Problem 70 - Chemistry The Science in Context

Question

Balloon angioplasty is a common procedure for unclogging arteries in patients suffering from arteriosclerosis. Iridium-192 therapy is being tested as a treatment to prevent reclogging of the arteries. In the procedure, a thin ribbon containing pellets of \(^{192} \mathrm{Ir}\) is threaded into the artery. The half-life of \(^{192} \mathrm{Ir}\) is 74 days. How long will it take for \(99 \%\) of the radioactivity from \(1.00 \mathrm{mg}\) of \(^{192}\) Ir to disappear?

Step-by-Step Solution

Answer

Answer: It takes approximately 246.2 days for 99% of the radioactivity from a 1.00 mg sample of Iridium-192 to disappear.

Step 1: Determine the remaining amount of Iridium-192

Since we are asked to find the time it takes for 99% of the radioactivity to disappear, we first need to figure out what 1% of the original 1.00 mg sample amounts to. This can be calculated as: 1% of 1.00 mg = 0.01 * 1.00 mg = 0.01 mg

Step 2: Using the half-life formula

We can use the half-life formula to relate the initial amount, remaining amount, half-life, and time elapsed. The formula is as follows: \(N(t) = N_0 \times \frac{1}{2}^{\frac{t}{t_{1/2}}}\) Where: \(N(t)\) is the remaining amount after time t, \(N_0\) is the initial amount, \(t\) is the time elapsed, \(t_{1/2}\) is the half-life of the radioactive substance (in this case, 74 days for Iridium-192)

Step 3: Plug values into the formula and solve for t

Given the initial amount (\(N_0 = 1.00 \:\text{mg}\)), the remaining amount (\(N(t) = 0.01 \:\text{mg}\)), and the half-life (\(t_{1/2} = 74 \:\text{days}\)), we can plug these values into the half-life formula and solve for t: \(0.01 = 1.00 \times \frac{1}{2}^{\frac{t}{74}}\) Now, we need to solve for t. We can use logarithms to isolate the variable t: \(\frac{0.01}{1.00} = \frac{1}{2}^{\frac{t}{74}}\) \(\log_2{\frac{1}{100}} = \frac{t}{74}\) \(t = 74 \times \log_2{100}\)

Step 4: Calculate the time

Using a calculator, we can now compute the value of t: \(t = 74 \times \log_2{100} \approx 246.2\) Therefore, it takes approximately 246.2 days for 99% of the radioactivity from a 1.00 mg sample of Iridium-192 to disappear.

Key Concepts

Understanding Radioactive Decay

Radioactive decay is a fundamental natural process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs spontaneously in a variety of forms, such as the emission of alpha particles, beta particles, or gamma rays. A key feature of radioactive substances is the half-life, which is the time required for half of the radioactive nuclei in a sample to decay.

To delve into the mechanics of radioactive decay, imagine a large pool of unstable atoms. These atoms all have a certain probability of decaying over a given time interval, but it's impossible to predict which ones will decay at any moment. Over time, as decays accumulate, the quantity of the radioactive substance decreases in a predictable way, described by its half-life.

It's essential to grasp that the half-life value is constant for a given isotope; this is what enables scientists and medical professionals to calculate how long it will take for a substance to become safe or to determine the appropriate dosage for medical treatments like Iridium-192 therapy.

Iridium-192 Therapy and Its Applications

Iridium-192 therapy is a medical treatment that harnesses the principles of radioactive decay to prevent the reclogging of arteries following a procedure known as balloon angioplasty. Iridium-192, designated by the isotopic symbol 192-Ir, has a half-life of 74 days and emits gamma rays, which minimizes the risk of nearby cells becoming damaged.

In Iridium-192 therapy, a thin ribbon embedded with pellets of the isotope is inserted into the artery. The emitted gamma rays help to prevent cells from proliferating too rapidly, thus reducing the risk of restenosis, or reclogging of the arteries. Understanding the half-life of Iridium-192 is crucial for medical professionals to calculate the exposure time needed to achieve the therapeutic effect while minimizing potential side effects.

Logarithmic Functions in Half-Life Calculations

Logarithmic functions are an indispensable part of solving half-life problems in radioactive decay scenarios. These functions can help us translate the exponential nature of decay into a linear form that can be easily understood and computed with.

Essentially, logarithms are the inverses of exponential functions. When dealing with half-lives, the decay equation often presents itself in an exponential form. To isolate the variable representing time, such as in the half-life calculation for Iridium-192 therapy, we employ the logarithm to 'undo' the exponent.

In the case of solving for the time it takes for a specific percentage of a radioactive sample to decay, we use a logarithmic function that corresponds to the base of the exponential decay (which is 2, because we're dealing with a 'half-life'). The function allows us to find the time required for a certain amount of decay to occur. This application of logarithmic functions is a powerful tool in the medical field, environmental science, and any other area where understanding the timing of decay is critical.