Would you be able to help me understand how to compare and contrast a 2p6 orbital to a 4s1 using quantum numbers to help?

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$2\mathrm{p}^6$, $4\mathrm{p}^1$ are not orbitals.

It is a partial description of an atom electron configuration, saying there are 6 electrons in 3 orbitals $2\mathrm{p}$, respectively 1 electron in the orbital $4\mathrm{s}$.

$2$ in $2\mathrm{p}$ means the (main, energy ) quantum number $n=2$. $n$ can have values 1, 2, .....

$p$ in $2\mathrm{p}$ means the ( orbital, orbital angular momentum) quantum number $\ell=1$. $\ell$ can have values 0, 1, ... , $n$ - 1.

The ( magnetic ) quantum number $m$ ( $-\ell \le m \le +\ell$ ) is for the axis projection of the vector of the electron orbital momentum ), determining e.g. there is

  • 1 orbital $n\mathrm{s}$

  • 3 orbitals $n\mathrm{p}$, called $\mathrm{p_x, p_y,p_z}$

  • 5 orbitals $n\mathrm{d}$, called $\mathrm{d_{{xy}^2}, d_{{xz}^2}, d_{{yz}^2}, d_{x^2-y^2}, d_{z^2}}$

  • 7 orbitals $n\mathrm{f}$, being seldom explicitly named.

See also Wikipedia.

In a hydrogen atom, or ions with just 1 electrons, orbitals with the same $n$ have the same energy(*).

In multielectron atoms, electrons in different orbitals have different interaction with other electrons and their energy differ. Their energy increases in order $\mathrm{s < p < d < f}$

Generally, but with multi exceptions, the energy ( and filling order) increases in this order:

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 4d 7p

The naming has historical origin in spectroscopy. s as sharp, p as principal, d as diffuse, f as fundamental.

  1. s means $\ell = 0$
  2. p means $\ell = 1$
  3. d means $\ell = 2$
  4. f means $\ell = 3$

Orbitals have 3 meanings:

  1. A wave function being a particular solution of the Schroedinger wave equation, with integer constants $n, l, m$ called quantum numbers.
  2. An electron quantum state with the particular $n, l, m$ values
  3. A 3D shape, describing the probable electron occurence, with borders defined by occurence probability threshold, corresponding to 1 and 2.

(*) Relativistic corrections for Schroedinger equation reveals and explain observations it is not fully true.

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