exponential decay ensures the remaining substance never truly reaches zero.
• The constant \( e \) in the formula is the base of the natural logarithm, approximately equal to 2.71828.

Understanding exponential decay helps scientists accurately determine how materials like Copper-64 will behave over periods, which is crucial for experiments and safety assessments.