The nitrogen atom in the following cyclic compounds can be removed as trimethylamine by successive Hoffmann eliminations (involving exhaustive methylation followed by heating with `AgOH`). The amine which will require a greater number of Hoffmann eliminations is:

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the cyclic compounds given and determine which amine will require a greater number of Hoffmann eliminations to remove the nitrogen atom as trimethylamine. The process involves exhaustive methylation followed by heating with AgOH.

### Step-by-Step Solution:

1. **Identify the Compounds**: We have four cyclic amines (let's denote them as A, B, C, and D). Each compound has a nitrogen atom that can be removed through Hoffmann elimination.

2. **Understand Hoffmann Elimination**: The Hoffmann elimination process involves two main steps:
- **Exhaustive Methylation**: The amine is treated with a methylating agent (like methyl iodide) to convert it into a quaternary ammonium salt.
- **Heating with AgOH**: The quaternary ammonium salt is then heated with silver hydroxide, leading to the elimination of nitrogen as trimethylamine.

3. **Count the Bonds to Break**: The number of Hoffmann eliminations required is directly related to the number of bonds that need to be broken to remove the nitrogen atom:
- For each bond broken, one Hoffmann elimination is required.

4. **Analyze Each Compound**:
- **Compound A**: Contains 2 bonds to nitrogen. Therefore, it requires 2 Hoffmann eliminations.
- **Compound B**: Contains 3 bonds to nitrogen. Therefore, it requires 3 Hoffmann eliminations.
- **Compound C**: Contains 1 bond to nitrogen. Therefore, it requires 1 Hoffmann elimination.
- **Compound D**: Contains 2 bonds to nitrogen. Therefore, it requires 2 Hoffmann eliminations.

5. **Determine the Maximum**: From the analysis:
- Compound A: 2 eliminations
- Compound B: 3 eliminations (maximum)
- Compound C: 1 elimination
- Compound D: 2 eliminations

6. **Conclusion**: The amine that will require a greater number of Hoffmann eliminations is **Compound B**, as it requires 3 eliminations.

### Final Answer:
The amine which will require a greater number of Hoffmann eliminations is **Compound B**.