How does the Wurtz reaction use sodium to make larger organic molecules?

The Wurtz reaction is a method used in organic synthesis to create larger hydrocarbons by coupling two molecules of an alkyl halide using sodium metal.12

The process relies on the high reactivity of sodium and the energetic favorability of forming sodium halides. During the reaction, two atoms of sodium facilitate the dehalogenation of two alkyl halide molecules, resulting in the formation of a new carbon-carbon bond and a sodium halide byproduct. The general chemical equation for this process is:21

$$2RCl+2Na\to R-R+2NaCl$$

In this reaction, the sodium effectively removes the halogen atoms, allowing the two organic radicals (R) to join together. For example, this method can be used to synthesize octane from bromobutane. A related process, the Wurtz-Fittig reaction, similarly employs sodium to synthesize alkyl-substituted aromatic hydrocarbons by reacting an alkyl halide with an aryl halide.23