Problem 39 - Physics

Question

Two radioactive nuclei \(A\) and \(B\) are present in equal numbers to begin with. Three days later, there are three times as many A nuclei as there are \(\mathrm{B}\) nuclei. The half-life of species \(\mathrm{B}\) is 1.50 days. Find the half-life of species \(\mathrm{A}\).

Step-by-Step Solution

Answer

The half-life of species A is approximately 2.37 days.

Step 1: Understanding the Problem

We have two types of radioactive nuclei, \(A\) and \(B\), which start off in equal quantities. After 3 days, \(A\)'s quantity is three times that of \(B\). We're given the half-life of \(B\) as 1.5 days and need to find the half-life of \(A\).

Step 2: Writing Initial Equations

Let the initial number of nuclei for both \(A\) and \(B\) be \(N_0\). After 3 days, \(N_A = 3N_B\). The decay formula is given by \(N(t) = N_0 e^{-\lambda t}\), where \(\lambda\) is the decay constant. Let \(\lambda_A\) and \(\lambda_B\) be the decay constants for \(A\) and \(B\), respectively.

Step 3: Calculating Decay Constant for B

The half-life formula is \(t_{1/2} = \frac{\ln(2)}{\lambda}\). For \(B\), \(1.5 = \frac{\ln(2)}{\lambda_B}\). Solving for \(\lambda_B\), we get \(\lambda_B = \frac{\ln(2)}{1.5}\).

Step 4: Expressing Nuclei Count After 3 Days

For \(B\), after 3 days, \(N_B = N_0 e^{-\lambda_B \times 3}\), where \(\lambda_B = \frac{\ln(2)}{1.5}\). Substitute this into the equation to express \(N_B\). Similarly, for \(A\), \(N_A = N_0 e^{-\lambda_A \times 3}\).

Step 5: Using Given Relationship Between A and B

From the problem, we know \(N_A = 3N_B\). Substituting the exponential decay expressions, we have \(N_0 e^{-\lambda_A \times 3} = 3N_0 e^{-\lambda_B \times 3}\). Simplifying, \(e^{-\lambda_A \times 3} = 3e^{-\lambda_B \times 3}\).

Step 6: Solving for Decay Constant of A

Taking natural logarithms on both sides of the equation \(e^{-\lambda_A \times 3} = 3e^{-\lambda_B \times 3}\), we get \(-\lambda_A \times 3 = \ln(3) - \lambda_B \times 3\). Now solve for \(\lambda_A\): \(\lambda_A = \frac{\ln(3) + 3 \cdot \lambda_B}{3}\).

Step 7: Calculating Half-life of A

Substitute \(\lambda_B = \frac{\ln(2)}{1.5}\) into the equation for \(\lambda_A\). Then calculate \(t_{1/2,A} = \frac{\ln(2)}{\lambda_A}\).

Step 8: Final Calculation

With all substitutions completed, calculate the numerical value for the half-life of \(A\). Carry out precise arithmetic operations to determine \(t_{1/2,A}\).

Key Concepts

Half-life

The concept of half-life is crucial in understanding radioactive decay. It refers to the time required for half of a given sample of radioactive nuclei to decay. In simpler terms, if you start with a certain amount of a radioactive substance, after one half-life, only half of it will remain unchanged. This process continues such that after two half-lives, only a quarter of the original amount is still intact, and so on.

Half-life is a constant for a given isotope. It’s a measure of how quickly a substance undergoes radioactive decay. For example, in our exercise, species B has a half-life of 1.5 days. This means every 1.5 days, half of the remaining B nuclei will decay. The half-life helps scientists predict how long a radioactive substance will remain active. Understanding this concept is essential for fields like nuclear medicine and radiocarbon dating.

• The half-life is independent of the initial amount of substance.
• It allows for easy prediction of decay over time.
• All radioactive isotopes have unique half-lives.

Decay Constant

The decay constant, denoted by the Greek letter lambda (\( \lambda \)), is a key component in understanding how quickly radioactive decay happens. It's directly related to the half-life of a radioactive substance. The relationship is given by the formula \( t_{1/2} = \frac{\ln(2)}{\lambda} \), where \( t_{1/2} \) is the half-life.

The decay constant represents the probability per unit time that a single radioactive nucleus will decay. For example, a high decay constant means the substance decays quickly, resulting in a short half-life.
To calculate the decay constant for any given isotope, you rearrange the half-life formula. For species B in our exercise, we calculated it as \( \lambda_B = \frac{\ln(2)}{1.5} \). This value helps in further calculations to find how the amount of radioactive material decreases over time.

• Decay constant is unique to each radioactive isotope.
• It helps determine the speed of decay.
• Inversely proportional to the half-life.

Exponential Decay

Exponential decay describes the process by which the quantity of a radioactive substance decreases over time at a rate proportional to its current value. This decay is mathematically expressed by \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the amount of substance remaining at time \( t \), \( N_0 \) is the initial amount, and \( \lambda \) is the decay constant.

In our exercise, this exponential model helped us determine how many nuclei of type A and B remain after a certain period. After 3 days, with known decay constants, we could explore how much of A and B still exists compared to their initial quantities.
The property that makes exponential decay unique is its characteristic timescale, provided by the half-life, over which the process occurs. This allows us to apply the model to predict future quantities of decaying substances efficiently.

• Exponential decay ensures a consistent proportional reduction over equal time intervals.
• Forms the basis of modeling in radiocaractive processes.
• Simplifies calculations in predicting the time evolution of substance decay.