Problem 47 - Chemical Principles

Question

Deoxyglucose labeled with fluorine- 18 is commonly used in PET scans to locate tumors. Fluorine- 18 has a half-life of \(109 \mathrm{~min}\). How long will it take for the level of fluorine- 18 in the body to drop to \(10 \%\) of its initial value?

Step-by-Step Solution

Answer

Fluorine-18 will take approximately 363 minutes to drop to 10% of its initial level.

Step 1: Understand the Half-Life Concept

The half-life of a substance is the time it takes for the substance's initial concentration to reduce to half of its original value. For fluorine-18, the half-life is 109 minutes.

Step 2: Set up the Decay Formula

The decay of a radioactive substance can be modeled by the exponential decay formula: \( N(t) = N_0 \times (1/2)^{\frac{t}{t_{1/2}}} \), where \( N(t) \) is the amount after time \( t \), \( N_0 \) is the initial amount, and \( t_{1/2} \) is the half-life of the substance. In this case, we are solving for \( t \) when \( N(t) = N_0 \times 0.10 \).

Step 3: Apply the Known Values and Solve for Time

Substituting \( N(t) = N_0 \times 0.10 \) and \( t_{1/2} = 109 \) minutes into the formula, we get \( 0.10 = (1/2)^{\frac{t}{109}} \). Now we need to solve for \( t \).

Step 4: Solve the Exponential Equation

To solve the equation, let's take the natural logarithm on both sides: \( \ln(0.10) = \ln((1/2)^{\frac{t}{109}}) \). By using the property of logarithms \( \ln(a^b) = b\ln(a) \), we can simplify our equation to: \( \ln(0.10) = \frac{t}{109}\ln(1/2) \). Solving for \( t \), we get: \( t = \frac{\ln(0.10)}{\ln(1/2)} \times 109 \) minutes.

Step 5: Calculate the Time

After calculating the above expression, you will get the time it takes for fluorine-18 to decay to 10% of its initial amount.

Key Concepts

Understanding Radioactive Decay

Radioactive decay is a spontaneous process where unstable atomic nuclei release energy in the form of radiation to become more stable. It's a random process at the level of single atoms, but for a large number of atoms, the decay rate is predictable. This rate is described by the half-life, which is the time required for half of the radioactive substance to decay.

For instance, fluorine-18, mentioned in the exercise, is a radioactive isotope used in medical imaging. Its decay helps in highlighting areas of the body, such as tumors, during a PET scan. Understanding this principle is crucial for health professionals to calculate the correct dosage and for patients to be aware of how long the radioactivity will be significant in their bodies.

Using the Exponential Decay Formula

The exponential decay formula is central to calculating how much of a radioactive substance remains over time. The formula, expressed as \( N(t) = N_0 \times (1/2)^{\frac{t}{t_{1/2}}} \), models the decay of a radioactive substance where \( N(t) \) is the remaining amount after time \( t \), \( N_0 \) is the initial amount, and \( t_{1/2} \) represents the half-life.

This formula is called 'exponential' because the quantity of the substance decreases at a rate proportional to its current value, leading to the characteristic 'exponential' decline. When using this model in medical settings, such as with fluorine-18 in PET scans, professionals can determine the time it will take for the radioactivity to reduce to a safe level, ensuring patient safety and effective imaging.

The Role of Natural Logarithm in Decay Calculations

The natural logarithm is a mathematical function that is the inverse of the exponential function. When dealing with exponential decay, the natural logarithm allows us to isolate the variable of interest, in this case time \( t \), from the equation.

By taking the natural logarithm of both sides of the equation, like in our decay formula, we can convert the exponent into a multiplication, which simplifies the process of solving for time. The property \( \ln(a^b) = b\ln(a) \) is particularly useful, as shown in the exercise solution. It's the key to finding out how long it takes for substances like fluorine-18 to decay to a certain level within the human body, essential information for the safe administration of radioactive materials in medical diagnostics.