To determine the half-life of argon-39 based on the information provided, we can use the exponential decay formula. Given the initial weight (Nâ) of the sample, the remaining weight (N), the time elapsed (t), and the half-life (Tâ/â), we can set this up:
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The decay formula can be expressed as: \[ N = Nâ \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
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Plugging in the values we have: \[ 394.5 = 1578 \left( \frac{1}{2} \right)^{\frac{538}{T_{1/2}}} \]
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First, we can calculate the ratio of the remaining mass to the initial mass: \[ \frac{394.5}{1578} \approx 0.25 \]
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This can be rewritten as: \[ 0.25 = \left( \frac{1}{2} \right)^{\frac{538}{T_{1/2}}} \]
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We recognize that \(0.25\) is equal to \(\left( \frac{1}{2} \right)^2\). Therefore: \[ \left( \frac{1}{2} \right)^{\frac{538}{T_{1/2}}} = \left( \frac{1}{2} \right)^2 \]
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Setting the exponents equal to each other: \[ \frac{538}{T_{1/2}} = 2 \]
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Solving for \(T_{1/2}\): \[ T_{1/2} = \frac{538}{2} = 269 \text{ years} \]
Thus, the half-life of argon-39 is 269 years.