Which is the correct order of bond lengths P,Q and R in
`Hoverset(p)(-)underset(H)underset(|)overset(H)overset(|)(C)-underset(H)underset(Q|)(C)=underset(H)underset(|)(C)-C-=Coverset(R)-H`

Options

A. `Pgt Q gt R`
B. `R gt Q gt P`
C. `Q gt P gt R`
D. `Q gt R gt P`

Explanation

To determine the correct order of bond lengths P, Q, and R in the given molecule, we need to analyze the hybridization of the carbon atoms involved in each bond and their corresponding s character.

### Step-by-Step Solution:

1. **Identify the Bonds**:
- The bonds in question are:
- P: C-H bond in the sp³ hybridized carbon.
- Q: C=C bond in the sp² hybridized carbon.
- R: C≡C bond in the sp hybridized carbon.

2. **Determine Hybridization**:
- For bond P (C-H):
- The carbon is bonded to four other atoms (three H and one C) with single bonds, indicating it is **sp³ hybridized**.
- For bond Q (C=C):
- The carbon is involved in a double bond (one sigma and one pi bond), indicating it is **sp² hybridized**.
- For bond R (C≡C):
- The carbon is involved in a triple bond (one sigma and two pi bonds), indicating it is **sp hybridized**.

3. **Calculate s Character**:
- **sp³ hybridization** (P):
- 1 s orbital and 3 p orbitals → Percentage of s character = \( \frac{1}{4} \times 100 = 25\% \)
- **sp² hybridization** (Q):
- 1 s orbital and 2 p orbitals → Percentage of s character = \( \frac{1}{3} \times 100 \approx 33\% \)
- **sp hybridization** (R):
- 1 s orbital and 1 p orbital → Percentage of s character = \( \frac{1}{2} \times 100 = 50\% \)

4. **Relate s Character to Bond Length**:
- The bond length is inversely related to the percentage of s character:
- **Less s character** → **Longer bond length**
- **More s character** → **Shorter bond length**

5. **Order the Bonds by Length**:
- Since bond P (sp³) has the least s character (25%), it will have the longest bond length.
- Bond Q (sp²) has a higher s character (33%) than P but lower than R.
- Bond R (sp) has the highest s character (50%) and will therefore have the shortest bond length.
- Thus, the order of bond lengths from longest to shortest is:
- \( P > Q > R \)

### Final Answer:
The correct order of bond lengths is \( P > Q > R \).