A sample of `._(53)I^(131)`, as `I^(ɵ)` ion, was administered to a patient in a carrier conissting `1.0 mg` of stable `I^(ɵ)` ion. After `4.0` days, `60%` of the initial radioactivity was detected in the thyroid gland of the patient. What mass of the stable `I^(ɵ)` ion had migrated to the thyroid gland? (Given: `t_(1//2)` of `I^(131) = 8` days)

Options

A. `0.65 mg`
B. `0.75 mg`
C. `0.85 mg`
D. `0.95 mg`

Explanation

To solve the problem step by step, we need to analyze the information provided and apply the principles of radioactive decay.

### Step 1: Understand the Given Information
- We have a sample of iodine-131 (`I^(131)`) administered as `I^(ɵ)` ion.
- The carrier contains `1.0 mg` of stable `I^(ɵ)` ion.
- After `4.0 days`, `60%` of the initial radioactivity was detected in the thyroid gland.
- The half-life (`t_(1/2)`) of `I^(131)` is `8 days`.

### Step 2: Calculate the Rate Constant (k)
The rate constant (k) for a first-order reaction can be calculated using the half-life formula:
\[ k = \frac{0.693}{t_{1/2}} \]
Substituting the half-life:
\[ k = \frac{0.693}{8 \text{ days}} = 0.086625 \text{ days}^{-1} \]

### Step 3: Calculate the Remaining Activity After 4 Days
Using the formula for radioactive decay:
\[ N = N_0 e^{-kt} \]
Where:
- \(N_0\) is the initial quantity of radioactive substance,
- \(N\) is the quantity remaining after time \(t\),
- \(k\) is the rate constant,
- \(t\) is the time in days.

However, we can also use the logarithmic form:
\[ \log \left(\frac{N_0}{N}\right) = kt \]
Rearranging gives:
\[ \frac{N_0}{N} = e^{kt} \]
Substituting \(k\) and \(t\):
\[ \frac{N_0}{N} = e^{0.086625 \times 4} \]
Calculating:
\[ \frac{N_0}{N} \approx e^{0.3465} \approx 1.414 \]
This means:
\[ \frac{N}{N_0} = \frac{1}{1.414} \approx 0.707 \]

### Step 4: Determine Initial Activity
Since we have rounded the initial activity to approximately `70%`, we can say:
\[ N_0 \approx 0.70 \]

### Step 5: Calculate the Activity Detected
Given that `60%` of the initial radioactivity is detected in the thyroid gland:
\[ \text{Activity in thyroid} = 0.60 \times N_0 \]
Substituting the initial activity:
\[ \text{Activity in thyroid} = 0.60 \times 0.70 = 0.42 \]

### Step 6: Calculate the Mass of Stable Iodine Migrated
Using the ratio of activities:
\[ \text{Mass migrated} = \frac{\text{Activity in thyroid}}{\text{Initial activity}} \times \text{Total mass of stable iodine} \]
Substituting the values:
\[ \text{Mass migrated} = \frac{0.60}{0.70} \times 1.0 \text{ mg} = \frac{6}{7} \text{ mg} \approx 0.857 \text{ mg} \]

### Final Answer
The mass of stable `I^(ɵ)` ion that had migrated to the thyroid gland is approximately `0.857 mg`.

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