Question

Radioisotopes are used to diagnose various illnesses. Iodine-131 (I-131) is administered to a patient to diagnose thyroid gland activity. The original dosage contains 280 MBq of I-131. If none is lost from the body, then after \(6 \mathrm{h}\) there are \(274 \mathrm{MBq}\) of I-131 in the patient's thyroid. What is the half-life of I-131, to the nearest day?

Short answer

The half-life of Iodine-131 is approximately 8 days.

Step-by-step solution

Identify and Write Down the Known Values

Given:- Initial activity, A_0 = 280 MBq- Activity after time t, A = 274 MBq- Time, t = 6 hours

Use the Radioactive Decay Formula

The formula for radioactive decay is \[ A = A_0 e^{-\frac{\text{ln}(2)}{T_{1/2}} t} \]Where A is the remaining activity, A_0 is the initial activity, T_{1/2} is the half-life, and t is the time.

Rearrange the Formula to Solve for the Half-life

Rearrange the equation to solve for the half-life, T_{1/2}.\[ \frac{A}{A_0} = e^{-\frac{\text{ln}(2)}{T_{1/2}} t} \]\[ \text{ln}\bigg(\frac{A}{A_0}\bigg) = -\frac{\text{ln}(2)}{T_{1/2}} t \]\[ T_{1/2} = -\frac{\text{ln}(2) \times t}{\text{ln}(\frac{A}{A_0})} \]

Plug in the Known Values

Substitute the known values into the rearranged formula:\[ T_{1/2} = -\frac{\text{ln}(2) \times 6}{\text{ln}(\frac{274}{280})} \]Calculate the natural logarithm values:- \text{ln}(2) \[ \text{ln}(2) \approx 0.693 \]- \text{ln}(274/280) \[ \text{ln}(0.9786) \approx -0.0217 \]Now substitute these results back into the equation:

Perform the Calculation

Perform the final calculation to find the half-life:\[ T_{1/2} = -\frac{0.693 \times 6}{-0.0217} \]\[ T_{1/2} \approx 191.65 \text{ hours} \]Convert the result from hours to days:\[ 191.65 \text{ hours} \div 24 \text{ hours/day} \approx 7.99 \text{ days} \]

Round the Result

Round the result to the nearest day:\[ T_{1/2} \approx 8 \text{ days} \]

Key concepts

radioactive decay

Radioactive decay is a natural process where an unstable atomic nucleus loses energy by emitting radiation. This emission leads to the transformation of the nucleus into a different state or a completely different element. Understanding radioactive decay is crucial in fields like medicine, archaeology, and nuclear physics. It helps us date ancient artifacts, diagnose medical conditions, and generate power.
One of the key terms in radioactive decay is the half-life, which is the time it takes for half of the radioactive atoms in a sample to decay. This decay is exponential and can be described using the formula:
\[ A = A_0 e^{-\frac{\text{ln}(2)}{T_{1/2}} t} \] where:

• \(A\): remaining activity
• \(A_0\): initial activity
• \(T_{1/2}\): half-life
• \(t\): time

Notice how the decay process doesn't proceed at a constant rate but rather slows down as fewer radioactive atoms remain.
Understanding these decay principles lets us predict the behavior of radioactive substances accurately.

logarithms

Logarithms are mathematical tools that help simplify the complexity of exponential relationships, like radioactive decay. They transform multiplicative processes into additive ones, making calculations much easier. In an equation such as:
\[ A = A_0 e^{-\frac{\text{ln}(2)}{T_{1/2}} t} \]
taking the natural logarithm of both sides can simplify solving for unknown variables:
\[ \text{ln}\bigg(\frac{A}{A_0}\bigg) = -\frac{\text{ln}(2)}{T_{1/2}} t \]
Here, \(\text{ln}(2)\) approximately equals 0.693, a constant value in logarithmic decimal form. Logarithms help us rearrange and inverse exponential equations, making the mathematical manipulation and solving for variables straightforward. Understanding logarithms is essential for working in precalculus, where solving exponential decay problems efficiently is critical.

precalculus

Precalculus bridges the gap between algebra and calculus, providing the skills necessary to tackle more complex mathematical problems. This field focuses on functions, their properties, and the various techniques needed to manipulate them. In the context of radioactive decay, precalculus includes exponential functions, logarithms, and their properties.
Let's break down the steps required to solve the radioactive decay problem:

• Identify initial activity \(A_0\) and remaining activity \(A\).
• Use the decay formula \( A = A_0 e^{-\frac{\text{ln}(2)}{T_{1/2}} t} \).
• Rearrange the formula to solve for half-life \(T_{1/2}\).
• Apply natural logarithms to simplify:
• Substitute known values and solve.

Precalculus concepts such as these ensure strong mathematical foundations, enabling students to approach and solve real-world problems effectively.Knowing precalculus makes it easier to understand and manipulate equations involving exponential growth and decay, making the math behind radioactive decay not just understandable but also practical.