Problem 67 - Chemistry The Science in Context

Question

Carbon-11 is an isotope used in positron emission tomography and has a half- life of 20.4 min. How long will it take for \(99 \%\) of the \(^{11} \mathrm{C}\) injected into a patient to decay?

Step-by-Step Solution

Answer

Answer: It takes approximately 68.32 minutes for 99% of Carbon-11 to decay.

Step 1: Understanding the radioactive decay formula

Radioactive decay can be modeled using the formula: \(N(t) = N_0 * e^{(-\lambda t)}\) Where: \(N(t)\) is the amount of substance present after time \(t\), \(N_0\) is the initial amount of substance, \(e\) is the base of the natural logarithm (approximately 2.718), \(\lambda\) is the decay constant, and \(t\) is the time Our goal is to find the time \(t\).

Step 2: Calculating the decay constant (\(\lambda\)) using the half-life

We can use the half-life information to calculate the decay constant (\(\lambda\)) with this formula: \(t_{1/2} = \dfrac{ln(2)}{\lambda }\) Now, we plug in the given half-life \(t_{1/2} = 20.4\) min: \(\lambda = \dfrac{ln(2)}{20.4}\)

Step 3: The Radioactive decay formula with 99% decayed

Now we need to adjust the radioactive decay formula to represent 99% decayed Carbon-11. If 99% of Carbon-11 has decayed, it means 1% of it is left. Therefore, we have: \(N(t) = 0.01 * N_0\) Now we can plug this into the radioactive decay formula: \(0.01 * N_0 = N_0 * e^{(-\lambda t)}\) Notice that \(N_0\) is on both sides of the equation, so we can divide both sides by \(N_0\) to remove it: \(0.01 = e^{(-\lambda t)}\)

Step 4: Solving for the time (\(t\))

Next, we want to find \(t\). We already have the decay constant \(\lambda = \dfrac{ln(2)}{20.4}\). Now we can solve for \(t\): \(0.01 = e^{(-(\dfrac{ln(2)}{20.4}) t)}\) To solve for \(t\), we can first take the natural logarithm of both sides: \(ln(0.01) = -(\dfrac{ln(2)}{20.4}) t\) Now we can isolate \(t\) by dividing both sides by \(-\dfrac{ln(2)}{20.4}\): \(t = \dfrac{ln(0.01)}{-(\dfrac{ln(2)}{20.4})}\) Finally, simplify to find the answer: \(t \approx 68.32\) (rounded to two decimal places) Thus, it will take approximately 68.32 minutes for 99% of the Carbon-11 to decay.

Key Concepts

Understanding Half-Life in Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics and chemistry, where an unstable atomic nucleus loses energy by emitting radiation. One of the key measures in understanding radioactive decay is the concept of 'half-life'. The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. It's a way to express the rate of decay: the longer the half-life, the slower the rate of decay.
For example, if we begin with 100 grams of a radioactive isotope with a half-life of 5 years, we would expect to have 50 grams remaining after 5 years. After another 5 years (making a total of 10 years), there would only be 25 grams left, and so on. This consistent halving is at the core of understanding how substances decrease over time due to radioactivity.
In practical terms, knowing the half-life aids in various fields, such as archaeology for carbon dating, medical treatments, and environmental studies where the decay of radioactive elements may pose a risk or benefit.

Positron Emission Tomography (PET)

Positron Emission Tomography, commonly known as PET, is a sophisticated medical imaging technique that reveals how tissues and organs are functioning. The process involves injecting a small amount of radioactive substance, called a radiotracer, into the body. The radiotracer, often an isotope like Carbon-11, decays by emitting positrons, which interact with electrons in the body to produce gamma rays.
These gamma rays are then detected by the PET scanner, which uses the data to construct detailed images of the body's internal function. PET scans are particularly useful for detecting cancer, monitoring heart disease, and evaluating brain disorders because they show how cells in the body are using nutrients like glucose, oxygen, and amino acids. This allows physicians to detect abnormalities in metabolic processes, leading to early diagnosis and treatment.
Half-life plays a crucial role in PET, as the radiotracer must remain active long enough to acquire the image but not so long as to expose the patient to unnecessary radiation. Therefore, isotopes with relatively short half-lives, like Carbon-11, are typically used for these tests.

The Role of the Natural Logarithm in Radioactive Decay

The natural logarithm is indispensable in various scientific and mathematical fields, particularly in the study of growth and decay processes. It is the logarithm to the base of the mathematical constant 'e', which is approximately equal to 2.718.
When tackling radioactive decay problems, we frequently encounter the natural logarithm, specifically when we need to invert the exponential decay equation to solve for time or decay constant. The natural logarithm allows us to transform an exponential decay equation into a linear form, which is much easier to handle algebraically.
For instance, in calculating how long it takes for a percentage of a radioactive substance to decay, we must reverse the process and determine the time from the remaining proportion. By taking the natural logarithm of both sides of the decay equation, we can isolate the time variable and solve for it.
In summary, without the natural logarithm, solving the intricacies of radioactive decay would be much more complex, and it underscores how intertwined mathematical concepts are with practical applications in science and engineering.