Arrange s, p and d sub-shells of a shell in the increasing order of effective nuclear charge (Zeff) experience

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Q. Arrange s, p and d sub-shells of a shell in the increasing order of effective nuclear charge (Z_eff) experienced by the electron present in them.

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Concept understanding — Effective Nuclear Charge

The Intuition: Why Don't Electrons Just Fly Away?

Imagine you're holding a magnet near a pile of paperclips. The closer the magnet, the stronger the pull. Now imagine you put a sheet of cardboard between the magnet and the paperclips. The pull weakens — the cardboard "shields" the paperclips from the full force of the magnet.

An atom works similarly. The nucleus (positive charge) pulls on the electrons (negative charge). But an electron is not alone — there are other electrons buzzing around between it and the nucleus. Those inner electrons act like the cardboard sheet: they shield or screen the outer electron from feeling the full positive charge of the nucleus.

So an outer electron doesn't "see" the full nuclear charge $Z$ (the atomic number). It sees a smaller, effective charge — the net positive pull after accounting for the repulsion from inner electrons.

That's effective nuclear charge, denoted $Z_{\text{eff}}$ .

The Precise Statement

Z_{\text{eff}} = Z - S

$Z_{\text{eff}}$ is always less than $Z$ (except for hydrogen, which has no other electrons to shield — there $Z_{\text{eff}} = Z$ ).

What Determines the Shielding Constant $S$ ?

Not all electrons shield equally. The key rules:

Inner electrons shield outer electrons very effectively. An electron in the $n=1$ shell completely blocks about 1 unit of charge from an electron in $n=2$ .
Electrons in the same shell shield poorly. They're at roughly the same distance, so they don't block much of the nucleus from each other.
Outer electrons do not shield inner electrons at all. An electron farther out cannot block the nucleus from one closer in.

There are detailed rules (Slater's rules) to calculate $S$ numerically, but the core idea is simple: the more electron shells between an electron and the nucleus, the more shielding, and the lower $Z_{\text{eff}}$ .

Why Does This Matter?

$Z_{\text{eff}}$ explains three fundamental patterns in the periodic table:

Trend	What happens to $Z_{\text{eff}}$	Why
Across a period (left to right)	Increases	Adding protons ( $Z$ up) but electrons go into the same shell (shielding roughly constant). Net pull on outer electrons gets stronger.
Down a group (top to bottom)	Stays roughly constant or decreases slightly	Adding a new shell means much more shielding. The extra protons are almost completely cancelled by the new inner electrons.
Atomic size	Larger $Z_{\text{eff}}$ → smaller atom	Stronger pull pulls electrons closer to nucleus.

Note

This is why fluorine is smaller than lithium, even though fluorine has more protons. The extra protons in fluorine are not fully shielded — the outer electrons feel a much stronger pull.

A Concrete Example: Sodium vs. Chlorine

Sodium ( $Z = 11$ ): Electron configuration $1s^2 2s^2 2p^6 3s^1$

The outermost electron (3s) is shielded by the 10 inner electrons ( $1s^2 2s^2 2p^6$ ). Roughly, $S \approx 10$ , so $Z_{\text{eff}} \approx 11 - 10 = 1$ . The outer electron feels a pull equivalent to just one proton. …

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The effective nuclear charge experienced by an electron depends on how much it is shielded by other electrons. Subshells with greater penetration power experience less shielding and thus a higher effective nuclear charge. The increasing order of effective nuclear charge is d < p < s.

The concept of effective nuclear charge ( $Z_{eff}$ ) is crucial for understanding atomic properties like ionization energy, electron affinity, and atomic size. It represents the net positive charge experienced by an electron in a multi-electron atom.

In an atom with multiple electrons, the electrons in the inner shells repel the electrons in the outer shells. This repulsion acts as a "shield" or "screen" between the outer electrons and the positively charged nucleus. As a result, the outer electrons do not experience the full attractive force of the nucleus. The effective nuclear charge is the actual nuclear charge ( $Z$ ) minus the shielding constant ( $S$ ), which accounts for this shielding effect.

Z_{eff}

The value of $S$ depends on the number and distribution of other electrons in the atom. For electrons within the same shell, the shielding effect varies for different subshells (s, p, d) due to their distinct shapes and spatial distributions, which in turn affects how close they can get to the nucleus.

Here is how we determine the order of effective nuclear charge:

Understanding Electron Penetration:

Electrons in different subshells within the same principal shell ( $n$ ) have different probabilities of being found close to the nucleus. This ability to approach the nucleus is called "penetration power." An electron that penetrates closer to the nucleus experiences a stronger attraction from the nucleus and is less shielded by the inner core electrons.
Order of Penetration Power:

For a given principal shell, the order of penetration power of the subshells is determined by their orbital shapes:

s-orbital > p-orbital > d-orbital > f-orbital

This means an s-electron has the highest probability of being found near the nucleus, followed by a p-electron, then a d-electron, and so on.
Relating Penetration to Shielding:
- An electron in a subshell with higher penetration power (e.g., an s-electron) spends more time closer to the nucleus. This means it is less effectively shielded by the inner core electrons because it frequently "gets inside" the region occupied by these core electrons. Consequently, it experiences less shielding from other electrons. …

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Q.Arrange s, p and d sub-shells of a shell in the increasing order of effective nuclear charge (Z_eff) experienced by the electron present in them.

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