Atomic Orbital- Shapes and Energy of Orbital

Orbitals are a mathematical function that showcases the wave-like behavior of dual electrons, nucleons, or electrons. This is a significant point of study in Chemistry and Quantum Mechanics since it is the three-dimensional space surrounding the nucleus with the highest probability of electron presence.

Another common name for orbitals is atomic orbitals or electrons. These combine to create molecule orbitals in s, p, f, and d subshells, and each atomic orbital contains a specific sum of energy. Notably, orbitals are available in different sizes and shapes, measurable through the wave function square (Ψ2).

Here, you would learn more about the structure of orbitals, types, and related details about orbital energy.

Atomic Orbitals- Overview and Relation to Quantum Numbers

In the context of atomic orbitals, three main quantum numbers are visible. According to the solution of Schrödinger’s Equation, it delivers the viable energy levels suitable for electrons to occupy. The solution also offers the secondary wave function(s) relative to all energy levels.

The structure of orbitals and their energy states appear with all three quantum numbers in sets. These include ‘n’, ‘ml’, and ‘ℓ’ for energy, shape, and angular momentum representation. Atomic orbitals are called wave function ‘ψ’ in terms of an atomic electron. It is possible to verify the electron-centric arrangement inside each atom with the quantum numbers.

Here, orbital energy is essential to consider. It refers to the overall energy an orbital-based electron needs to go to infinity, or the released energy when an infinity-based electron joins the orbital.

The shells and subshells determine their value; all atomic orbitals in one subshell have one energy.

Types of Atomic Orbitals

Understanding the different variations available is crucial to understanding the shapes-of-orbitals and other types. Here are the main versions available.

s Orbitals

The s orbital shape is present in the symmetric spherical structure surrounding the atom’s nucleus. The further one goes from the atomic nucleus, the higher the energy level. Therefore, the size of orbitals follows the sequence  1s< 2s< 3s< 4s.

It is simple to find electrons in 1s; the probability decreases in the next jump. There is a sharp decline in probability density ψ2 (r) in 2s orbital, denoting ‘0’; later, it rises. If the r-value grows, the probability density ψ2 (r)reduces again at the small maximum point.

First two graphs represent the plots of orbital wave functionψ(r) and next  two graphs represent the variation of probability density ψ2 (r) as a function of distance r of the electron from the nucleus for 1s and 2s orbitals.

The point at which zero electron probability is known as a nodal point in the study of atomic orbitals and their energy. Nodes come in two types in s orbitals and other types:

•       Angular nodes- represents a direction
•       Radical nodes- denotes the distance to the atomic nucleus

Quantity of angular nodes = l

Quantity of radial nodes = n – l – 1

Total quantity = n – 1

p Orbitals

The p orbital shape appears as a dumbbell. Here, the node appears inside the nucleus centre.

p orbitals take up around six electrons in one section maximum since three orbitals are present, each at a 90-degree angle. The size changes based on the principal quantum number or ‘n’.

Here, 4p > 3p > 2p.

The p orbitals have specific sections on both sides of their nodal plane, i.e., lobes. This passes from the middle of the nucleus. Here, the possibility of finding any electron is non-existent (0) at the point of intersection between two lobes. All three atomic orbitals hold the same shape, size, and energy. The only factor of distinction is present in the lobe’s orientation, i.e., alongside the x/y/z axis.

The formula for calculating the number of nodes here is n -2.

d Orbitals

The shape of d orbitals appears as two dumbbells or a cloverleaf in one place. For d orbital the value of l=2. thus the value of principal quantum number n is 3. Additionally, the l value cannot be greater than n-1.

Plus, the d orbitals are 5 in number. This is because the ml value in association with the d orbitals is (–2, –1, 0, +1, and +2) in the context of l = 2. The designations of these five atomic orbitals are dxy / dyz / dxz / dx 2– y 2 / dz2. Only the last one (dz2) does not have the same shape as the former four.

Here, the probability density stands at 0.

f Orbitals

The shape of the f orbital is diffused.For f orbital the value of l=3, thus the minimum value of principal quantum number n is 4.Therefore, the principal quantum number holds a minimum value of 4 for these atomic orbitals. To note, the types of f orbitals are seven in total. 

Calculating Orbital Energy

The energies of atomic orbitals are quantifiable with only some levels available that electrons can stay in. Two electrons present in same orbital has same energy . When the electron drops to one of the lower-level atomic orbitals, there is release of electromagnetic radiation. While the electron rises to a higher orbital level, that energy gets absorbed.

In terms of calculating the energy levels of single electrons in hydrogen-type atoms, the principal quantum number is the main factor to focus on. Under the Aufbau Principle in Chemistry, the atomic orbitals with the lower levels come first.

•   If two atomic orbitals have a same  n+l value,then the orbitals with  a lower value of principle quantum number will have  low energy .
• The stability of atoms depends on the attraction between the nucleus (positive charge), electrons, and electron-centric repulsive force. Stability is maintained when the total attraction interaction passes the repulsive interactive sum.

Conclusion

To understand the placement of electrons around a nucleus, atomic orbitals are mathematical functions to consider. This defines the wave-type behavior and location of electrons. To measure atomic orbitals, understanding the quantum numbers like azimuthal quantum number (l) and principal quantum number (n).

To measure the shape and energy of electrons in atomic orbitals, it is important to understand the types of orbitals first. To understand the atomic energy between the orbitals and the location of electrons, use the quantum numbers and nodes to calculate the energy levels.