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Polyethylene is a polymeric carbon chain that consists of repeating ethyl units (two carbons each bonded to two hydrogens). It is produced from the monomer ethylene (more properly called ethene) which consists of two carbon atoms connected to each other via a double bond. The double bond can be broken down to a single bond and used to form a new bond to a new ethylene compound, with the process repeating to form the polyethylene polymer. This polymerization reaction takes a highly flammable gas (ethylene) and turns it into a malleable solid (polyethylene) which has an enormous number or uses (packaging and tubing to name a couple).
A polymerization reaction is one in which a large group of smaller chemical precursors called monomers are added together in a repeating fashion to produce a larger, long chained compound called a polymer. The monomers are chemically reactive species that usually require some type of catalyst or initiator to begin the polymerization reaction, which can continue as long as monomer is present to produce the desired amount of polymer.
The reactions between ethene molecules as they for polythene is known as an ADDITION reaction. The monomers (ethene in this case) must have a carbon-carbon double bond, which is critical for the polymerization, and the finished polymer no longer has this double bond in its "repeating unit" (into which the monomer transforms). The addition polymer is DEFINED BY the fact that ALL the atoms present in the initial monomers are still to be found in the final, polymer, product. A second category of polymer, the "condensation" polymer, DOES lose a set of a few specific atoms from the monomers as it polymerizes, but that's not specifically pertinent to your question...
Imagine, or better yet, draw two ethene molecules side by side with the double bonds horizontal. (I judge, from the wording of your question, that you know how to do this). You know that each covalent bond in a molecule is formed by 2 paired electrons. Thus, the double bond(s) have 2 sets paired electrons. One of those pairs (the pi bond) can get involved in reactions WITHOU breaking the overall carbon-carbon bond on each molecule. Let's erase one bond in each of the double bonds and replace each with 2 dots representing electrons. (Same thing, right? One bond = 2 electrons.) Now move one of these electrons from each of the molecules to the space between them. 2 electrons = 1 bond! You now have a bond LINKING these two monomers into an incipient monomer! Those single electrons you left behind at the (former) double bonds? The one on the left reacts to for a bond with a 3rd monomer. The one on the right likewise reacts with a 4th... and so on as the polymer grows.
Last thing: By IUPAC rules you properly name this polymer as "polyethene". It is, however FAR more commonly known as "polyethylene".
Polyethylene (polyethene) is a polymer, or chain, of ethylene (ethene) molecules. Ethylene consists of two carbon atoms with a double bond between them along with four hydrogens, two bonded (via single bonds) to each carbon.
The synthesis reaction is initiated with the addition of a very reactive radical with an electron to "give". This radical donates its electron to one of the carbon atoms by forming a bond with that carbon atom. This breaks the double bond between the two carbons because each carbon can only have 4 bonds at a time. The electron from the broken bond is transferred to the other carbon atom.
This free electron on the carbon makes it highly reactive, much like the radical added to initiate the reaction. This free electron reacts with another ethylene molecule in the solution. The electron reacts with one of the two carbons, breaking the double bond between the two carbons on the second ethylene molecule. The electron from the broken bond goes to the other carbon atom, once again creating a reactive molecule.
This process continues (theoretically) until it is terminated or there are no more ethylene molecules to be attached to the growing chain. However, it is common that additional free radicals have to be added to polymerization reaction to keep them going.
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