Find 4 Quantum Numbers with examples

What are the 4 quantum numbers?

1.      Principal quantum number (n)

2.     Momentum quantum number (ℓ)

3.     Magnetic quantities number (mℓ)

4.     Electron spin quantum numbers (ms)

Quantum numbers

To know how electrons will be arranged we must examine the energy levels in the atom by studying the following quantum numbers:

1. Principal quantum number (n)

Energy levels in atom are determined by the principal quantum number and the larger value of n the higher value of energy level and farther the electron from the nucleus. The value of n determines the size of the level and n take integers 1, 2, 3, etc.

2. Momentum quantum number (ℓ)

Determines the shape of orbital in which the electron is likely to be present and it is resulting from the movement of the electron around the nucleus. That each principal level n consists of one or more sub - (secondary) levels. The number of secondary levels in any orbital is equal to the number of principal quantum number:

·       n=1 consists of one secondary level of energy s.

·       n=2  contains two secondary levels is s and p, and the third principal level,

·       n=3, contains three secondary energy levels are s, p and d

·       n=4 contains four secondary levels of quantum numbers are s, p, d, and f.

Each n value corresponds to a specific value for the quantum secondary number ℓ are integers starting with zero and ending with ( n-1). If n=1, that ℓ will start with zero and end with (1 - 1) so there is one value for ℓ= zero but if n= 2 it is start with zero and end with (1 - 2) by1 so there are two values for ℓ is zero and one. So when n = 3 there are three values 0, 1, 2 and when n= 4 there are four values: 0, 1, 2, 3 and so on (Table 1-1).

Table (1-1): The values of ℓ and the symbols of the secondary levels indicated

In our study we will only consider secondary levels s, p, d, f because they are only ones that are preoccupied with electrons in atoms in ground state. To determine the secondary level from any n principal level symbolically writes the value of n for the n principal level then the character as- signed to the secondary level, for example the secondary level s from the second principal level has the symbol 2s and has values(n=2, ℓ = 0). The secondary level d of the third principal level is 3d and has quantum values (n = 3, ℓ = 2) and so on.

3. Magnetic quantities number (mℓ)

Atomic orbitals can take the same shape around the nucleus but in different directions. Magnetic quantities number indicates the direction of the orbital around the nucleus where each secondary level consists of one or more orbitals.

This number was used to explain the appearance of extra lines in the spectrum when atom placed in a magnetic field. A spatial distribution of an electron, spherical symmetric, ie, the probability of its existence is the same in all trends from the nucleus.

On the other hand the probability the presence of the electron p in some directions of the nucleus is more than in other. In fact, the probability of distribution for the electron p of two diffuse lobes to some extent, one on each side of nucleus.

The p-level consists of three orbitals the secondary level d consists of five orbitals and the secondary level f consists of seven orbitals and their spatial distribution is more complex largely.

Each value of ℓ matched with the values of mag- natic quantum numbers, which are positive and negative integers. When the value of ℓ:

·       ℓ=0 there is one value for mℓ which is (0).

·       ℓ = 1, the values of mℓ are (+1, 0,-1)

·       ℓ= 2 the values of mℓ is (+2, +1, 0, -1, -2).

·       ℓ=3 then the value of mℓ are (+3, +2, +1, 0, -1, -2,-3).

Table (1-2) summarizes quantum numbers of primary, secondary and magnetic.

Table (1-2): Values of quantum numbers of primary, secondary and magnetic.

4. Electron spin quantum numbers (ms)

We explained earlier that there is one orbital for the secondary level S, three orbitals for secondary level p, five for secondary level d and seven for the secondary level f and where these secondary levels can accommodates 2, 6, 10 and 14 electrons respectively. It follows that any orbital can accommodate two electrons but electrons differ in the same orbital by one important thing is that it has opposite spindle rotation. The reason for talking about the spin of the electron comes from sightings the magnetic behavior of the material can be obtained from the behavior of magnetic for single atoms, by Ottostron experiment.

Figure 1: Ottostron experiment.

In this experiment [Figure 1], a beam of silver atoms Neutralization (resulting from the evaporation of silver) passed between two magnetic electrodes. It was found that the beam splits into two separate beams, that is, half the atoms deflects in a certain direction and the rest deflects in the opposite direction.

To explain this behavior that each electron behaves like a fine magnet. This magnetism is produced by the spin of the negative charge because it is known that the spindle rotation of any charge is generated magnetic field. There are two opposite directions of the spin; we expect each electron to attract each other but that disability of the attraction between the electron orbital repulsion in their charge.

Since the movement of the rotation of the two electrons is confined in only two directions so we have two values of the spindle quantum ms are +1/2and -1/2.

It can summarize the characteristics of the electron in the atom As follows:-

1.      The principal quantum number n and this characteristic indicates the order of the electrons configuration and increase the distance from the nucleus.

2.     The momentum quantum number mℓ describes the type of orbital that the electron operates from, where its spatial distribution is similar (for example, the electron s has spherical distributions are symmetrical and the electrons p have symmetrical distributions along separated directions in the space).

3.     The magnetic quantum number mℓ and this characteristic defines any orbital from orbitals of the secondary level in which an electron is likely to be present.

4.     The electron spin quantum number (ms) and this characteristic determines either of the two possible directions of the spin of the electron.

5.     When the four characteristics of the electron are determined in an certain atom, it will discover that it cannot exist in the same one atom another electron that has a set similar to those of its the four characteristics. This fundamental definition is known as (the Puli exclusion principle).

This definition states that it cannot two electrons in the same atom have the same values for all quantum four number.

Example 1

Set the four quantum values of the last electron for each of the following element atoms:

Xe= 54

V= 23

Cl= 17

Li= 3

Solution:

·       n=2 because the electronic configuration ended in the second level 2

·       ℓ=0 because the electronic configuration ended with the secondary level s

·       mℓ = 0 because the last electron is located in the secondary level s

·       ms = +1/2 Because the last electron rotates clock wise (the first electron of the Orbital).

·       n=3 because the configuration ended with the third principal level 3

·       ℓ=1 because the configuration ended with the secondary level p

·       mℓ = 0 because the last electron is located in the orbital which has a value of ℓ = 0

·       ms = -1/2 Because the last electron is the second electron in the orbital.

·       n = 3 because the highest principal level contains electrons at the third

·       ℓ=2 because the electronic configuration ended with the secondary level d

·       mℓ = 0 because the last electron is located in the orbital whose value is m ℓ = 0

·       ms = +1/2 because the last electron is the first electron in the orbital.

·       n = 5 Because the highest principal level contains electrons is the fifth

·       ℓ=1 because the electronic configuration ended with the secondary level p

·       mℓ = -1 because the last electron is located in the orbital whose value is m ℓ = -1

·       ms = -1/2 Because the last electron is the second electron in the orbital.

Example 2

Compare the four quantum numbers of the last electron to each of the two element atoms Li= 3 and Na= 11.

Solution:

The difference is only in the principal quantum number.

Example 3

Write the electronic configuration of the Mn= 25 atom and write the quantum number of the electrons, for the last secondary level.

Solution:

Example 4

If the last electron of an atom of an element has the following four quantum numbers:

·       n = 3

·       ℓ = 2

·       m ℓ = +1

·       ms = -1/2

What is the atomic number of this element?

Solution:

n=3 the principal level will be the third

ℓ =2 the secondary level is the level and contains five orbitals

mℓ = +1

Located in the orbital indicator with the letter X

ms = -1/2 is the second electron

Therefore, from previous information the last secondary level in the electronic configuration will end with a secondary 3d level which contains seven electrons then become an electronic configuration of secondary levels in this atom as follows:

Adding the number of electrons above the secondary levels in the electronic configuration and the sum value represents the atomic number of this element.

So     Atomic number = 27