An isotope decays to `1//16^(th)` of its mass in 1h. What is the half-life period of the isotope?

Options

A. 15 min
B. 30 min
C. 12 min
D. 10 min

Answer

To find the half-life of an isotope that decays to \( \frac{1}{16} \) of its mass in 1 hour, we can follow these steps:

### Step 1: Understand the decay process
The decay of an isotope can be described using the concept of half-lives. After each half-life, the remaining mass of the isotope is halved.

### Step 2: Determine the number of half-lives
If the isotope decays to \( \frac{1}{16} \) of its original mass, we can express this mathematically. Let \( N_0 \) be the initial mass. After \( n \) half-lives, the remaining mass \( N \) can be expressed as:
\[ N = \frac{N_0}{2^n} \]
We know that after 4 half-lives, the mass becomes \( \frac{N_0}{16} \):
\[ \frac{N_0}{2^n} = \frac{N_0}{16} \]
This implies:
\[ 2^n = 16 \]
Since \( 16 = 2^4 \), we find that \( n = 4 \). Therefore, it takes 4 half-lives to decay to \( \frac{1}{16} \) of its original mass.

### Step 3: Relate the number of half-lives to time
We know from the problem that this decay occurs in 1 hour (or 60 minutes). Since we have established that it takes 4 half-lives to reach this point, we can set up the equation:
\[ 4 \times \text{half-life} = 60 \text{ minutes} \]

### Step 4: Solve for the half-life
To find the half-life, we divide both sides by 4:
\[ \text{half-life} = \frac{60 \text{ minutes}}{4} = 15 \text{ minutes} \]

### Conclusion
Thus, the half-life of the isotope is **15 minutes**.

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