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Answer: BF3 is nonpolar because all of the fluorine atoms are arranged in a trigonal planar fashion around the molecule, thereby cancelling out each other's pull for electrons. 

Although the difference between Boron's electronegativity (2.04) and Fluorine's (3.98) is great enough to make the individual bonds polar covalent, the symmetrical arrangement of atoms around the central Boron atom cancels out these possible dipole moments. This results in an overall structure that is generally nonpolar in character. 

You may have observed that this structure for BF3 seems to violate the octate rule specifically for the central Boron atom which only has six electrons in its valence shell. These are generally called incomplete octets and are formed due to the low electronegativity of Boron. Since double-bonding one of the fluorine's to complete the octet would leave the fluorine with a formal charge of +1, it becomes quickly apparent the infeasibility of this structure since fluorine is the most electronegative element and would not simply lose an electron. Therefore, it is simply assumed that the structure with the incomplete octet is a more stable configuration.

However, when these kinds of elements are handled in real life they are often unstable due to the incomplete octet. BF3 for example will try to form an octet on the central atom by converting to BF4-.  Other elements that may at times form incomplete octets include Beryllium and Aluminum. 


BF3 Ball and Stick Model
BF3 Ball and Stick Model. Created with MolView.
How does BF3 behave in the real world?

Typically this compound is highly reactive, being able to corrode metals including stainless steel. It is utilized as a catalyst in a wide variety of organic chemistry reactions. BF3 typically behaves as an acid since the presence of fluorines produces an "electron-deficient" structure. This character is evident in many of the different organic reactions which utilize BF3.

Electronegativity Chart: https://i1.wp.com/chemistry.com.pk/wp-content/uploads/2018/09/electronegativity-chart.jpg?ssl=1

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