Problem 103 - Chemistry: An Atoms-Focused Approach

Question

Rhodium-105 is an isotope currently under investigation in diagnostic applications. The half-life of \(^{105} \mathrm{Rh}\) is \(35.4 \mathrm{h}\) which is sufficiently long for transport from the supplier a hospital. A supplier ships \(250 \mathrm{mg}\) of \(^{105} \mathrm{RhCl}_{3}\) overnight (12 hours). a. What percentage of the \(^{105}\) Rh remains upon arrival? b. How long will it take for \(95 \%\) of the \(^{105} \mathrm{Rh}\) to decay?

Step-by-Step Solution

Answer

Short Answer: After 12 hours of transport, approximately 82.86% of the initial Rhodium-105 isotope remains. It takes about 116.6 hours for 95% of the Rhodium-105 to decay, at which point only 5% remains.

Step 1: Relationship between half-life and decay constant

The half-life is the time it takes for half of the isotope to decay, and it can be used to find the decay constant, which is a measure of how quickly the isotope decays. The relationship between half-life (T) and decay constant (k) is given by: \[ T = \frac{\ln(2)}{k} \]

Step 2: Calculate the decay constant

Given the half-life as \(35.4 \mathrm{h}\), we can use the relationship in step 1 to find the decay constant (k): \[ k = \frac{\ln(2)}{35.4} \]

Step 3: Use exponential decay formula to find remaining percentage

The exponential decay formula for the remaining percentage of an isotope is: \[ N(t) = N_0 e^{-kt} \]Where \(N(t)\) is the amount remaining at time \(t\), \(N_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is the number of time units (in this case hours) passed. In this exercise, we need to find the remaining percentage after 12 hours with the initial amount of \(250\ \mathrm{mg}\). \[ N(12) = 250 e^{-\frac{\ln(2)}{35.4} \times 12} \]Calculate the amount remaining and divide it by the initial amount to get the percentage remaining.

Step 4: How long will it take for 95% to decay

We need to find the time it takes for only 5% to remain (since 95% has decayed). Using the exponential decay formula, we can set up the equation: \[ N(t) = 0.05N_0 \]Where \(t\) is the time it takes for 95% to decay. In this case, rearrange the equation to solve for time \(t\): \[ t = \frac{\ln(\frac{N(t)}{N_0})}{-k} \]Substitute \(N(t)=0.05N_0\) and the decay constant calculated in step 2, and solve for the time \(t\).

Key Concepts

Half-Life

The "half-life" of an isotope is the time it takes for half of the material to decay away. This means if you had 100 grams of a substance with a half-life of 10 hours, you would have 50 grams remaining after 10 hours. It is a critical factor in understanding radioactive decay because it helps determine how long an isotope remains active or useful. When calculating, remember that the decay process continues, so after another half-life, you would have 25 grams left and so on. The half-life, denoted by \( T \), is always constant for a given isotope, and it’s unique to each one, providing a reliable measure for calculations related to decay over time.

Exponential Decay Formula

The "exponential decay formula" describes how the amount of a radioactive substance decreases over time. In mathematical terms, it is represented as: \[ N(t) = N_0 e^{-kt} \] Here, \( N(t) \) is the quantity remaining after time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( e \) is the base of the natural logarithm. This formula is essential for predicting how much of a material will remain at any given time, which is useful for applications like medical diagnostics or nuclear energy operations. To calculate, you'll substitute the given values for initial amount, the decay constant, and the time elapsed to find the remaining quantity. Much like a bank account earning interest (but in reverse), this formula can show rapid decreases in quantity even if the half-life seems long.

Decay Constant

The "decay constant" \( k \), connects the half-life of an isotope with its rate of decay. It is determined using the formula: \[ k = \frac{\ln(2)}{T} \] where \( T \) is the half-life. The decay constant is pivotal as it indicates how quickly a material decreases in radioactive strength. The natural logarithm \( \ln(2) \) is approximately 0.693, which is a constant involved in these calculations. A larger value of \( k \) means a quicker decay, while a smaller one implies a slower process. Understanding the decay constant helps scientists and engineers to plan for how long a substance can remain effective, whether for scientific experiments, medical treatments, or storage of nuclear materials. It’s a crucial number that translates half-life into everyday usability and applicability in decay equations.