Problem 1
Question
The equation \(F=e^{-k t}\) describes the fraction \(F\) of an original isotope remaining after a period of \(t\) years; the exponent is negative because it refers to a decrease over time. The constant \(k\) provides a measure of how rapidly the original isotope decays. For the decay of carbon- 14 to nitrogen- \(14, k=0.00012097\). To find \(t\), rearrange the equation by following these steps: (a) Take the natural logarithm of both sides of the equation: \(\ln (F)=\) In(e \({ }^{-k t}\) ). Rewrite the right side of this equation by applying the following rule: \(\ln \left(e^{x}\right)=x \ln (e)\). (b) Since \(\ln (e)=1\), simplify the equation. (c) Now solve for \(t\) and write the equation in the form
Step-by-Step Solution
Verified Answer
The time \( t \) is given by the equation \( t = \frac{\ln(F)}{-k} \).
1Step 1: Apply Natural Logarithm
We start with the equation \( F = e^{-k t} \). To linearize it, we apply the natural logarithm to both sides: \( \ln(F) = \ln(e^{-k t}) \). This is the first step in rearranging the equation to solve for \( t \).
2Step 2: Simplify the Right Side of the Equation
Next, use the logarithmic identity \( \ln(e^x) = x \ln(e) \) to simplify \( \ln(e^{-k t}) \). This becomes \( -k t \ln(e) \). Since \( \ln(e) = 1 \), the equation simplifies to \( \ln(F) = -k t \cdot 1 \) or \( \ln(F) = -k t \).
3Step 3: Solve for \( t \)
Now rearrange the simplified equation \( \ln(F) = -k t \) to solve for \( t \). Divide both sides by \( -k \) to isolate \( t \): \( t = \frac{\ln(F)}{-k} \). This is the equation that gives \( t \) in terms of \( F \) and \( k \).
Key Concepts
Isotope DecayCarbon-14 DatingNatural LogarithmsRadioactive Decay Constant
Isotope Decay
Isotope decay is a natural process where unstable isotopes transform into more stable ones over time. This process can take days, months, or even millions of years, depending on the isotope in question. As isotopes decay, they emit particles or energy, gradually changing into a different element or a stable isotope of the same element. The rate at which an isotope decays is described by its decay constant, often denoted by the symbol \( k \). The larger the decay constant, the faster the decay process.
An important aspect of isotope decay is its predictability; the fraction of the remaining isotope can be graphically represented by an exponential decay curve. This curve shows a rapid decline that slows over time, eventually approaching zero. Exponential models, like \( F = e^{-kt} \), are used to calculate the fraction of the isotope remaining after a set time \( t \). Understanding the decay process helps in various fields such as archaeology, geology, and even medical diagnostics.
An important aspect of isotope decay is its predictability; the fraction of the remaining isotope can be graphically represented by an exponential decay curve. This curve shows a rapid decline that slows over time, eventually approaching zero. Exponential models, like \( F = e^{-kt} \), are used to calculate the fraction of the isotope remaining after a set time \( t \). Understanding the decay process helps in various fields such as archaeology, geology, and even medical diagnostics.
Carbon-14 Dating
Carbon-14 dating is a technique used to determine the age of ancient artifacts and geological findings. This method relies on measuring the amount of Carbon-14, a radioactive isotope, remaining in a sample. Living organisms continuously exchange Carbon-14 with their environment; however, when they die, this exchange stops, and the Carbon-14 starts to decay.
To estimate the age of a sample, scientists measure how much Carbon-14 it still contains and use the decay model \( F = e^{-k t} \). Here, \( k \) represents the decay constant specific to Carbon-14, with a known value of approximately 0.00012097. The age \( t \) can be solved as \( t = \frac{\ln(F)}{-k} \) if the fraction \( F \) of Carbon-14 remaining is known. This method is pivotal in understanding historical timelines and developing an accurate chronology of ancient events.
To estimate the age of a sample, scientists measure how much Carbon-14 it still contains and use the decay model \( F = e^{-k t} \). Here, \( k \) represents the decay constant specific to Carbon-14, with a known value of approximately 0.00012097. The age \( t \) can be solved as \( t = \frac{\ln(F)}{-k} \) if the fraction \( F \) of Carbon-14 remaining is known. This method is pivotal in understanding historical timelines and developing an accurate chronology of ancient events.
Natural Logarithms
Natural logarithms are a type of logarithm with the base \( e \), where \( e \) is a mathematical constant approximately equal to 2.71828. They are commonly used in scientific and mathematical calculations involving growth and decay processes. A special property of natural logarithms is captured by the identity \( \ln(e^x) = x \), which greatly simplifies exponential equations.
In decay models, applying the natural logarithm allows for the linearization of exponential equations, making them easier to manipulate and solve. For example, in the equation \( \ln(F) = \ln(e^{-kt}) \), the natural logarithm helps isolate the variables to solve for time \( t \). By converting exponential equations into linear ones, natural logarithms transform complex calculations into more manageable forms. They are particularly useful in contexts like Carbon-14 dating to find the time elapsed since an organism's death.
In decay models, applying the natural logarithm allows for the linearization of exponential equations, making them easier to manipulate and solve. For example, in the equation \( \ln(F) = \ln(e^{-kt}) \), the natural logarithm helps isolate the variables to solve for time \( t \). By converting exponential equations into linear ones, natural logarithms transform complex calculations into more manageable forms. They are particularly useful in contexts like Carbon-14 dating to find the time elapsed since an organism's death.
Radioactive Decay Constant
The radioactive decay constant \( k \) is a fundamental part of understanding how rapidly a radioactive isotope decays. This constant is unique to each isotope and determines the speed at which the decay process occurs. A higher \( k \) value means a faster decay rate, whereas a smaller \( k \) suggests a slower process.
In the context of Carbon-14 dating, the decay constant \( k = 0.00012097 \) is used to calculate the time \( t \) since a sample stopped exchanging carbon with the environment. The relationship between decay constant and the amount of isotope remaining is described by \( F = e^{-kt} \). By analyzing the fraction \( F \) of the original radioactive isotope left in a sample, scientists can use the equation \( t = \frac{\ln(F)}{-k} \) to determine the age of an artifact or fossil. This unique characteristic bridges a crucial link between physics and other scientific disciplines like archaeology and geology.
In the context of Carbon-14 dating, the decay constant \( k = 0.00012097 \) is used to calculate the time \( t \) since a sample stopped exchanging carbon with the environment. The relationship between decay constant and the amount of isotope remaining is described by \( F = e^{-kt} \). By analyzing the fraction \( F \) of the original radioactive isotope left in a sample, scientists can use the equation \( t = \frac{\ln(F)}{-k} \) to determine the age of an artifact or fossil. This unique characteristic bridges a crucial link between physics and other scientific disciplines like archaeology and geology.