### Quantum number orientation of the orbitals

Fine-structure of spectrum lines in quantum physics suggested that the energy levels of an electron are more complex than a consideration of the electrostatic interactions from the Bohr hydrogen energy level and the Sommerfeld model. But **Quantum numbers** **orbitals diagram** provide the energy of electrons which participate in this interaction

Magnetic interactions comparatively small in magnitude are responsible for the fine structure defects of an atom.

Thus the closer examination of the atomic spectra demanded further refinement of the atomic orbitals. It was found that under high resolution the spectral lines of alkali metals had a fine structure of an atom.

For the study the fine structure of an atom four quantum number needed to explain the various spectra.

These quantum numbers are the identification numbers for an individual electron in an atom to describe the position and energy level of an atom.

In order to study the size, shape, orientation of orbital four quantum numbers are necessary. These quantum numbers are

- Principal quantum number.
- Azimuthal quantum number.
- Magnetic quantum number.
- Spin quantum number.

#### How to find the principal quantum number?

The principal quantum number describes the energy level or the principal shell to which an electron can stay. It is denoted by n.

Hence the primary importance of the principal quantum number for determining the size of an atom and energy of an electron.

From Bohr’s model of the hydrogen atom, the energy value of the hydrogen energy level fixed by the fixed value of n.

But for the multielectron atom energy value of each electron depends mostly on the principal quantum number of an orbital.

As the value of the principal quantum number increases the atomic radius or nucleus electron separation increases and the energy also raised.

The principal quantum number always an integer and can assume the value, 1, 2, 3, 4…. but not zero.

n = 1, 2, 3, 4, 5, …………… ∞

#### Azimuthal quantum number

The azimuthal quantum number was introduced by Sommerfeld in his atomic model and gives the angular momentum of an electron in its elliptical movement around the nucleus of an atom and the fine structure of the hydrogen spectrum.

Thus the general geometric shapes of an electron cloud or orbital described by the azimuthal quantum number or angular momentum quantum number. Permitted values of l for a given value of n has 0 to (n-1).

∴ l = 0, 1, 2, 3…..(n-1)

The total number of different values of l equal to n.

- n = 1, l = 0 (1S-orbital).
- For 2S and 2P-orbital, n = 2, l = 0, 1.
- n = 3, l = 0, 1, 2 (3S, 3P, and 3d-orbital).

#### Magnetic quantum number of an atom

Bohr’s model hydrogen atom could not explain the splitting of a single spectral line into a number of closely spaced lines in presence of magnetic field or presence of the electric field.

Thus the presence of more lines in the spectrum of the magnetic field or electric field indicates the energy levels are further subdivided by the additional quantum number called magnetic quantum number.

The magnetic quantum number associated with the orientation of the electron cloud with respect to a given direction, usually that of a strong magnetic field. This denoted by m_{l}.

For a given value of the azimuthal quantum number, the magnetic quantum number can have any integral value between +1 to -1.

∴ m_{l} = + l, (l – 1), … 0 … – 2, – l

#### Spin quantum number in chemistry

When spectral lines of hydrogen, lithium, sodium, and potassium observed by the instrument of high resolving power, each of the lines of the spectral series was found to consist of a pair of lines known double line structure.

Thus to describe these double lines of the fine structure another fourth quantum number necessary and it is known as a spin quantum number.

But the electron itself regarded as a small magnet. A beam of a hydrogen atom can split into two beams by a strong magnetic field.

Thus this indicates that there are two kinds of spin that can be differentiated on the basis of their behavior in a magnetic field.

Hence the electron has either spin clockwise or counterclockwise. And it is represented as(↑↓).

The spin quantum number independent of the other three numbers. Because two directions of spin have two possible spin quantum values (+ ½) and (- ½). And these values depending on the direction of rotation of the axis of the electron.

#### How to find quantum numbers of an element?

Question

What are the four quantum numbers of the 19th electron of chromium?

Answer

The atomic number of chromium 24. Thus the electron configuration of chromium

1S^{2} 2S^{2} 2P^{6} 3S^{2} 3P^{6} 4S^{1} 3d^{5}

Hence 19th electron means 4S^{1} electron and the quantum number for this electron

n = 4, l = 0, m = 0, and s = +½

Question

Find out the correct set of four quantum numbers for the valence electron of rubidium.

Answer

The correct set of four quantum numbers for the valence electron of the rubidium atom

5, 0, 0, +½

Question

How many electrons in an atom have the following quantum numbers n = 4 and l = 1?

Answer

6 electrons in an atom have the following quantum numbers n = 4 and l = 1.

Question

How many possible numbers of orbitals of an atom when n = 4?

Answer

Number of possible orbitals when n = 4,

[1(4S) + 3(4P) + 5(4d) + 7(4f)]

= 16.

Question

How many possible orbitals are there when n = 3, l = 1, and m_{l} = 0 ?

Answer

The number of possible orbitals = 1.

### Shapes of atomic orbitals

Atomic orbitals are the basic building blocks of the orbital diagram or alternatively known as the electron cloud or wave mechanics model.

According to the electron cloud model, an orbital is a region in space where the probability of finding an electron maximum.

Thus the probability of finding the electron cloud in 1S orbital of the hydrogen atom at certain positions near the nucleus of the hydrogen atom. And electron density maximum in the region just surrounded the nucleus of an atom.

But according to the electron wave model, the wave function of the electron in an atom is called orbital.

Thus the probability of finding an electron in space around the nucleus involves two aspects,

- Radial probability.
- Angular probability.

It is not possible to represent an orbital completely in one diagram on paper. Thus an angular probability distribution mutually combined to form the overall shape of the orbital.

#### The shape of S-orbital in an atom

The angular probability distribution greater interacts and important for S-orbital. But the S-electron has no angular dependence because the relevant wave function is independent of angles θ and Φ.

Thus the sphere of a definite radius represents the probability of finding the electron cloud in S-orbital.

The electron cloud distribution in S-subshell has a spherically symmetrical probability distribution.

#### The shape of p orbital in chemistry

The magnetic quantum number of p orbital 1, 0, -1. Hence P-orbitals are three orientations in space and these orientations represented as P_{x}, P_{y}, and P_{z}.

So these subshells mutually perpendicular and concentrated along the respective coordinate axis X, Y and Z.

But unlike the S-orbital, the angular part of the P-wave dependent on θ and Φ and P-subshell shielding by S-electron of an atom.

#### The shape of d-subshell of an atom

When n =3, the orbitals start with the 3rd main energy level of an atom.

l = 2(d – orbital),

m_{l} = -2, -1, 0, 2, 1

Thus d-orbital has five orientations in space represents

But the absence of a magnetic field all these five d orbitals are equivalent. Thus the quantum energy level of these d-orbitals is forming a five-fold degenerate energy level of an atom.