In electrolysis of NaCl. How does CL become 1/2CL2?

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ihaveabutt
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Na+ + CI- → Na + 1/2C12

How is it Cl becomes Cl2? (what about conservation?) and why does it become diatomic?
 
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I'm not sure about the conservation you're mentioning, but I believe Cl becomes Cl2 because they are each short of an octet by one electron. So, each Cl gives up one electron to form a bond between them and the octets are satisfied.