Why do substances need to be either molten or dissolved in water for electrolysis to occur?

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Electrolysis works by seperating oppositely charged ions with electricity. Positive ions are attracted to the negative terminal whereas negative ions are attracted to the positive terminal. 

For this effect to be acheived, the ions must be free to move, when molten or dissolved in water the ionic bonds are broken, as a resullt the ions are free to move around the solution and towards the terminals.

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