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Is bicyclo[1.1.0]butane chiral? I think it isn't, but if one of the hydrogen atoms bound to the tertiary or secondary carbon atoms is replaced with e.g. chlorine, the formed compound will be chiral, right?

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Bicyclo[1.1.0}butane is an achiral molecule.

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Any molecule containing a plane of symmetry must be achiral and bicylco[1.1.0]butane has 2 planes of symmetry. One plane bisects the molecule and passes through the 2 tertiary carbons, the other plane bisects the molecule and contains the 2 secondary carbons.

Even replacing one of the secondary or tertiary hydrogens with a substituent does not make the molecule chiral. Such a substitution will remove one plane of symmetry, but not the other one.

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In the above bicylco[1.1.0]butane where we have substituted a chlorine for one of the tertiary hydrogens, the plane containing the 2 bridgehead carbons and the chlorine still remains. However, if you replace one of the tertiary hydrogens and one of the secondary hydrogens, the resultant disubstituted-bicylco[1.1.0]butane will be chiral since both symmetry planes have been destroyed and no other plane or axis of symmetry exists.

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