The half life of radioactive substance is T. Then the fraction of the substance that has decayed in time t is-

Options

A. (t/T)
B. `2^(t//T)`
C. `(1//2)^(t//T)`
D. `1-(1//2)^(t//T)`

Explanation

To solve the problem of finding the fraction of a radioactive substance that has decayed in time \( t \) given its half-life \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life \( T \) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. After one half-life, half of the original substance remains.

2. **Determine the Remaining Fraction**:
The fraction of the substance that remains after time \( t \) can be expressed using the formula:
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^{n} \]
where \( n \) is the number of half-lives that have passed in time \( t \).

3. **Calculate the Number of Half-Lives**:
The number of half-lives \( n \) that have passed in time \( t \) is given by:
\[ n = \frac{t}{T} \]
where \( T \) is the half-life of the substance.

4. **Substitute \( n \) into the Remaining Fraction**:
Now, substituting \( n \) into the remaining fraction formula:
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^{\frac{t}{T}} \]

5. **Calculate the Decayed Fraction**:
The fraction of the substance that has decayed is the complement of the remaining fraction:
\[ \text{Fraction decayed} = 1 - \text{Remaining fraction} \]
Thus,
\[ \text{Fraction decayed} = 1 - \left( \frac{1}{2} \right)^{\frac{t}{T}} \]

### Final Answer:
The fraction of the radioactive substance that has decayed in time \( t \) is:
\[ \text{Fraction decayed} = 1 - \left( \frac{1}{2} \right)^{\frac{t}{T}} \]