Animated Solution for Chemistry - Atomic Structure: Among the following, the energy of 2s-orbital is lowest in

Imagine you are an electron residing in the $2s$ orbital of an atom. Your entire existence is governed by a delicate tug-of-war: the positive nucleus is constantly pulling you in, while other electrons are trying to push you away. This problem asks us a fascinating question: in which atom—Hydrogen, Lithium, Sodium, or Potassium—is the energy of this $2s$ orbital the lowest?

To answer this, we need to understand what "lowest energy" actually means in the quantum world.

The energy of an electron in a multi-electron atom is dictated by the effective nuclear charge ($Z_{eff}$). The mathematical relationship is beautifully simple yet profound:

Here, $n$ is the principal quantum number, and $Z_{eff}$ is the net positive charge the electron actually "feels" from the nucleus after accounting for the shielding effect of inner electrons.

Notice the negative sign! Energy is defined as zero when the electron is infinitely far away and completely free. As the electron gets captured by the nucleus, it releases energy, falling into a "potential well." Therefore, a stronger pull means a deeper well, resulting in a more negative (and thus lower) energy.

The question specifically asks about the $2s$ orbital across different elements. This is a crucial detail. Because we are strictly looking at the $2s$ orbital, the principal quantum number is locked at $n=2$.

Since $n$ is constant, the denominator in our energy equation doesn't change. The energy now depends entirely on the numerator: the effective nuclear charge, $Z_{eff}$.

Let's line up our contenders and look at their atomic numbers ($Z$): Hydrogen (H): $Z=1$ Lithium (Li): $Z=3$ Sodium (Na): $Z=11$ Potassium (K): $Z=19$

As we move from Hydrogen to Potassium, the atomic number $Z$ increases dramatically. This means the nucleus is packing more and more protons. While it's true that more electrons are also added (which increases shielding), the increase in the actual nuclear charge $Z$ outpaces the shielding effect for a fixed inner orbital like the $2s$.

Therefore, as $Z$ increases, the effective nuclear charge $Z_{eff}$ experienced by the $2s$ electrons also increases.

Potassium has a whopping 19 protons in its nucleus. Compared to the other atoms in the list, Potassium's nucleus exerts a massive, overpowering attractive force on its $2s$ electrons.

Because the $2s$ electrons in Potassium are pulled so tightly toward the nucleus, they reside in the deepest potential well. Their energy is the most negative, which mathematically means it is the lowest.