To determine how many days have elapsed between measurements of radioactive iodine-131, we can use the half-life decay formula:
\[ A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
Where:
- \(A\) is the remaining quantity of the substance,
- \(A_0\) is the initial quantity (250 g in this case),
- \(t\) is the total time elapsed,
- \(T_{1/2}\) is the half-life (8 days for iodine-131).
Given that \(A = 15.625\) g, let's set up the equation:
\[ 15.625 = 250 \left( \frac{1}{2} \right)^{\frac{t}{8}} \]
First, we can simplify the equation:
\[ \frac{15.625}{250} = \left( \frac{1}{2} \right)^{\frac{t}{8}} \]
Calculating \(\frac{15.625}{250}\):
\[ \frac{15.625}{250} = 0.0625 \]
Now our equation is:
\[ 0.0625 = \left( \frac{1}{2} \right)^{\frac{t}{8}} \]
Next, we can express \(0.0625\) as a power of \(\frac{1}{2}\):
\[ 0.0625 = \frac{1}{16} = \left( \frac{1}{2} \right)^4 \]
So we have:
\[ \left( \frac{1}{2} \right)^{\frac{t}{8}} = \left( \frac{1}{2} \right)^4 \]
This means the exponents must be equal:
\[ \frac{t}{8} = 4 \]
Now we can solve for \(t\):
\[ t = 4 \times 8 = 32 \text{ days} \]
Therefore, the number of days that have elapsed between measurements is \(\boxed{32}\).
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