A certain substance decays to 1/32 of its initial activity in 25 days. Calculate its half-life.

Options

A. 4 days
B. 5 days
C. 3 days
D. 6 days

Answer

To solve the problem of finding the half-life of a substance that decays to 1/32 of its initial activity in 25 days, we can follow these steps:

### Step 1: Understand the decay process
The decay of a radioactive substance can be described using the concept of half-life, which is the time taken for half of the substance to decay. If the substance decays to 1/32 of its initial activity, we need to determine how many half-lives it takes to reach this fraction.

### Step 2: Determine the number of half-lives
The fraction of the substance remaining after a certain number of half-lives can be expressed as:
\[ \frac{1}{2^n} \]
where \( n \) is the number of half-lives.

In this case, we want to find \( n \) such that:
\[ \frac{1}{2^n} = \frac{1}{32} \]

### Step 3: Solve for \( n \)
We know that:
\[ 32 = 2^5 \]
Thus, we can equate:
\[ 2^n = 2^5 \]
This implies that:
\[ n = 5 \]
So, it takes 5 half-lives for the substance to decay to 1/32 of its initial activity.

### Step 4: Calculate the half-life
We know that the total time taken for this decay process is 25 days. Since this time corresponds to 5 half-lives, we can find the half-life (\( t_{1/2} \)) using the formula:
\[ t_{1/2} = \frac{\text{Total time}}{n} \]
Substituting the values we have:
\[ t_{1/2} = \frac{25 \text{ days}}{5} = 5 \text{ days} \]

### Final Answer
The half-life of the substance is **5 days**.

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