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How can we justify the longer length of the dative bond by bridge Cl with Al? Is it because to prevent steric hindrance or repulsions the Cl atom is farther away from other Al as compared to normal Cl?

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We know that the number of electrons in both the bonds are same. One of the bond involves a covalent bond $\ce{Al-Cl}$ where as the other is a 3-centred bond $\ce{Al-Cl\bond{->}Al}$. Thus while providing a dative bond, the effective atomic charge of bridged chlorine decreases in comparison with that at the terminal position. Due to this difference in atomic charge, the terminal ones attract more and closer that the bridges.

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