Which of the following orbitals are degenerate? 3dₓy, 4dₓy, 3dz2, 3dyz, 4dyz, 4dz2

NCERT Exemplar · Q25

Q. Which of the following orbitals are degenerate? 3d_xy, 4d_xy, 3d_z2, 3d_yz, 4d_yz, 4d_z2

Odisha ChseShort· 2mImportance★★★★★est

79% · 110/140 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Imagine you're climbing a smooth ramp. You can stop at any height — 1 metre, 1.5 metres, 2.1 metres — anywhere you like. That's how we intuitively think about energy in everyday life: continuous, like a slide.

Now imagine a staircase. You can stand on step 1, step 2, or step 3 — but you cannot stand halfway between step 2 and step 3. There's no such place. The steps are discrete, not continuous.

Energy level quantization is the idea that in the microscopic world of atoms and molecules, energy behaves like a staircase, not a ramp. Electrons in an atom cannot have just any energy — they can only occupy specific, allowed energy levels. Everything else is forbidden.

Why does this happen? The core intuition

In the classical world, an electron orbiting a nucleus would continuously radiate energy, spiral inward, and crash — atoms would be unstable. But atoms are stable. Nature solved this problem by imposing a rule: the electron's angular momentum (and therefore its energy) can only take certain discrete values.

Think of a guitar string. It can only vibrate at specific frequencies — its fundamental and harmonics. You can't pluck it to produce a frequency halfway between two harmonics. The string's vibration is quantized by its boundaries. Similarly, an electron bound to a nucleus is confined in space, and that confinement forces its energy to be quantized.

Note

Quantization is not a mysterious extra rule — it emerges naturally whenever a wave (like an electron's matter wave) is confined. Confinement creates standing waves, and standing waves only exist at specific frequencies.

The precise statement

For a bound system (like an electron in an atom), the total energy $E$ of the system can only take certain discrete values:

E = E_1, E_2, E_3, \dots

where each $E_n$ is a specific, fixed number. The integer $n$ (1, 2, 3, …) is called the principal quantum number. The lowest energy level ( $n=1$ ) is the ground state; higher levels ( $n>1$ ) are excited states.

For the hydrogen atom, the allowed energies are given by:

E_n = -\frac{13.6 \text{ eV}}{n^2}

So:

$n=1$ : $E_1 = -13.6$ eV (ground state)
$n=2$ : $E_2 = -3.4$ eV
$n=3$ : $E_3 = -1.51$ eV
and so on, approaching 0 eV as $n \to \infty$ (the ionization limit)

Important

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free (ionized). The more negative the energy, the more tightly bound the electron.

How do we know this is real?

The most direct evidence comes from atomic spectra. When an electron jumps from a higher energy level to a lower one, it emits a photon of light with energy exactly equal to the difference:

\Delta E = E_{\text{higher}} - E_{\text{lower}} = hf

where $h$ is Planck's constant and $f$ is the frequency of the emitted light. …

Start your 14-day free trial to unlock the full solution →

Orbitals are degenerate if they have the same energy. In multi-electron atoms, orbitals within the same subshell (same principal quantum number $n$ and azimuthal quantum number $l$ ) are degenerate. Therefore, the $3d_{xy}$ , $3d_{z^2}$ , and $3d_{yz}$ orbitals are degenerate with each other, and the $4d_{xy}$ , $4d_{yz}$ , and $4d_{z^2}$ orbitals are degenerate with each other.

The concept of degenerate orbitals is fundamental to understanding atomic structure and electron configurations. Orbitals are said to be degenerate if they possess the same energy. The factors determining an orbital's energy depend on whether we are considering a hydrogenic (single-electron) atom or a multi-electron atom.

In a hydrogenic atom (like H, He $^+$ , Li $^{2+}$ ), the energy of an orbital depends only on the principal quantum number $n$ . For example, in a hydrogen atom, the $2s$ and $2p$ orbitals have the same energy, and the $3s$ , $3p$ , and $3d$ orbitals all have the same energy. This is because there is only one electron, and there are no electron-electron repulsions or shielding effects.

However, for multi-electron atoms, the situation changes due to electron-electron repulsions and shielding. In these atoms, the energy of an orbital depends on both the principal quantum number $n$ and the azimuthal (or angular momentum) quantum number $l$ . This is why, for example, the $2s$ orbital is lower in energy than the $2p$ orbital, and the $3s < 3p < 3d$ .

Important

For a given multi-electron atom, all orbitals belonging to the same subshell (i.e., having the same $n$ and $l$ values) are degenerate. For instance, the three $2p$ orbitals ( $2p_x, 2p_y, 2p_z$ ) are degenerate, and the five $3d$ orbitals ( $3d_{xy}, 3d_{yz}, 3d_{xz}, 3d_{x^2-y^2}, 3d_{z^2}$ ) are degenerate.

Let's apply this understanding to the given orbitals.

Identify the quantum numbers for each orbital:

The notation for an orbital is $n l_m$ , where $n$ is the principal quantum number, $l$ corresponds to the subshell type ( $s, p, d, f$ ), and the subscript indicates the specific orientation (magnetic quantum number $m_l$ ).
- For $s$ orbitals, $l=0$ .
- For $p$ orbitals, $l=1$ .
- For $d$ orbitals, $l=2$ .
- For $f$ orbitals, $l=3$ .
Let's list the $n$ and $l$ values for each given orbital:
- $3d_{xy}$ : $n=3$ , $l=2$
- $4d_{xy}$ : $n=4$ , $l=2$
- $3d_{z^2}$ : $n=3$ , $l=2$
- $3d_{yz}$ : $n=3$ , $l=2$
- $4d_{yz}$ : $n=4$ , $l=2$
- $4d_{z^2}$ : $n=4$ , $l=2$
Group orbitals by their $n$ and $l$ values: …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ24 — Nickel atom can lose two electrons to form Ni²+ ion. The atomic number…→Next questionQ26 — Calculate the total number of angular nodes and radial nodes present i…

Q.Which of the following orbitals are degenerate? 3d_xy, 4d_xy, 3d_z2, 3d_yz, 4d_yz, 4d_z2

🎓Unlock everything free for 14 days

Q.Which of the following orbitals are degenerate?
3d_xy, 4d_xy, 3d_z2, 3d_yz, 4d_yz, 4d_z2

Unlock everything free for 14 days