Question

Why is the s-orbital always spherical in shape?

Answer 1

Because its wave function has no angular dependence.

By definition, an `s` orbital has zero angular momentum, and `l = 0`. Any nonzero angular momentum leads to atomic orbitals having non-spherical shapes.

Some explicit wave functions for the hydrogen atomic orbitals are:

`psi_(1s)(r,theta,phi) = 1/(sqrtpi) (1/a_0)^(3//2) e^(-r//a_0)`

`psi_(2s)(r,theta,phi) = 1/(4sqrt(2pi)) (1/a_0)^(3//2) (2 - r/a_0)e^(-r//2a_0)`

`psi_(3s)(r,theta,phi) = 1/(81sqrt(3pi)) (1/a_0)^(3//2) [27 - 18(r/a_0) + 2(r/a_0)^2]e^(-r//3a_0)`

The main thing you should notice is that all of these `s` orbital wave functions have no `theta` or `phi` in them, which are angles in spherical coordinates.

That means there is no way the angles could deviate from a straight integration in spherical coordinates at a constant radius (giving a spherical integration path).