There are six different, optically active, dichloroalkanes that would be formed in the free radical chlorination reaction of butane. Let's understand why.

Free Radical Chlorination Reaction
The free radical chlorination reaction involves the substitution of hydrogen atoms in an alkane molecule with chlorine atoms, resulting in the formation of dichloroalkanes. Butane (C4H10) is a hydrocarbon with four carbon atoms and ten hydrogen atoms.

Mechanism of Free Radical Chlorination
The reaction proceeds through a radical chain mechanism, which involves three steps: initiation, propagation, and termination.

1. Initiation:
In the initiation step, a chlorine molecule (Cl2) is split homolytically by ultraviolet (UV) light, resulting in the formation of two chlorine radicals (Cl•).

2. Propagation:
In the propagation step, the chlorine radical reacts with butane to form a butyl radical (C4H9•) and HCl. This butyl radical can undergo further reactions to form different products.

3. Termination:
In the termination step, the radical chain reaction is terminated by the combination of two radical species, resulting in the formation of non-radical products.

Formation of Dichloroalkanes
In the case of butane, there are six different hydrogen atoms available for substitution, resulting in the formation of six different dichloroalkanes. Each of these dichloroalkanes has a chiral center (asymmetric carbon atom), making them optically active.

The six different dichloroalkanes formed from butane can be classified based on the carbon atom position of the substituted chlorine atoms:

1. 1,1-Dichlorobutane:
In this compound, both chlorine atoms are present on the same carbon atom in the butane chain.

2. 1,2-Dichlorobutane:
In this compound, one chlorine atom is present on the first carbon atom, and the other chlorine atom is present on the second carbon atom in the butane chain.

3. 1,3-Dichlorobutane:
In this compound, one chlorine atom is present on the first carbon atom, and the other chlorine atom is present on the third carbon atom in the butane chain.

4. 1,4-Dichlorobutane:
In this compound, one chlorine atom is present on the first carbon atom, and the other chlorine atom is present on the fourth carbon atom in the butane chain.

5. 2,2-Dichlorobutane:
In this compound, both chlorine atoms are present on the second carbon atom in the butane chain.

6. 2,3-Dichlorobutane:
In this compound, one chlorine atom is present on the second carbon atom, and the other chlorine atom is present on the third carbon atom in the butane chain.

Therefore, a total of six different, optically active, dichloroalkanes would be formed in the free radical chlorination reaction of butane.