Question

The half-life of \({ }^{27} \mathrm{Mg}\) is \(9.50 \mathrm{~min} .\) (a) Initially there were \(4.20 \times 10^{1227} \mathrm{Mg}\) nuclei present. How many \({ }^{27} \mathrm{Mg}\) nuclei are left 30.0 min later? (b) Calculate the \({ }^{27} \mathrm{Mg}\) activities (in Ci) at \(t=0\) and \(t=30.0\) min. (c) What is the probability that any one \({ }^{27} \mathrm{Mg}\) nucleus decays during a 1 -s interval? What assumption is made in this calculation?

Short answer

About 5.38% of initial nuclei remain; activity at t=0 is ~0.76 Ci. Probability of decay in 1 s is ~0.0012.

Step-by-step solution

Understand the Half-Life Formula

The half-life of an element is the time taken for half of its nuclei to decay. The formula to calculate the remaining nuclei after time \( t \) is given by the exponential decay formula: \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]where \( N_0 \) is the initial number of nuclei and \( t_{1/2} \) is the half-life.

Calculate the Remaining Nuclei After 30 Minutes

Given that \( t = 30 \) minutes, \( t_{1/2} = 9.50 \) minutes, and \( N_0 = 4.20 \times 10^{12} \), substitute these values into the formula:\[ N(30) = 4.20 \times 10^{12} \times \left( \frac{1}{2} \right)^{\frac{30}{9.5}} \].Calculate \( \frac{30}{9.5} \approx 3.16 \) and then raise \( \frac{1}{2} \) to this power, followed by multiplying with \( N_0 \).

Calculate Activities at t=0 and t=30 minutes

Activity \( A \) is related to the number of nuclei by \( A = \lambda N \), where \( \lambda = \frac{\ln(2)}{t_{1/2}} \).(a) At \( t = 0 \):\[ A_0 = \left( \frac{\ln(2)}{9.50 \times 60} \right) \times 4.20 \times 10^{12} \].(b) At \( t = 30 \): Use the \( N(30) \) found earlier to compute \( A(30) = \lambda N(30) \).

Calculate the Decay Probability Over 1 Second

The probability of decay over a small interval \( \Delta t \) for a single nucleus is \( P = \lambda \Delta t \).Plug in \( \lambda = \frac{\ln(2)}{9.50 \times 60} \) and \( \Delta t = 1 \) second to find \( P \).

Discuss Assumptions

The calculation assumes constant decay rate during the small time interval (1 second). This is valid as the half-life is much larger than the interval duration so the decay rate doesn't change significantly.

Key concepts

Half-Life

Understanding the concept of half-life is crucial when dealing with radioactive decay. Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. For example, the given half-life of ^{27} \text{Mg} is \(9.50 \text{ min}\). It means that every 9.50 minutes, the number of original magnesium nuclei is reduced to half.
This predictable pattern allows us to calculate the remaining number of nuclei at any given time using the exponential decay formula.
By knowing the half-life, you can determine how long it will take for radioactive material to decrease to a safe level. It also helps to predict when an isotope might become useful for specific applications or, conversely, reach a state where it is no longer useful.

Exponential Decay

Exponential decay is a process by which a quantity decreases at a rate proportional to its current value. This is a fundamental characteristic of radioactive decay.
The general formula for exponential decay is N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}}, where \(N(t)\) represents the number of nuclei remaining after time \(t\), \(N_0\) is the initial quantity of nuclei, and \(t_{1/2}\) is the half-life.
This formula ensures that the decay process is predictable, demonstrating how half of the remaining nuclei decay with each half-life passed. Understanding this concept is essential in fields like nuclear physics, medicine, and environmental science.

Activity Calculation

The activity of a radioactive sample denotes how many radioactive decays are occurring in it per unit of time, typically measured in curies (Ci). This helps us understand the intensity of the radioactivity.
To find the activity \(A\), we use the equation \(A = \lambda N\), where \(\lambda\) is the decay constant, and \(N\) represents the number of radioactive nuclei.

• At time \(t=0\), the activity can be calculated using the initial number of radioactive nuclei. You simply plug \(N_0\) into the equation along with \(\lambda\) to find \(A_0\).
• Similarly, at any time \(t\), you use \(N(t)\) to find the current activity \(A(t)\).

Knowing the initial and current activities can give insights into the rate at which the sample becomes less radioactive, which is particularly important in fields like nuclear medical treatments and radiation safety.

Decay Constant

The decay constant \(\lambda\) is a vital parameter in understanding how quickly a radioactive substance will decay. It is inversely related to half-life and is expressed as \(\lambda = \frac{\ln(2)}{t_{1/2}}\). This constant provides a rate of decay that is independent of the sample size or initial quantity.
When calculating the probability of a single nucleus decaying within a small time frame, like 1 second, you use this decay constant: \(P = \lambda \Delta t\). This reveals how the decay constant is a direct measure of the likelihood of decay occurring within that particular timeframe.
Understanding the decay constant allows scientists to predict and model decay processes with precision. This is particularly valuable in both theoretical studies and practical applications involving radioactivity and safety assessments.