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Be and Br have similar electronegativities. Since the electronegativity values of Be and Br are so close, they both pull the electron with nearly the same strength, meaning the electron is closer to the middle and is shared between both Be and Br
The difference in electronegativity is what determines an ionic bond. Professor Lavelle said that the difference in electronegativity would approximately be 2 in order for it to be an ionic bond. Any less would be a covalent bond.
In order for a compound to be considered an ionic compound (remember that all compounds have both ionic and covalent properties) the difference in electronegativity must be greater than 2. For BeBr2, this condition is not met. (Side note: BeBr2 is a nonpolar molecule as well).
Here, the thing to remember is that even though we normally think of ionic bonds as between a non-metal and a metal, as in BeBr2, we have to consider the relative electronegativity of the atoms involved in the bond and the electronegativity difference between them. Usually, the electronegativity difference between a non-metal and a metal is enough to be considered ionic, but this case would be an exception.
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