The bond length between hybridised carbon atom and other carbon atom is minimum in

The correct option is Propyne.

To understand why propyne has the minimum bond length between the hybridized carbon atom and the other carbon atom, we need to analyze the hybridization of the carbon atoms involved in the carbon-carbon bonds of each given molecule.

First, let us look at the types of carbon-carbon bonds present in each option:

1. Propane ($C_3{H}_8$):
Propane is an alkane. All the carbon atoms in propane are single-bonded to each other and are $s{p}^3$ hybridized.
The C—C bond is formed by the overlap of two $s{p}^3$ hybridized carbon orbitals (an $s{p}^3—s{p}^3$ single bond).

2. Butane ($C_4{H}_{10}$):
Like propane, butane is also an alkane. All carbon atoms are $s{p}^3$ hybridized, resulting in $s{p}^3—s{p}^3$ single bonds.

3. Propene ($C_3{H}_6$):
Propene is an alkene containing one double bond. The structure is $C{H}_3—CH=C{H}_2$.
Here, the single bond between the methyl carbon and the double-bonded carbon is formed by the overlap of an $s{p}^3$ orbital and an $s{p}^2$ orbital ($s{p}^3—s{p}^2$ bond). The double bond itself is an $s{p}^2—s{p}^2$ bond.

4. Propyne ($C_3{H}_4$):
Propyne is an alkyne containing a triple bond. The structure is $C{H}_3—C\equiv CH$.
In propyne, the carbon-carbon triple bond ($C\equiv C$) is formed between two $sp$ hybridized carbon atoms.

Now, let us relate the hybridization to the bond length. The bond length depends directly on the size of the hybrid orbitals, which is determined by the percentage of s-character in the hybrid orbital:
- $s{p}^3$ hybrid orbital: 25% s-character (larger, more diffused)
- $s{p}^2$ hybrid orbital: 33.3% s-character
- $sp$ hybrid orbital: 50% s-character (smaller, closer to the nucleus)

As the s-character increases, the hybrid orbital becomes smaller and holds the shared electrons closer to the carbon nuclei, which shortens the bond length. Furthermore, multiple bonds (double and triple bonds) pull the nuclei closer together than single bonds. Specifically:
- Carbon-carbon single bond ($C—C$) length is approximately 1.54 Å.
- Carbon-carbon double bond ($C=C$) length is approximately 1.34 Å.
- Carbon-carbon triple bond ($C\equiv C$) length is approximately 1.20 Å.

Since propyne contains a carbon-carbon triple bond ($sp—sp$ overlap), it has the greatest s-character and the shortest distance between the hybridized carbon nuclei, resulting in the minimum carbon-carbon bond length among the options.