All the boron trihalides except `BI_(3)` may be prepared by direction between the elements. Boron trihalides consist of trigonal-planar `BX_(3)` molecules. Unlike the halides of the other elements in the group they are monomeric in the gas, liquid and solid states, `BF_(3)` and `BCl_(3)` are gases, `BBr_(3)` is a volatile liquid and `BI_(3)` is a solid. Boron trihalides are Lewis acids because they form simple Lewis complexes with suitable bases, as in the reaction:
`BF_(3)(g)+NH_(3)(g)toF_(3)B-NH_(3)(s)`
However, boron chlorides, bromides and iodides are susceptible (sensitive) to protolysis by mild proton sources such as water, alcohols and even amines, for example `BCl_(3)` undergoes rapid hydrolysis:
`BCl_(3)(g)+3H_(2)O(l)toB(OH)_(3)(aq)+3HCl (aq)`
It is supposed that the first step in the above reaction is the formation of the complex `Cl_(3)B larr OH_(2)` which then eliminates `HCl` and reacts with water.
Which of the following is the correct prediction about observed `B-X` bond length, in `BX_(3)` molecules?

Options

A. `B-F` bond length in `BF_(3)` is found to be less than theoretical value because the electronegativity values of `B(2.04)` and `F(4.0)` suggest the bond to be ionic and hence the attraction between oppositely charged ions must decrease the bond length
B. `BF_(3)` and `[BF_(4)]^(-)` have equal `B-F` bond length
C. The decrease in the `B-F` bond length in `BF_(3)` is due to delocalised `p_(pi)-p_(pi)` bonding between vacant `'2p'` orbital of `B` and filled `'2p'` orbital of `F`.
D. The correct `B-X` bond length order is `B-FgtB-ClgtB-BrgtB-I`

Answer

To solve the question regarding the bond lengths in boron trihalides (BX₃), we need to analyze the properties and bonding characteristics of these compounds. Here’s a step-by-step breakdown:

### Step 1: Understand the Structure of Boron Trihalides
Boron trihalides (BX₃) have a trigonal planar geometry. This means that the boron atom is at the center with three halogen atoms (X) at the corners of an equilateral triangle. The bond angles are approximately 120 degrees.

**Hint:** Remember that the geometry of a molecule can affect its bond lengths due to steric and electronic factors.

### Step 2: Analyze the Bonding in BF₃
In BF₃, boron has an empty p-orbital that can accept electron density from the filled p-orbitals of fluorine through back-bonding. This pπ-pπ back-bonding decreases the bond length because the effective nuclear charge experienced by the bonding electrons increases, pulling the atoms closer together.

**Hint:** Look for the presence of back-bonding in covalent compounds, especially when there are empty orbitals involved.

### Step 3: Compare with Other Boron Trihalides
For other boron trihalides like BCl₃, BBr₃, and BI₃, the extent of back-bonding decreases as the size of the halogen increases. Chlorine, bromine, and iodine have larger atomic radii than fluorine, leading to weaker pπ-pπ back-bonding interactions. Therefore, the bond lengths in BCl₃, BBr₃, and BI₃ will be longer compared to BF₃.

**Hint:** Consider how atomic size affects bond length and the strength of back-bonding.

### Step 4: Evaluate the Statements
Now, let's evaluate the statements given in the question:

1. **Statement A:** Incorrect - BF₃ is covalent, not ionic.
2. **Statement B:** Incorrect - BF₃ and BF₄⁻ do not have equal bond lengths due to the absence of back-bonding in BF₄⁻.
3. **Statement C:** Correct - The decreasing bond length in BF₃ is due to pπ-pπ back-bonding.
4. **Statement D:** Incorrect - The correct order of bond lengths is BI > BBr > BCl > BF.

### Conclusion
The correct prediction about the observed B-X bond length in BX₃ molecules is that the bond length decreases from BI₃ to BF₃ due to the increasing effectiveness of back-bonding with smaller halogens.

**Final Answer:** The correct option is **C**.