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Because in order for something to conduct electricity, it must be able to transport electrons or charge.

Ionic substances such as NaCl cannot conduct electricity in their solid state since the Ions (Na+ + Cl- ) are not free to move around and therefore cannot carry electrons. When you dissolve them, or melt them, the Ions are free to move and so can carry electrons from the cathode (negative end) to the anode (positive end).

Electrons leave the cathode and enter the solution via the Positive Na+:

Na+ + e- = Na which reacts with chlorine gas formed at the anode, which is where the electrons leave the system:

2Cl- - 2e- = Cl2

And hence the NaCl is recovered:

2Na + Cl2 = 2NaCl

So the electrons have moved from the cathode into and through the solution via the ions and left via the anode, hence the solution/liquid conducts electricity.

This is also the principle behind electrolysis, except you remove the products formed at the cathode and anode, rather than allowing them to recombine

2NaCl(l) + electricity = 2Na + Cl2

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