A compound can be divided into two equal halves & contains even 'n' asymmetric carbon atoms. The number of optical isomer is –

Options

A. `2^(n)`
B. `2^(n-1)`
C. `2^((n//2-1))`
D. `2^(n-1) + 2^((n//2-1))`

Answer

To solve the problem of determining the number of optical isomers for a compound with an even number of asymmetric (chiral) carbon atoms that can be divided into two equal halves, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Chiral Carbons (n)**:
- The problem states that the compound contains an even number of asymmetric carbon atoms. Let's denote this number as \( n \).

2. **Determine the Type of Compound**:
- Since the compound can be divided into two equal halves, it indicates that it has a plane of symmetry. This means that the compound has meso forms.

3. **Use the Formula for Optical Isomers**:
- For compounds with chiral centers that can be divided into two equal halves (and have an even number of chiral centers), the formula to calculate the number of optical isomers is:
\[ \text{Number of Optical Isomers} = 2^{n-1} \]
- Here, \( n \) is the number of chiral carbon atoms.

4. **Calculate the Number of Optical Isomers**:
- Since \( n \) is even, we can substitute \( n \) into the formula. For example, if \( n = 2 \), the number of optical isomers would be:
\[ 2^{2-1} = 2^1 = 2 \]
- If \( n = 4 \), the number of optical isomers would be:
\[ 2^{4-1} = 2^3 = 8 \]
- This pattern continues for any even \( n \).

5. **Conclusion**:
- Thus, the number of optical isomers for a compound with an even number of asymmetric carbon atoms that can be divided into two equal halves is given by \( 2^{n-1} \).

### Final Answer:
The number of optical isomers is \( 2^{n-1} \).