The technology of lithium-ion batteries is a huge area of research and, although what follows is only a surface explanation of what is happening, it may already be out of date by the time you read this.
A lithium-ion cell contains a positive end and a negative end which are connected by an electrolyte mixture that allows lithium ions to move from one end to the other through a separator that does not allow the flow of electrons.
The positive end contains one of several compounds such as \(LiCoO_2\). Because this is an ionic compound and because lithium ions are very soluble, the connection between the \(CoO_2^{-1}\) and the \(Li^{+1}\) ion is loose and tentative.
The negative end of the cell has a matrix that can hold lithium ions. This matrix is often made of graphite.
An uncharged (completely used up) lithium-ion cell looks like this:
Charging a lithium-ion battery involves connecting the battery to a power source that can create a large negative charge on the graphite end of the battery and a positive charge on the Co end. This is done by moving electrons through the charger from the Co end to the graphite end.
As a result of this, the \(Li^{+1}\) ions are attracted to the graphite end. They move through the separator and lodge in the graphite matrix.
The result of this process is a charged battery. The power is stored in the potential energy found between the \(Li^{+1}\) ions and the \(CoO_2^{-1}\) ions.
When the battery is placed in an open circuit (when your phone is on, for instance) the lithium ions can flow back to the Co side of the battery which causes electrons to move through the external circuit (your phone).
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