So now you've become a sodium ion (Na+). Now you have ten electrons. That's the same number as neon (Ne). But you aren't neon (Ne). Since you're missing an electron you aren't really a complete sodium (Na) atom either. You are now something completely new. An ion. Your whole goal as an atom was to become a "happy atom" with completely filled electron orbitals. Now you have those filled shells. You are stable. What do you do that's so special now? Now that you have given up the electron, you are quite electrically attractive. Other electrically charged atoms (ions) are now looking at you and seeing a good partner to bond with. That's where chlorine comes in.
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