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Graphite, a covalent molecule conducts current in solid state.
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The structure of graphite consists of many flat layers of hexagons. Each carbon atom in the layer is joined by strong covalent bonds to only three other carbon atoms. Graphite conducts electricity, due to delocalization of the pi bond electrons above and below the planes of the carbon atoms. These electrons are free to move, so are able to conduct electricity. However, the electricity is only conducted along the plane of the layers.

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