Among following molecule `N-Si` bond length is shortest:

Options

A. `N(SiH_(3))_(3)`
B. `NH(SiH_(3))_(2)`
C. `NH_(2)(SiH_(3)`
D. All have equal `N-Si` bond length

Explanation

To determine which molecule has the shortest N-Si bond length among the given options, we can analyze the bonding characteristics and hybridization of nitrogen and silicon in each case.

### Step-by-Step Solution:

1. **Identify the Molecules**: We need to consider the different nitrogen-silicon compounds provided. Let's assume we have the following compounds for analysis:
- A: N-SiH3
- B: N-Si2H6
- C: N-Si (with lone pair on nitrogen)
- D: N-Si (with no lone pair)

2. **Analyze the Hybridization**:
- For **N-SiH3**: Nitrogen is likely to be sp3 hybridized due to the presence of one lone pair and three bonds with silicon. The bond character is primarily single.
- For **N-Si2H6**: Nitrogen here can also be considered sp3 hybridized. However, the presence of two silicon atoms may allow for some back-bonding, but the overall bond character remains single.
- For **N-Si (with lone pair)**: The lone pair on nitrogen can participate in back-bonding with the empty d-orbitals of silicon, leading to a stronger bond character.
- For **N-Si (without lone pair)**: This bond would be primarily a single bond without any additional interactions.

3. **Consider Back-Bonding**:
- Back-bonding occurs when the lone pair of electrons on nitrogen can be donated to the empty d-orbitals of silicon. This interaction can strengthen the bond and decrease the bond length.
- In the case of **N-Si (with lone pair)**, the back-bonding effect is significant, leading to a shorter bond length compared to the other compounds.

4. **Evaluate Electronegativity and Hybridization**:
- The bond length is inversely related to the percentage of s-character in the hybridization. More s-character means a shorter bond length.
- In the case of **N-Si (with lone pair)**, the sp2 hybridization of nitrogen (due to back-bonding) leads to a higher s-character (33.3%) compared to sp3 hybridization (25% s-character).

5. **Conclusion**:
- Based on the analysis, the N-Si bond in the compound with back-bonding (N-Si with lone pair) will have the shortest bond length due to the increased bond character from back-bonding and higher s-character from hybridization.

### Final Answer:
The molecule with the shortest N-Si bond length is the one where nitrogen has a lone pair and can participate in back-bonding with silicon.