Problem 40 - Physics for Scientists and Engineers with Modern Physics

Question

(I) A radioactive material produces 1280 decays per minute at one time, and \(3.6 \mathrm{~h}\) later produces 320 decays per minute. What is its half-life?

Step-by-Step Solution

Answer

The half-life is approximately 1.8 hours.

Step 1: Understand the decay process

Radioactive decay follows an exponential pattern, described by the formula \( N(t) = N_0 \cdot e^{-\lambda t} \), where \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, and \( \lambda \) is the decay constant.

Step 2: Establish initial conditions

At \( t = 0 \), the decay rate is 1280 decays per minute. Thus, \( N_0 = 1280 \). After \( t = 3.6 \) hours, the decay rate becomes 320 decays per minute. Therefore, \( N(3.6) = 320 \).

Step 3: Set up the decay equation

Using the decay equation: \( 320 = 1280 \cdot e^{-\lambda \cdot 3.6} \).

Step 4: Solve for the decay constant \( \lambda \)

First, simplify the equation: \( \frac{320}{1280} = e^{-3.6\lambda} \). Then \( 0.25 = e^{-3.6\lambda} \). Take the natural logarithm of both sides to solve for \( \lambda \): \( \ln(0.25) = -3.6\lambda \). So \( \lambda = \frac{-\ln(0.25)}{3.6} \).

Step 5: Calculate \( \lambda \)

Calculate \( \ln(0.25) \approx -1.3863 \). Substitute into the equation: \( \lambda \approx \frac{1.3863}{3.6} \approx 0.3851 \).

Step 6: Use \( \lambda \) to find the half-life

The half-life \( T_{1/2} \) is determined by the relation \( T_{1/2} = \frac{\ln(2)}{\lambda} \). Substitute \( \lambda \approx 0.3851 \) to find \( T_{1/2} \).

Step 7: Calculate the half-life

Calculate \( \ln(2) \approx 0.6931 \). Then \( T_{1/2} = \frac{0.6931}{0.3851} \approx 1.8 \) hours.

Key Concepts

Radioactive Decay

Radioactive decay occurs when an unstable atomic nucleus loses energy by emitting radiation. This process is spontaneous and continues until the substance transforms into a stable form. In simple terms, radioactive materials have nuclei that undergo changes and release energy in the form of particles or electromagnetic waves. This process is random, but it averages out over large numbers of atoms according to a predictable pattern defined by its half-life.

A half-life is the time it takes for half of the radioactive material to decay. Different substances have different half-lives, ranging from fractions of a second to millions of years, making it a crucial parameter in understanding radioactive decay. The half-life helps to explain how quickly or slowly a material changes over time.

• Random but measurable process
• Transforms unstable to stable forms
• Key concept in nuclear physics

The unit of measurement for radioactive decay is 'decays per minute' or 'disintegrations per minute,' which tells us how many nuclear transformations occur in a given time period.

Decay Constant

The decay constant, represented as \( \lambda \), is a fundamental property of a radioactive isotope that describes the speed at which it decays. This constant is crucial because it helps calculate the rate of decay at any given time. It appears as a negative exponent in the radioactive decay equation.

In the equation \( N(t) = N_0 \cdot e^{-\lambda t} \), \( N(t) \) is the number of undecayed nuclei at time \( t \), \( N_0 \) is the initial number of nuclei, \( e \) is the base of the natural logarithm, and \( \lambda \) is the decay constant. The value of \( \lambda \) is directly proportional to the rate of decay—the larger the \( \lambda \), the faster the decay.

• A key parameter for determining decay rate
• Directly affects the half-life
• Determines how quickly particles decay

Understanding the decay constant allows scientists to predict the behavior of radioactive materials over time, providing valuable information for applications such as carbon dating, medical diagnostics, and nuclear power.

Exponential Decay

Exponential decay is a mathematical concept describing the process by which quantities decrease at a rate proportional to their current value. In the context of radioactive decay, this means that the rate of decay is always a fixed percentage of the remaining quantity, rather than a fixed amount.

In the formula \( N(t) = N_0 \cdot e^{-\lambda t} \), exponential decay is evident. Here, the rate of decay decreases as the quantity decreases, forming a downward curve when graphed over time. The exponential function \( e^{-\lambda t} \) plays a key role, ensuring each step of time reduces the quantity by the same proportion.

• Describes decreasing quantities over time
• Characterized by a constant percentage rate
• Leads to smooth, continuous decay curves

This concept also finds its applications beyond radioactivity, including in fields like finance (to model interest decay), biology (to model populations), and chemistry (to model the loss of reactants). The key takeaway is that while the quantity decreases, the process remains proportional, providing a predictable decay pattern.