Problem 17 - Algebra and Trigonometry

Question

\(17-24\) . These exercises use the radioactive decay model. Radioactive Radium The half-life of radium- 226 is 1600 years. Suppose we have a 22 -mg sample. (a) Find a function \(m(t)=m_{0} 2^{-4 / h}\) that models the mass remaining after \(t\) years. (b) Find a function \(m(t)=m_{0} e^{-r t}\) that models the mass remaining after \(t\) years. (c) How much of the sample will remain after 4000 years? (d) After how long will only 18 mg of the sample remain?

Step-by-Step Solution

Answer

(a) \( m(t) = 22 \times 2^{-t/1600} \); (b) \( m(t) = 22e^{-\left(\frac{\ln(0.5)}{1600}\right)t} \); (c) 3.89 mg remains after 4000 years; (d) It will take about 3792 years for the mass to reduce to 18 mg.

Step 1: Determine the Decay Rate

First, we need to understand that the decay rate is based on the half-life of the radioactive substance. The half-life of radium-226 is given as 1600 years.

Step 2: Model Using Half-Life Function

(a) Use the function \( m(t) = m_0 \cdot 2^{-t/h} \) to model the mass. Here, \( m_0 = 22 \) mg and \( h = 1600 \) years. The function becomes \[ m(t) = 22 \cdot 2^{-t/1600} \].

Step 3: Model Using Exponential Decay

(b) The exponential decay function \( m(t) = m_0 e^{-rt} \) requires the decay constant \( r \). Given the half-life, we find \( r \) using \( e^{-r \cdot 1600} = \frac{1}{2} \). Taking the natural logarithm on both sides gives \( -r\cdot 1600 = \ln(0.5) \), hence \( r = \frac{\ln(0.5)}{-1600} \). The function becomes \[ m(t) = 22e^{-\left(\frac{\ln(0.5)}{1600}\right)t} \].

Step 4: Calculate Remaining Mass After 4000 Years

(c) To find the remaining mass after 4000 years, substitute \( t = 4000 \) in one of the models (e.g., from Step 2): \[ m(4000) = 22 \cdot 2^{-4000/1600} = 22 \cdot 2^{-2.5} \]. Evaluating the expression, we find \( m(4000) \approx 3.89 \) mg.

Step 5: Calculate Time for Mass to Reach 18 mg

(d) To find how long it takes for the mass to be 18 mg, set \( m(t) = 18 \) and solve: \[ 18 = 22 \cdot 2^{-t/1600} \]. Solving the equation involves rearranging for \( t \): \( \frac{18}{22} = 2^{-t/1600} \). Take the logarithm: \( \ln(\frac{18}{22}) = -\frac{t}{1600} \ln(2) \), giving \( t \approx 3792 \) years.

Key Concepts

Half-Life

The concept of half-life is crucial to understanding how radioactive materials decay over time. A half-life is the time it takes for half of the radioactive substance to decay. In the case of radium-226, its half-life is 1600 years. This means that

• After 1600 years, only half of the radium-226 will remain.
• After another 1600 years, half of that remaining half will have decayed, leaving a quarter.

This pattern continues, resulting in an exponential decrease of the material over time. Half-life is a fixed property of each radioactive element, making it a fundamental concept in radioactive decay studies.

Exponential Decay

Exponential decay describes the decreasing quantity of a substance over time, often represented by a mathematical formula. This type of decay happens at a rate proportional to its current value, which is why it is called exponential. In terms of radium-226, the mass decreases in such a way that it halves at regular intervals, described by

• the equation: \( m(t) = m_0 \, 2^{-t/1600} \)

Here, \( m(t) \) represents the mass at time \( t \), \( m_0 \) is the initial mass, and the exponent \( -t/1600 \) reflects how time affects the decay. Given the vast potential for applications, from dating archaeological finds to medical treatments, understanding this exponential process is extremely valuable.

Decay Constant

The decay constant, often represented by the symbol \( r \), characterizes the rate at which a radioactive substance undergoes exponential decay. It is part of the formula \( m(t) = m_0 e^{-rt} \), which offers another way to express radioactive decay. The decay constant provides insights such as:

• A larger \( r \) indicates a faster decay rate.
• It is calculated using the half-life of the substance.

In mathematical terms, \( r \) is derived from the relation \( e^{-r \cdot \text{half-life}} = \frac{1}{2} \), solving which helps us understand how quickly half the substance will decay. For radium-226, if you compute \( r \) using its half-life of 1600 years, you gain a deeper understanding of its decay behavior over time.

Radium-226

Radium-226 is a naturally occurring radioactive element found in uranium ores. It has some distinct properties:

• A relatively long half-life of 1600 years, meaning it decays slowly.
• It emits alpha particles as it decays, which can be hazardous if ingested or inhaled.

Due to its radioactive nature, radium-226 was historically used in luminescent paints and is now primarily of interest for scientific research. Understanding how radium-226 decays is important for managing radioactive waste and studying geological and archaeological samples.