Problem 25 - Calculus with Concepts in Calculus

Question

Suppose a "parent" substance decays exponentially into a "daughter" substance so that the amount \(P(t)\) of the parent remaining after \(t\) years is given by $$ P(t)=P(0) e^{-\lambda t} $$ The amount \(D(t)\) of the daughter substance is given by $$ D(t)=P(0)-P(t) $$ Show that $$ t=\frac{1}{\lambda} \ln \left(\frac{D(t)}{P(t)}+1\right) $$ (This formula can be used to determine the age of an object; see Exercise 26.)

Step-by-Step Solution

Answer

The formula for \(t\) is derived as \(t = \frac{1}{\lambda} \ln\left(\frac{D(t)}{P(t)} + 1\right)\).

Step 1: Understanding the Problem

We have a parent substance that decays exponentially over time, and we are given expressions for the amounts of parent substance, \(P(t)\), remaining and the daughter substance, \(D(t)\), formed. We need to derive the expression for \(t\) when given \(D(t)\) and \(P(t)\).

Step 2: Express \(D(t)\) in Terms of \(P(t)\) and \(P(0)\)

We know \(D(t) = P(0) - P(t)\). Hence:\[ P(t) = P(0) - D(t) \]

Step 3: Substitute P(t) in the Exponential Decay Formula

Substitute \(P(t)\) from the exponential decay formula:\[ P(t) = P(0) e^{-\lambda t} \] into the expression:\[ P(0) e^{-\lambda t} = P(0) - D(t) \]

Step 4: Solve for the Exponential Term

Rearrange and solve:\[ P(0) e^{-\lambda t} + D(t) = P(0) \]\[ e^{-\lambda t} = \frac{P(0) - D(t)}{P(0)} \]

Step 5: Isolate \(-\lambda t\)

Taking the natural logarithm on both sides:\[ -\lambda t = \ln\left(\frac{P(0) - D(t)}{P(0)}\right) \]

Step 6: Solve for \(t\)

Rearrange to solve for \(t\):\[ t = -\frac{1}{\lambda} \ln\left(\frac{P(0) - D(t)}{P(0)}\right) \]Use \(\frac{P(0) - D(t)}{P(t)} = \frac{D(t)}{P(t)} + 1\) to substitute:\[ t = \frac{1}{\lambda} \ln\left(\frac{D(t)}{P(t)} + 1\right) \]

Step 7: Conclusion

We have derived the expression for \(t\) in terms of \(\lambda\), \(D(t)\), and \(P(t)\). Thus, showing how the given formula tells us the age of a sample given the amounts of parent and daughter substances.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This phenomenon happens over time and results in the transformation of one element, known as the "parent" into another, called the "daughter." The process by which these transformations occur is described mathematically by exponential decay formulas.
- Exponential decay indicates that the quantity of the parent substance decreases at a rate that is proportional to its current amount.- Over time, as more of the parent substance decays, the amount remaining becomes smaller, and this attrition follows an exponential pattern.In the context of our problem, the parent substance decays and forms the daughter substance. The amount of the parent substance remaining after a time 't' years is denoted as \( P(t) = P(0)e^{-\lambda t} \), where \( P(0) \) is the initial amount and \( \lambda \) is the decay constant. The amount of daughter substance formed is \( D(t) = P(0) - P(t) \). Understanding this relationship helps us to calculate the age of a sample by measuring the remaining parent and formed daughter substances.

Natural Logarithm

The natural logarithm is a mathematical function that is widely used in calculations involving exponential decay, particularly in physics and chemistry.
- It is the inverse operation of exponentiation, specifically with the base of the natural exponential function, Euler's number \( e \).- The natural logarithm is denoted as \( \ln \), and it transforms multiplication into addition, which simplifies complex exponential expressions. In our decay problem, the natural logarithm is employed to isolate the variable \( t \) when solving the decay equation for time. By taking the natural logarithm on both sides of the equation, we can simplify the expression \( e^{-\lambda t} = \frac{P(0) - D(t)}{P(0)} \) into \( -\lambda t = \ln \left(\frac{P(0) - D(t)}{P(0)}\right) \). This conversion enables clear calculation and derivation of time within the context of radioactive decay.

Rate of Decay

The rate of decay in exponential decay problems like radioactive decay refers to how quickly the parent substance is transformed into the daughter substance. This transformation rate is influenced by several factors, most notably the decay constant \( \lambda \).
- The decay constant \( \lambda \) is a positive value that characterizes how fast a radioactive substance undergoes decay.For a given decay process, the relationship between time and the amount of parent substance follows the formula \( P(t) = P(0)e^{-\lambda t} \). Here, \( \lambda \) helps determine how fast the parent amount decreases over time.
- A larger \( \lambda \) means a quicker decay, and therefore, a faster reduction of the parent substance.- In our exercise, the derived formula \( t = \frac{1}{\lambda} \ln \left(\frac{D(t)}{P(t)} + 1\right) \) utilizes this constant to calculate the time elapsed accurately, providing insights into the sample's age based on the current quantities of parent and daughter substances. Recognizing the importance of \( \lambda \) helps students appreciate the dynamic nature of radioactive materials and their real-world implications in dating and other applications.