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*Quantum number and orbitals*

- A wave function represents an electron is the product of two parts, a radial part, and an angular part.

- The square of the radial part of the wave function indicates the probability of finding the electron at any distance r from the nucleus.

- The square of the angular part of the wave function gives the probability of finding an electron in a particular direction from the nucleus.

- The radial dependence and angular dependence of wave function taken together, tell us that a three-dimensional standing electron wave (orbital) can be a picture to have size, shape, and orientation of an orbital.

- In order to describe the size, shape, orientation of an orbital four quantum number are necessary. These quantum numbers are designed as,

**Principal quantum numbers***Azimuthal quantum numbers**Magnetic quantum numbers**Spin quantum numbers*

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*Principal quantum numbers(n)*

- The principal quantum numbers(n) is of primary importance in determining the size and hence the energy of an electron.

- For hydrogen, the energy is fixed by the value of the principal quantum number(n). In other multi-electron atoms, the energy of each electron depends on the value of the principal quantum number of the electron.

- As the value of the principal quantum number(n) increases the radius (nucleus electron separation increases that are the size of the orbital increases).

- The energy also raised, principal quantum numbers(n) is always an integer and can assume the value,1,2,3,4.... but not zero.

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*Azimuthal quantum numbers(l)*

*Azimuthal quantum numbers(l)*

- The general geometric shapes of an electron wave (orbital) are described by the azimuthal quantum numbers. This quantum number related to for the electron in that state.

Thus, l = 0, 1, 2, 3.....(n-1)

- Therefore an electron having principal quantum number n, assumed the values l is 0 to (n-1).

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*Magnetic quantum numbers(m)*

- The magnetic quantum number associated with the orientation of the electron wave with respect to a given direction, usually that of a strong magnetic field. This quantum number hasn't effect on the shape of orbital or on the energy of an electron.

For a given Value of l, ml can have any integral value between +1 to -1.

∴ m

∴ m

_{l}= + l, (l - 1), (l - 2), (l - 3) ..... 0 ..... - 1, - 2, - l###
*Spin quantum number(s)*

- Besides the three quantum numbers, also has forth quantum number namely spin quantum number (s), was necessary to completely describe an electron in a particular shell.

- The electron itself regarded as a small magnet. A beam of a hydrogen atom can be split into two beams by a strong magnetic field. This indicates that there are two kinds of a hydrogen atom in which can be differentiated on the basis of their behavior in a magnetic field.

- It has been postulated that each electron spin around its axis like a lope and they behave like a magnet. A spinning electron can have only two possibilities.

- The electron can either spin clockwise or counterclockwise. The two directions of spin represent as(↑↓).

- This four quantum number s = 1/2 is independent of other three quantum numbers.

- Two directions of spin are represented as (↑↓) can have two possible ms values +1/2 and -1/2 depending on the direction of rotation of the electron about its axis.

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*How to find quantum numbers?*

Quantum number orbitals |

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*Definition of orbitals*

- An orbital is a region in space where there is a high probability of finding an electron.

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*Shapes of orbitals S, P, d*

- The S orbitals penetrate the nucleus most while the P and d orbitals cannot penetrate the nucleus.

- This means that S orbital electron can efficiently screen the nuclear charge from other electrons compared to the other P and d electrons.

- The wave function of the electron in an atom is called orbital. The wave function is plotted against distance and space in three dimensional marked by a curve will give the shape of the orbitals.

- The probability of finding an electron in space around the nucleus involves two aspects, radial probability, and angular probability.

- It is not possible to represent completely in one diagram on paper the directional properties of electron orbitals. An angular probability distribution must be combined mentally to have an overall shape of the orbital.

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*Shape of S orbital*

- The angular probability distribution is greater interest and importance, an S orbital electron has no angular dependence because the relevant wave function is independent of angles Î¸ and Î¦.

- There is, therefore, an equal chance of discovering the electron in any direction of the nucleus.

Shape of S orbital |

- With the nucleus at the origin of the Cartesian axes, the sphere of the radius(r) represents the probability of finding the electron in S orbital.

- An S orbital electron has a spherically symmetrical probability distribution.

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*Shape of P orbitals*

- The P orbitals are three orientations is represents as, Px, Py, and Pz. The orientations of the orbital plane correspond to ml = 1, 0, -1 is mutually at the right angles.

- So the orbitals designated Px, Py and Pz are mutually perpendicular and they are concentrated along the respective coordinate axis X, Y and Z. Unlike the S orbitals, the angular part of the P wave function is dependent on Î¸ and Î¦.

Shape of p orbitals |

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*Shape of d-Orbital*

*Shape of d-Orbital*

- These orbital arises when n =3 (M - shell), that is the orbitals start with the 3rd main energy level.

- When l = 2(d - orbital), m = -2, -1, 0, 2, 1 indicating that d - orbitals have five orientation, that is, dxy, dxz, dyz, dx² - y² and dz².

- All these five d - orbitals, in the absence of magnetic field, are equivalent in energy and are, therefore, said to be five-fold degenerate.

Shape of d orbitals |

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*Quantum number orbitals questions answers*

*Question*

- What are the four quantum numbers of the 19th electron of Cr atom?

*Answer*- The atomic number of Cr atom is 24. Thus the electronic configuration of Cr atom is,

1S² 2S² 2P⁶ 3S² 3P⁶ 4S¹ 3d⁵

- 19th electron means 4S¹ electron. Thus for 4S electron n =4, and l =0 and for S¹ electron m = 0 and s = +½.

- ∴ The four quantum numbers of the 19th electron of Cr atom are 4, 0, 0, +½

*Question*- Write 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 Rubidium atom are,

5, 0, 0, +½

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

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

*Question*- How many possible orbitals are there for n = 4?

- Thus the number of possible orbitals when n = 4 is [1(4S) + 3(4P) + 5(4d) + 7(4f)] = 16

*Question*- How many possible orbitals are there for n = 3, l =1, and m

_{l}=0 ?

*Answer*- Thus the number of orbitals is 1, 3S orbital.