Tetrahedral geometry creates a tetrahedron which is a four-faced triangular pyramid with bond angles of 109.5 degrees between each of the hydrogens. To minimize these repulsions between the hydrogens the methane adopts a tetrahedral geometry. These hydrogen atoms each have electron clouds around them which are negative and repel each other. An sp3 hybridized carbon like methane has four bonds each going to a single hydrogen atom. And a dashed line means a bond going away from you into the plane of the page. A shaded triangle (or wedge) means a bond coming toward you out the plane of the page. A straight line (or solid line) represents a bond that is part of the plane of the page. It's meant to show the 3-D shape of bonds in molecules like the sp3 hybridized bonds in methane. They can overlap, but in different ways, and the bonds thus formed are not called sigma bonds but pi bonds. Since the other orbitals are not oriented along the bond axis, they cannot overlap "head on". pz orbital, along the z axis, and any s orbital (which is spherical) can overlap to form a sigma bond. If the z-axis is taken as the bond axis, only orbitals with the central axis along z-axis can form sigma bond. Head on overlap is actually a layman's term to specify the requirement of specific symmetries in combining atomic orbitals. A sigma bond involves head on overlap of atomic orbitals. The second question can be much more satisfactorily answered. In that concept, there is no explanation as to why we do not include the inner orbitals, but by not including them we get the right answers, and hence that became a so called "rule" of hybrid orbitals. Actually, there are no hybrid orbitals and hybridisation concept, introduced by Pauling is now obsolete and replaced with the superior molecular orbital model, which answers all the shortcomings of hybridisation, one of which you just mentioned. A god question, but unfortunately no simple answer.
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