To determine the age of the rock containing Lokuim and its daughter isotope DOL, you can follow these steps:
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Understand the Relationship Between Parent and Daughter Isotope: The element Lokuim (parent) decays into DOL (daughter) over time. In this scenario, you have a sample that contains 75% DOL and 25% Lokuim, meaning that for every 1 part of Lokuim, there are 3 parts of DOL in the sample.
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Determine the Ratio of Parent to Daughter Isotope: Since the sample has 25% Lokuim remaining, this indicates that 75% of the original Lokuim has decayed into DOL. This gives a ratio of Lokuim to DOL that can be expressed as: \[ \text{Ratio} = \frac{\text{Amount of Lokuim}}{\text{Amount of DOL}} = \frac{25%}{75%} = \frac{1}{3} \]
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Use the Decay Formula: The relationship between the parent isotope and the daughter isotope can be described using the decay equation: \[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \] where \(N\) is the remaining quantity of the parent isotope (Lokuim), \(N_0\) is the initial quantity, \(t\) is the time elapsed (age of the rock), and \(t_{1/2}\) is the half-life of Lokuim (3500 years).
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Calculate the Age: Rearranging the decay equation to find time: \[ \text{Given that } \frac{N_0}{N} = 4 \quad \text{(from Lokuim to DOL ratio)} \] This indicates that the initial amount of Lokuim was 4 times the current amount (since it has decayed to 1/4 of its original amount). Applying the equation: \[ \frac{N_0}{N} = 4 = \left(\frac{1}{2}\right)^{\frac{t}{3500}} \] Taking the log base 2 of both sides: \[ 4 = 2^2 \Rightarrow \frac{t}{3500} = 2 \Rightarrow t = 2 \times 3500 = 7000 \text{ years} \]
Thus, the age of the rock is 7000 years.
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