To calculate the decay of a radioactive isotope with a given half-life, you can use the exponential decay formula. The amount of a radioactive substance remaining after a certain amount of time can be calculated using the following formula:
\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
Where:
- \( N(t) \) is the quantity of the substance remaining after time \( t \).
- \( N_0 \) is the initial quantity of the substance.
- \( T_{1/2} \) is the half-life of the substance.
- \( t \) is the elapsed time.
In your case, the half-life (\( T_{1/2} \)) is 30 seconds.
Steps to calculate the decay:
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Identify your initial amount \( N_0 \). For example, let's say you start with 100 grams of the isotope.
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Choose the time \( t \) that you want to calculate the decay for. For example, if you want to find out how much is left after 90 seconds, you would set \( t = 90 \) seconds.
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Calculate the remaining quantity: \[ N(90) = 100 \times \left( \frac{1}{2} \right)^{\frac{90}{30}} \] This simplifies as follows:
- \( \frac{90}{30} = 3 \)
- \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
- Thus, \[ N(90) = 100 \times \frac{1}{8} = 12.5 \text{ grams} \]
So, after 90 seconds, 12.5 grams of the radioactive isotope would remain.
Generalization
You can apply this method for any time period or initial quantity using the same equation. Just substitute the values of \( N_0 \), \( t \), and \( T_{1/2} \) appropriately.
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