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In terms of bonding and properties, why doesn't Chlorine conduct electricity either in the solid state or in solution/ molten state?
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Well, chlorine in (for example) water solution can conduct electricity but that conductivity is poor.
In water solution chlorine undergo reaction:
Cl2 + 3H2O <=> ClO- + Cl- + 2H3O+
And because of presence of these ions - water solutions of chlorine can conduct electricity.