How many `1^(@)` amine of the formula `C_(5)H_(13)N` can be used to resolve reacemic 2-methylbutonic acid?

Explanation

To determine how many primary amines of the formula C₅H₁₃N can be used to resolve racemic 2-methylbutanoic acid, we can follow these steps:

### Step 1: Understand the Structure of 2-Methylbutanoic Acid
2-Methylbutanoic acid has the following structure:
- It consists of a four-carbon chain (butanoic acid) with a methyl group attached to the second carbon.
- The molecular formula is C₅H₁₀O₂.

### Step 2: Identify the Required Primary Amines
We are looking for primary amines (1° amines) with the formula C₅H₁₃N. A primary amine has the structure R-NH₂, where R is an alkyl group.

### Step 3: Count the Possible Structures for C₅H₁₃N
1. **Positioning the Amine Group**: The amine group can be placed at different positions on the carbon chain. We will consider the following configurations:
- **1st Position**: CH₃-CH₂-CH₂-CH₂-NH₂ (Pentylamine)
- **2nd Position**: CH₃-CH(CH₃)-CH₂-CH₂-NH₂ (2-Methylbutylamine)
- **3rd Position**: CH₃-CH₂-CH(CH₃)-CH₃-NH₂ (3-Methylbutylamine)
- **4th Position**: CH₃-CH₂-CH₂-CH(CH₃)-NH₂ (4-Methylbutylamine)

2. **Chirality Consideration**:
- For the 2nd position (2-Methylbutylamine), there is a chiral center, which means it can exist as two enantiomers (R and S forms).
- For the 4th position (4-Methylbutylamine), there is also a chiral center, which can also exist as two enantiomers.

### Step 4: Calculate the Total Number of Primary Amines
- **1st Position**: 1 amine (Pentylamine) - no chirality
- **2nd Position**: 2 amines (2-Methylbutylamine) - chiral
- **3rd Position**: 1 amine (3-Methylbutylamine) - no chirality
- **4th Position**: 2 amines (4-Methylbutylamine) - chiral

### Step 5: Summarize the Total Count
Adding these together:
- 1 (from 1st position) + 2 (from 2nd position) + 1 (from 3rd position) + 2 (from 4th position) = **6 primary amines**.

### Conclusion
Thus, there are a total of **6 primary amines** of the formula C₅H₁₃N that can be used to resolve racemic 2-methylbutanoic acid.

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