The half life of a radioactive isotope is 44 days . In how many days `1.0` g will be reduced to `62.5` mg ?

Options

A. 275 days
B. 704 days
C. 353 days
D. 176 days

Explanation

To solve the problem of how many days it will take for 1.0 g of a radioactive isotope to decay to 62.5 mg, we can follow these steps:

### Step 1: Understand the half-life concept
The half-life of a radioactive isotope is the time taken for half of the radioactive sample to decay. In this case, the half-life is given as 44 days.

### Step 2: Convert the initial and final amounts to the same unit
We start with 1.0 g of the sample. To make calculations easier, we convert this to milligrams:
1.0 g = 1000 mg

We want to find out how long it takes for this 1000 mg to reduce to 62.5 mg.

### Step 3: Determine the number of half-lives required
To find out how many half-lives it takes to go from 1000 mg to 62.5 mg, we can use the formula:

\[ N = N_0 \left(\frac{1}{2}\right)^n \]

Where:
- \( N \) is the final amount (62.5 mg)
- \( N_0 \) is the initial amount (1000 mg)
- \( n \) is the number of half-lives

Rearranging the formula gives us:

\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \]

Substituting the values:

\[ \frac{62.5}{1000} = \left(\frac{1}{2}\right)^n \]

Calculating the left side:

\[ \frac{62.5}{1000} = 0.0625 \]

### Step 4: Solve for \( n \)
Now we need to express 0.0625 as a power of 1/2:

\[ 0.0625 = \left(\frac{1}{2}\right)^4 \]

This means:

\[ n = 4 \]

### Step 5: Calculate the total time taken
Since each half-life is 44 days, we can find the total time taken by multiplying the number of half-lives by the duration of each half-life:

\[ \text{Total time} = n \times \text{half-life} = 4 \times 44 \text{ days} \]

Calculating this gives:

\[ \text{Total time} = 176 \text{ days} \]

### Final Answer
It will take **176 days** for 1.0 g of the radioactive isotope to decay to 62.5 mg.

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