Ionic compounds have giant structures in which many strong electrostatic forces hold the ions together.
It takes a lot of energy to break the bonds which hold a giant ionic lattice together, so all the ionic compounds have high melting and boiling points, and are solids at room temperature.
However, when an ionic compound has been melted the ions are free to move. This allows them to carry electrical charge, so the liquids conducts electricity.
Some ionic solids dissolve in water because water molecules can split up the lattice. The ions are free to move in the solutions and so they also conduct electricity.
Ionic compounds cannot conduct electricity when solid because the ions can only vibrate about fixed positions; they cannot move around.
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