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BeCl2 is covalent while MgCl2 is ionic.
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Be2+ ion has very small ionic size and therefore, it has very high charge density. Due to this, it has high tendency to distort the electron cloud around the negatively charged chloride ion (Cl) which is larger in size. This results in partial covalent character of the bond in BeCl2. Mg2+ ion has very less tendency to distort the electron cloud of Cl due to the bigger size of Mg2+ as compared to Be2+.

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