Decay constant of a radioactive substance is `69.3 sec^(-1)`, find `t_(1//16)` of the same substance.

Options

A. `4xx10^(-2)` sec
B. `2xx10^(-2)` sec
C. `1xx10^(-2)` sec
D. none of these

Explanation

To find \( t_{1/16} \) of a radioactive substance given its decay constant \( k = 69.3 \, \text{sec}^{-1} \), we can follow these steps:

### Step 1: Understand the relationship between decay constant and half-life
The decay constant \( k \) is related to the half-life \( t_{1/2} \) of a radioactive substance by the formula:
\[ t_{1/2} = \frac{\ln(2)}{k} \]
where \( \ln(2) \approx 0.693 \).

### Step 2: Calculate the half-life
Using the given decay constant:
\[ t_{1/2} = \frac{0.693}{69.3} \]
Calculating this gives:
\[ t_{1/2} = 0.01 \, \text{seconds} \]

### Step 3: Determine the relationship for \( t_{1/16} \)
Since \( t_{1/16} \) represents the time taken for the substance to decay to \( \frac{1}{16} \) of its original amount, we can relate this to the half-life:
\[ \frac{1}{16} = \left(\frac{1}{2}\right)^4 \]
This means that it takes 4 half-lives to decay to \( \frac{1}{16} \).

### Step 4: Calculate \( t_{1/16} \)
Thus, we can calculate \( t_{1/16} \) as:
\[ t_{1/16} = 4 \times t_{1/2} \]
Substituting the value of \( t_{1/2} \):
\[ t_{1/16} = 4 \times 0.01 = 0.04 \, \text{seconds} \]

### Final Answer
Therefore, the value of \( t_{1/16} \) is:
\[ t_{1/16} = 0.04 \, \text{seconds} \quad \text{or} \quad 4 \times 10^{-2} \, \text{seconds} \]