# Why does the size of hybrid orbitals vary as sp³>sp²>sp?

In my text book it said that

Size of hybrid orbitals vary as sp³>sp²>sp

Does this size variation in hybid orbitals means that S orbitals are smaller than P orbitals?

But we know that larger the quantum number "n" is, the larger the orbital. I guess it has nothing to do with type of orbital.

What could be the reason for this variation in sizes?

• S orbitals are smaller than p orbitals. – Avnish Kabaj Jan 24 '18 at 5:32
• $n$ is identical for the valence s and p orbitals of the same element. – Karl Jan 24 '18 at 10:02

Looking at the shapes of atomic orbitals shown in many textbooks can be deceptive. The probability density of the radial part of the wavefunction ($r^2|R_{n,l}(r)|^2$) shows that the maximum probability for 2s orbitals is noticeably greater than for 2p, and for d orbitals the same applies with p greater than d. (As the principle quantum number increases so does the position of the maximum probability density of an orbital, i.e. increasing $n$ increases the size of the orbital.)
The hybrid orbitals form from the atomic orbitals and are added together in different proportions. So the position of maximum probability is a type of average value of the s and p. The sp hybrid has $2s \pm 2p$, the $sp^2$ in the z direction $2s+\sqrt{2}.2p$ and the $sp^3$ also in the z direction just as for the sp orbital. It seems to be that the maximum position of probability taken along, say, the z direction is the same in each case.
• Re: "maximum probability is a type of average value of the s and p." in your next step, have you done vector addition of the $s$ and $p$ orbitals? – Gaurang Tandon Feb 4 '18 at 16:22
• Can you please illustrate with a diagram? Or anything you deem more useful? For someone who hasn't seen those diagrams before, this addition of radial functions is not obvious at all, and could use a bit of detailing. [Besides that, another thing, that $sp^2$ orbitals are (trigonal) planar, so won't they have a zero addition value along an axis perpendicular to their plane?] – Gaurang Tandon Feb 5 '18 at 2:33