The half life of a radioactive element is 8 hours. A given number of nuclei of that element is reduced to `1//4` of that number after two hours.

Answer

To solve the problem, we need to analyze the decay of a radioactive element using the concepts of half-life and the decay constant. Here’s a step-by-step solution:

### Step 1: Understand the given information
- Half-life (T_half) of the radioactive element = 8 hours.
- After 2 hours, the number of nuclei is reduced to \( \frac{1}{4} \) of the original number.

### Step 2: Define the initial number of nuclei
Let the initial number of nuclei be \( N_0 \).

### Step 3: Determine the number of nuclei after 2 hours
According to the problem, after 2 hours, the number of nuclei \( N \) is given by:
\[ N = \frac{N_0}{4} \]

### Step 4: Use the radioactive decay formula
The formula for radioactive decay is:
\[ N = N_0 e^{-\lambda t} \]
where \( \lambda \) is the decay constant and \( t \) is the time elapsed.

### Step 5: Relate the decay constant to half-life
The decay constant \( \lambda \) is related to the half-life by the formula:
\[ \lambda = \frac{\ln(2)}{T_{half}} \]
Substituting \( T_{half} = 8 \) hours:
\[ \lambda = \frac{\ln(2)}{8} \]

### Step 6: Substitute values into the decay formula
Substituting \( N = \frac{N_0}{4} \), \( \lambda = \frac{\ln(2)}{8} \), and \( t = 2 \) hours into the decay formula:
\[ \frac{N_0}{4} = N_0 e^{-\left(\frac{\ln(2)}{8}\right) \cdot 2} \]

### Step 7: Simplify the equation
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{4} = e^{-\left(\frac{\ln(2)}{8}\right) \cdot 2} \]
This simplifies to:
\[ \frac{1}{4} = e^{-\frac{\ln(2)}{4}} \]

### Step 8: Rewrite the exponential equation
Using the property of logarithms, we can rewrite \( e^{-\frac{\ln(2)}{4}} \):
\[ e^{-\frac{\ln(2)}{4}} = \frac{1}{e^{\frac{\ln(2)}{4}}} = \frac{1}{2^{1/4}} \]
Thus, we have:
\[ \frac{1}{4} = \frac{1}{2^{1/4}} \]

### Step 9: Verify the equality
Since \( \frac{1}{4} = \frac{1}{2^2} \), we can see that:
\[ 2^2 = 2^{1/4} \quad \text{(This is not true)} \]
This indicates that the statement in the problem is incorrect.

### Conclusion
The statement that the number of nuclei is reduced to \( \frac{1}{4} \) after 2 hours is incorrect based on the calculations.

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