If `T` is the half-life of a radioactive material, then the fraction that would remain after a time `(T)/(2)` is

Options

A. `1/sqrt2`
B. `(sqrt2-1)/(sqrt2)`
C. `1/2`
D. `3/4`

Answer

To solve the problem, we need to find the fraction of radioactive material that remains after a time of \( \frac{T}{2} \), where \( T \) is the half-life of the material.

### Step-by-step Solution:

1. **Understand the Concept of Half-Life:**
The half-life \( T \) of a radioactive material is the time taken for half of the radioactive nuclei to decay. After one half-life, the remaining fraction of the material is \( \frac{1}{2} \).

2. **Use the Exponential Decay Formula:**
The number of remaining radioactive nuclei after time \( t \) can be expressed as:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

3. **Relate Half-Life to Decay Constant:**
The relationship between half-life \( T \) and decay constant \( \lambda \) is given by:
\[ T = \frac{\ln 2}{\lambda} \]
This means:
\[ \lambda = \frac{\ln 2}{T} \]

4. **Calculate Remaining Nuclei After \( \frac{T}{2} \):**
We want to find the remaining nuclei after a time of \( \frac{T}{2} \). Substitute \( t = \frac{T}{2} \) into the exponential decay formula:
\[ N\left(\frac{T}{2}\right) = N_0 e^{-\lambda \left(\frac{T}{2}\right)} \]

5. **Substitute for \( \lambda \):**
Substitute \( \lambda \) with \( \frac{\ln 2}{T} \):
\[ N\left(\frac{T}{2}\right) = N_0 e^{-\left(\frac{\ln 2}{T}\right) \left(\frac{T}{2}\right)} = N_0 e^{-\frac{\ln 2}{2}} \]

6. **Simplify the Expression:**
Using the property of exponents:
\[ e^{-\frac{\ln 2}{2}} = e^{\ln(2^{-1/2})} = 2^{-1/2} = \frac{1}{\sqrt{2}} \]
Therefore:
\[ N\left(\frac{T}{2}\right) = N_0 \cdot \frac{1}{\sqrt{2}} \]

7. **Calculate the Fraction Remaining:**
The fraction of the radioactive nuclei that remains after time \( \frac{T}{2} \) is:
\[ \frac{N\left(\frac{T}{2}\right)}{N_0} = \frac{N_0 \cdot \frac{1}{\sqrt{2}}}{N_0} = \frac{1}{\sqrt{2}} \]

### Final Answer:
The fraction of radioactive material that remains after a time of \( \frac{T}{2} \) is \( \frac{1}{\sqrt{2}} \).