Problem 53 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The activity of a sample drops by a factor of 4.0 in 8.6 minutes. What is its half-life?

Step-by-Step Solution

Answer

The half-life is 4.3 minutes.

Step 1: Understanding Exponential Decay

In radioactive decay, the activity of a sample decreases over time following an exponential relationship. If the activity drops by a factor of 4, it means that the original activity is now one-fourth of its initial value.

Step 2: Using the Decay Formula

The decay of a radioactive sample can be represented by the formula \( A = A_0 e^{-kt} \), where \( A \) is the activity at time \( t \), \( A_0 \) is the initial activity, and \( k \) is the decay constant. Given that \( A = \frac{1}{4}A_0 \) when \( t = 8.6 \) minutes, we can rewrite the equation as \( \frac{1}{4} = e^{-8.6k} \).

Step 3: Solving for the Decay Constant k

To solve for \( k \), take the natural logarithm of both sides: \( \ln\left(\frac{1}{4}\right) = -8.6k \). Simplifying, we find \( k = -\frac{\ln(\frac{1}{4})}{8.6} \). Re-evaluating, \( \ln(\frac{1}{4}) = -\ln(4) \), hence \( k = \frac{\ln(4)}{8.6} \).

Step 4: Calculating Half-Life

The half-life \( T_{1/2} \) is related to the decay constant by the formula \( T_{1/2} = \frac{\ln(2)}{k} \). Substitute \( k = \frac{\ln(4)}{8.6} \) into the formula: \( T_{1/2} = \frac{\ln(2)}{\frac{\ln(4)}{8.6}} \). Simplifying, the half-life \( T_{1/2} = \frac{8.6 \times \ln(2)}{\ln(4)} \).

Step 5: Simplifying the Expression

Since \( \ln(4) \) can be written as \( 2\ln(2) \), substitute this into the expression: \( T_{1/2} = \frac{8.6 \times \ln(2)}{2\ln(2)} = \frac{8.6}{2} \). Thus, \( T_{1/2} = 4.3 \) minutes.

Key Concepts

Half-Life Calculation

The concept of half-life is pivotal when dealing with radioactive decay. Half-life, denoted as \( T_{1/2} \), represents the time it takes for a given sample to reduce to half its original activity due to radioactive decay. Knowing the half-life of a substance helps scientists predict how long a radioactive material will remain active.

To calculate half-life, we use the relationship with the decay constant \( k \). The formula linking these two is \( T_{1/2} = \frac{\ln(2)}{k} \). The natural logarithm of 2, \( \ln(2) \), approximately equals 0.693. This steady ratio reflects the exponential nature of radioactive decay, showing how half-lives remain constant for each particular isotope, regardless of the initial amount.

By understanding this relationship, we can determine the rate at which a radioactive sample decays and predict future activity levels, which is essential in fields like nuclear medicine, archaeology (carbon dating), or nuclear power generation.

Exponential Decay Formula

Radioactive decay is a natural, spontaneous process where unstable nuclei lose energy by emitting radiation. This process can be mathematically expressed using the exponential decay formula, \( A = A_0 e^{-kt} \).

In this formula:

• \( A \) is the activity of the radioactive sample at any time \( t \).
• \( A_0 \) represents the initial activity of the sample.
• \( k \) is the decay constant, specific to each radioactive substance.
• \( t \) is the time elapsed.

The exponential decay formula shows how the activity decreases exponentially over time. For example, in our exercise, the activity dropped to one-fourth of its original value in 8.6 minutes. Plugging these values into the formula allows us to solve for the decay constant and, subsequently, determine the half-life.

Understanding this formula allows scientists to make accurate predictions about the decay behavior of radioactive materials.

Decay Constant

The decay constant, denoted as \( k \), is a fundamental parameter in the study of radioactive decay. It quantifies the rate at which a radioactive substance disintegrates. Each radioactive isotope has a unique decay constant.

The relation between the decay constant and half-life is given by \( T_{1/2} = \frac{\ln(2)}{k} \), which allows precise computation of how long a certain isotope will take to decay to half of its initial activity.

To find \( k \), you often use the rearranged exponential decay formula: \( k = \frac{\ln(A_0/A)}{t} \). In our specific problem, since the sample's activity reduced to one-fourth in 8.6 minutes, we applied this formula first to solve for \( k \) and then used \( k \) to find the half-life of the substance. The knowledge of \( k \) is crucial, especially in applications like nuclear waste management, radiotherapy, and radiocarbon dating where understanding exact decay rates is essential.