Gadolinium-`153`, which is used to detect osteoporoiss, has a life of `242` days. After how many days, on an average, a nuclide can be conisdered absent form patient's system?

Options

A. A. Infinity
B. `B. 350` days
C. `C. 242` days
D. `D. 4 xx 10^(-3)` days

Explanation

To solve the problem, we need to determine how many days it takes for Gadolinium-153 to be considered absent from a patient's system based on its half-life. Here's a step-by-step solution:

### Step 1: Understand the Half-Life
The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life of Gadolinium-153 is given as 242 days.

### Step 2: Calculate the Decay Constant (k)
The decay constant (k) for a first-order reaction can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} \]
where \( t_{1/2} \) is the half-life.

Substituting the given half-life:
\[ k = \frac{0.693}{242} \]

### Step 3: Calculate k
Now, we perform the calculation:
\[ k \approx \frac{0.693}{242} \approx 0.002864 \text{ days}^{-1} \]

### Step 4: Calculate the Average Lifetime (t_average)
The average lifetime (t_average) of a radioactive substance can be calculated using the formula:
\[ t_{average} = \frac{1}{k} \]

Substituting the value of k:
\[ t_{average} = \frac{1}{0.002864} \approx 349.5 \text{ days} \]

### Step 5: Conclusion
On average, Gadolinium-153 can be considered absent from the patient's system after approximately 350 days.

### Final Answer
**Approximately 350 days.**

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