Problem 63 - Chemistry The Science in Context

Question

A 1.00 -mg sample of \(^{192}\) Ir was inserted into the artery of a heart patient. After 30 days, 0.756 mg remained. What is the half-life of \(^{192}\) Ir?

Step-by-Step Solution

Answer

Answer: The half-life of \(^{192}\)Ir is approximately 15.6 days.

Step 1: Identify given values and plug into the formula

We are given \(N_0 = 1.00\) mg, \(N(t) = 0.756\) mg, and \(t = 30\) days. We will plug these values into the half-life decay formula: $$ 0.756 = 1.00 \cdot (0.5)^{\frac{30}{t_{1/2}}} $$

Step 2: Solve for \(t_{1/2}\)

Now we need to solve for \(t_{1/2}\). To do this, we will first isolate the term with \(t_{1/2}\) by dividing both sides by 1.00: $$ 0.756 = (0.5)^{\frac{30}{t_{1/2}}} $$ The next step is to take the logarithm of both sides, which allows us to solve for the exponent. In this case, we will use the base 2 logarithm since we are dealing with a half-life: $$ log_2 (0.756) = \frac{30}{t_{1/2}} $$ Now, we can isolate \(t_{1/2}\) by multiplying both sides by \(t_{1/2}\) and then dividing by \(log_2 (0.756)\): $$ t_{1/2} = \frac{30}{log_2 (0.756)} $$ Finally, calculate \(t_{1/2}\) using a calculator: $$ t_{1/2} \approx 15.6 \ \text{days} $$

Step 3: State the half-life of \(^{192}\)Ir

The half-life of \(^{192}\)Ir is approximately 15.6 days.

Key Concepts

Nuclear Decay

Nuclear decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. This can occur in several ways, such as through the emission of alpha particles, beta particles, or gamma rays.

In the context of the half-life problem of a medical radioactive isotope like Iridium-192 (¹⁹²Ir), which is used in medical treatments, we can understand this decay as a transformation from a less stable form to a more stable one. It's crucial for medical professionals to know the half-life of such isotopes to plan treatments correctly and ensure patient safety. The half-life specifically measures the time it takes for half of the original amount of a radioactive substance to decay.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are atoms with an unstable nucleus that gain stability by undergoing radioactive decay. These isotopes have numerous applications across different fields, including medicine, archaeology, and environmental science.

Iridium-192 is a prime example, used in brachytherapy to treat various forms of cancer. Understanding the half-life of radioactive isotopes is critical for their safe and effective use. In the given exercise, the remaining amount of ¹⁹²Ir after a certain period is integral to determining its half-life and, consequently, its suitability for specific medical procedures.

Exponential Decay

Exponential decay models the process by which the quantity of a substance decreases at a rate proportional to its current value. It's a continuous, rapid decline in the observable amount of a substance, such as a radioactive isotope.

The formula to describe this rate is often written as N(t) = N₀e^-kt, where N(t) is the quantity at time t, N₀ is the original amount, e is the base of the natural logarithm, and k is a constant representing the decay rate. For half-life calculations, this model simplifies to using a factor of 0.5 raised to the power of time divided by the half-life, as seen in our example using ¹⁹²Ir.

Logarithmic Functions

Logarithmic functions are the mathematical tools we use to undo exponential operations. In the context of half-life calculations, we apply logarithms to both sides of the equation to solve for the unknown half-life when dealing with exponential decay data.

Using a logarithmic function allows us to isolate the exponent when the equation is in the form of an exponential decay formula. This isolation is essential to find the half-life of a substance, stepping away from the continuous rate of decay and allowing for a discrete and understandable measurement, which is crucial for practical applications such as determining how long a radioactive isotope will be effective in a medical treatment scenario.