Problem 8 - Calculus with Concepts in Calculus

Question

Suppose an unknown radioactive substance produces 4000 counts per minute on a Geiger counter at a certain time, and only 500 counts per minute 4 days later. Assuming that the amount of radioactive substance is proportional to the number of counts per minute, determine the half-life of the radioactive substance.

Step-by-Step Solution

Answer

The half-life of the radioactive substance is approximately \( \frac{4}{3} \) days.

Step 1: Define the Problem

We are given an initial count rate of 4000 counts per minute and a count rate of 500 counts per minute 4 days later. We need to find the half-life of the substance, which is the time for the count rate to reduce to half its initial amount.

Step 2: Use the Exponential Decay Formula

We use the exponential decay formula, which is given by \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the initial amount, \( N(t) \) is the amount at time \( t \), and \( k \) is the decay constant.

Step 3: Calculate Decay Constant

Substitute the values into the decay formula: \( 500 = 4000 e^{-4k} \). Rearrange and solve for \( k \): \( \frac{500}{4000} = e^{-4k} \). Simplifying gives us \( \frac{1}{8} = e^{-4k} \). Taking the natural log on both sides, \( \ln\left(\frac{1}{8}\right) = -4k \). Thus, \( k = -\frac{\ln(\frac{1}{8})}{4} \).

Step 4: Express Half-life with Decay Constant

The half-life \( T_{1/2} \) is related to the decay constant by the formula \( T_{1/2} = \frac{\ln(2)}{k} \).

Step 5: Calculate Half-life

Substitute \( k \) into the half-life formula: \( T_{1/2} = \frac{\ln(2)}{-\frac{\ln(\frac{1}{8})}{4}} \). Simplifying, we get \( T_{1/2} = \frac{4 \ln(2)}{\ln(8)} \). Since \( \ln(8) = 3 \ln(2) \), \( T_{1/2} = \frac{4}{3} \text{ days} \).

Step 6: Verify the Calculation

The derived half-life of \( \frac{4}{3} \) days should work if we plug it back and verify if after \( \frac{4}{3} \) days, the count rate reduces to half initially provided values. It correctly checks out with the calculations done here.

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept in calculus that describes the process where a quantity decreases at a rate proportional to its current value. This can be mathematically expressed using an exponential function of the form \( N(t) = N_0 e^{-kt} \).
Here, \( N(t) \) represents the amount of substance remaining at time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant. What makes exponential decay interesting is that it applies to a wide range of natural phenomena, including population decay and cooling processes.

• The rate of decay is directly proportional to the amount present.
• As time goes on, the substance decreases more slowly due to less of it remaining to decay.
• The function never actually reaches zero, describing a smooth, continuous decay over time.

Using calculus, one can differentiate this function to find instantaneous rates of change, providing insights into how rapidly the quantity decreases at any point in time.

Radioactive Decay

Radioactive decay is a specific type of exponential decay where unstable atomic nuclei lose energy by emitting radiation. This process results in the transformation of the material into a different substance.
Each type of radioactive material has its own unique rate of decay, often characterized by a decay constant \( k \) in the formula \( N(t) = N_0 e^{-kt} \).

• Radioactive decay is stochastic; it happens at random intervals, but can be modeled statistically over time.
• Examples include processes like carbon-14 decay, used in radiocarbon dating.
• Safety protocols in handling radioactive materials often rely on understanding their decay rates.

In practice, scientists measure the activity of a radioactive sample (often in counts per minute) using instruments like Geiger counters, which can detect radiation as it decays.

Half-life

Half-life is the time required for a quantity to reduce to half its initial value in a process following exponential decay, such as radioactive decay. It's an important measure because it provides a tangible way to understand the rate and extent of decay in a given material.
The half-life \( T_{1/2} \) is related to the decay constant by \( T_{1/2} = \frac{\ln(2)}{k} \).

• Half-life is used extensively in fields like archaeology and medicine for dating artifacts and understanding drug efficacy respectively.
• It provides a consistent metric for comparing rates of decay across different substances.
• This measure is stable over time and is independent of initial amounts.

Understanding half-life allows scientists and engineers to predict how long a substance will remain active or hazardous.

Decay Constant

The decay constant \( k \) is a parameter in the exponential decay equation \( N(t) = N_0 e^{-kt} \) that describes the rate of decay of a substance. It's a critical value as it helps determine how quickly a substance decreases over time.
The decay constant is directly related to the half-life, offering a way to convert between the two measures.

• A larger \( k \) indicates a faster decay process.
• It's derived from the natural logarithm of the ratio of remaining to initial substance over time.
• The decay constant has units of time inverse (e.g., 1/seconds), reflecting its role in the rate measure.

Determining \( k \) accurately is essential for predicting how long it takes for a substance to decay to a particular amount, which is crucial in applications such as nuclear safety and pharmacokinetics.