## WAVE MECHANICS

Wave mechanics deal with electron occupying space. The electrons revolve the nucleus through orbit and occur in the orbital.

Orbital:Is a region within an atomic sublevel that can be occupied by maximum two electrons that have opposite spin. Any orbital can accommodate a maximum of two electrons.

The energy level of atoms is a specific region around the nucleus electrons can occupy in atoms. A shell is defined as a complete group of orbital possessing the quantum number. The

electrons arranged in the atom starting from orbit from nucleus toward the orbit of highest energy level. The following include position of locating electrons in the atom.

i. Principal quantum number (n)

The atom shells called principal quantum number (main energy level). The electrons filled in atom in order of increasing the energy from the nucleus. Each shell have specific constant number or electrons obtained taking

e = 2n^{2}

Old name | 1 | 2 | 3 | 4 |

New name | K | L | M | N |

** (ii)Subsidiary quantum number (L)**

Subsidiary quantum number is a sub energy level. The sub energy level known as azimuthal quantum number, sub shell or sub energy level. The shell divides to form sub shell. The total number of sub shells in each shell is equal to the number of shell from the nucleus. This describes shapes of orbitals (sub-energy level).Values of subsidiary quantum number start from 0——-n-1

The names of sub shell include the following;

iii. Magnetic quantum number (m)

This is a quantum number which describes the orbital of each sub shell. This sub shell divided and form orbital. Each sub shells have specific number of orbital.

(iv) Spin Quantum number (s)

These quantum numbers describe the spinning of electrons in the orbital. Maximum number of electrons in the orbital in only two which spines in opposite direction. Since each orbital has two maximum electrons result the sub shell to have constant total number of electrons which accommodated in all shells S^{2}, p^{6}, d^{10}, f^{14}, g^{18}, h^{22} and i^{26}. In order to write electrons of element consider the knowledge of all four quantum numbers.

The following series of sub shell used to write the electrons structure.

This arrangement used to write the electronic configuration.

Arrangement used to write the electronic configuration has some irregularities or abnormality. This irregularity occurs because there is overlapping of orbital. The orbital of low energy jump toward highest energy level and orbital of high energy drop down towards lowest energy level. Example 4s arranged first before 3d orbital.

RULES WHICH GOVERN ARRANGEMENT OF ELECTRONS IN ORBITALS

1. Aufbau’s principle – State that “in the ground state of orbital the lowest energy is filled by electrons before filling orbital of the highest energy”.

This means that electron filled in order of increasing energy level. The orbital of lowest energy when is full of electrons then the electron filled on the next orbital of highest energy level. Consider the following arrangement

2. Hund’s rule of maximum multiplicity

State that, “The electrons pair in the orbital of the same sub shell is allowed if all orbital of the same energy are occupied by electrons of maximum multiplicity. This means that the degenerate orbital are not allowed to pair up unless each orbital is singly occupied. The incoming pairing is done in an opposite spin. Consider a case with 5 electrons in valency shell.

3. Paul’s exclusive principle

States that, “Two electrons in an atom can never have exactly similar set of all quantum numbers.” This means that the maximum number of electrons which is occupied in the orbital is only two. Two electrons may occupy the same orbital only on condition that they have their spin in the opposite direction. Two electrons may have three quantum numbers the same; n, l and ml but the fourth, ms must be different.

Sometimes the electrons structure or electronic diagram of ions is written according to what kind of ions given.

There are two kinds of ions which include the following;-

Cation or metallic ion is formed through losing of electrons. The electrons which are lost occur in the orbital out the of the noble gas structure. The electron removed starting from orbital of low energy toward the orbital of highest energy.

Anion or non-metallic ion – is formed by gaining electrons. Electrons which gain is added in the orbital having unpaired electrons starting from orbital of lowest energy to the orbital of highest energy level

QUANTUM NUMBER

Is a number which describe the characteristics of electrons or is a number which is used to characterize electrons as they occupy orbitals in different energy levels. The quantum number is a number which describe main energy level, orientation of orbitals and spinning of electrons.

**There are four type of quantum number which includes the following:-**

**1.Principle quantum number (n)**: Is a number which specifies the location and the energy of electron. This is a number which describe the main energy level of electrons. The principle quantum number called shell orbit, main energy level and stationary state. The principle quantum number have specific name from the nucleus. K, L, M, N etc.

**2. Subsidiary principle quantum number (L)**: Is a number which specifies the shape of orbitals. Subsidiary principle quantum number is a sub energy level. The sub energy level is known as azimuthal number sub shell or sub energy level. The shell is divided to form sub shell. The total number of sub shells in each shell is equal to the number of shells from the nucleus. The names of sub shells include the following; s, p, d, f, g, h, i. The value of sub energy level i.e. 0, 1, 2, …L = n – 1.The value of L must be smaller that n because L = n – 1

n | Value of L | Sub shell |

1 | 0 | 1s |

2 | 0, 1 | 2s2p |

3 | 0, 1, 2 | 3sp3d |

4 | 0, 1, 2, 3 | 4s4p4d4f |

5 | 0, 1, 2, 3, 4 | 5s5p5d5f5f |

** 3. Magnetic quantum number**: Is a number which specifies the number of orbital present in a given value of subsidiary quantum number. This is a quantum number which describes the orientation of orbital. The magnetic quantum number is a number of orbital. When the sub shell divides form orbital, these orbitals are called Magnetic quantum number.

1s – Orbital S = X

**4. Spin quantum number (s)** – This is a quantum number which describes the direction in which the electrons spin. There are two maximum electrons, these electrons are moving in opposite direction. They revolve clockwise + ½ and another revolve anti – clock wise – ½

**APPLICATION OF QUANTUM NUMBER**

Quantum number is applied to find the number of sub shell (L), number of orbital (m) and total number of electrons (s) of the given quantum number (n). There are two methods which are applied to find total sub shells, orbital and electrons.

Find total number of electrons in the principle quantum number 2.

(By using tree diagram)

Total shells = 2 (s, p)

Orbitals = 4 (0, 0, +1, -1)

Electrons = 8

Also: by using tabular form

(b) Find the total number of electrons in the principle quantum number 3

Soln: (by using tree diagram)

**Example**

(a) The modern theory of electron behavior is based on two assumptions. State them

(b) Define the following terms

i. Orbital.

ii. Energy levels of atoms.

iii A shell.

iv. Quantum.

**Solution:**

(a) The modern theory of electron behavior is based on two assumptions.

(i) An electron has a dual nature i.e. it behaves both as a particle and wave.

(ii) It is practically impossible to determine simultaneously both the position and momentum of an electron with any degree of precision.

(b) (i) An Orbital is a region within an atomic sublevel that can be occupied by a maximum of two electrons that have opposite spin. There are s,p,d, and f orbitals is a volume of space surrounding the nucleus within which there is a more than 95% chances of finding electrons. Any orbital can accommodate a maximum of two electrons.

(ii) Energy level of atoms in specific region around the nucleus that electrons can occupy in atoms.

(iii) A shell is defined as a complete group of orbital possessing the same quantum number.

(iv) Quantum is the smallest countable, discrete packet or increment of radiant energy that can be absorbed or emitted.

**Example**

Explain the meaning of the following terms:

(a) Quantization of energy and radiation.

(b) Wave particles duality of matter.

(c) Quantum number.

**Solution**

(a)Light and other forms of electromagnetic radiations are not emitted continuously but in discrete “packets” called photons. Energy of an electromagnetic radiation is also not emitted continuously but in “packets” called quanta. A photon of radiation carries a quantum of energy. Energy and radiation are quantized in that electron shall fall from one energy level to another and not anywhere else in between.

(b) According to de Broglie, every sub – atomic particles has both wave and particle properties. Photons of light and other electromagnetic radiations, are regarded and wavelengths. They are regarded as waves because they have specific frequencies. Although electrons have extremely small mass, they have both velocity and momentum as particles.

(c) Quantum numbers are used to characterize electrons as they occupy orbitals in different energy levels. Each electron is characterized by four quantum numbers:-

The principal quantum number, n, which specifies the location and energy of the electron.

The azimuthal, subsidiary or angular momentum quantum number, l, which specifies the shapes of the orbitals.

The magnetic quantum number, m, which specifies the orbital orientation in space.

The spin quantum number, s, which specifies the direction in which the electron is spinning. These quantum numbers completely describe the stationary stage of the electron.

**Example**

(a) Distinguish between the following terms:-

(i) An orbital and orbit.

(ii) S – Orbital and P – orbital.

(iii) A quantum of light and quantization of energy.

(iv) Quantum shell and quantum number.

(b) Explain briefly the meaning of the following quantum numbers:-

(i) n

(ii) l

(iii) m

(iv) ms

(c) In a tabular form specify all the four quantum number for each electron in an atom whose n value is 2. Given all the orbitals are full of electrons.

(d) Given the value of the quantum numbers n, l and m for the electron with the highest energy in sodium atom in the ground state.

(e) Write down all the quantum number of all the electrons in the ground state of nitrogen atom.

(f) Give the values of all the four quantum number for 2p electrons in Nitrogen.

(g) Briefly explain why the following quantum numbers are not allowed.

i) n = 1, l = 1, m = 0

ii) n = 1, l = 0, m = 2

iii) n = 4, l = 3, m = 4

iv) n = 0, l = -1, m = 1

v) n = 2, l = -1, m = 1

**Solution**

(a) (i) An orbit is a well define circular path in which electrons were assumed to revolve around the nucleus. Whereas an orbital is a three dimensional region of space around the nucleus whereby there is high probability of finding an electron.

(ii) S–Orbital is spherical and hence non directional whereas p-orbital is dumbbell in shape and it is directional.

(iii) A quantum of light is a photon of radiation emitted when an electron jumps from one energy level to another whereas quantization of energy means energy emitted or absorbed following electron transition between one energy level and another is not continuous rather it is in form of small packet called quanta.

(iv) A quantum shell is the energy level of an electron in a given atom whereas quantum number refers to specifies way of defining an electron in a given atom whereas quantum number refers to specified way of defining an electron in a given orbital of an atom using n, l, m and ms values.

(b) (i) n – represents the principle quantum number;

It specifies the location and the energy of an electron.

It is a measure of the volume of the electrons cloud.

As the value of n increases or becomes less negative it means the energy levels of the electrons get further away from the nucleus n can have only integers 1, 2, 3, 4 to infinite represented by K,L,M,N etc. Also called azimuthal quantum number or subsidiary quantum number, specifies the shapes of the orbital, when n = 1 there’s only l s orbital where n = 2, there’s 2s and 2p orbital, thus for a given value of n, l, are all from 0, 1, 2…up to l = n – 1. Thus when n = 4 values are 0, 1, 2 and 3. Which represents 4s, 4p, 4d and 4, â„“ sub–levels m Magnetic quantum number specifies the number of orbital present in a given value of â„“.

It specifies the orientating i.e. the direction of the orbital to magnetic field in which it’s placed.

It accounts for the splitting of the spectral lines observed when an atom emitting radiation is placed in a magnetic field, ms – Refers to spin quantum number.

(i) It indicates the direction in which the electron spins.

(ii) Only two values are allowed for an electrons i.e. electrons may spin clockwise, shown as +1/2 or may spin anticlockwise as shown as ½ or

(c)

Principle (n) | Azimuthal | Magnetic quantum | Spins ms |

n = 1 | â„“ = 0 | m = 0 | ^{+1}/_{2 }, ^{–1}/_{2} |

n = 2 | â„“ = 0
â„“ = 1 |
m = 0
m = -1 m = 0 m = 1 |
^{+1}/_{2} , ^{–1}/_{2}
Total of 10 electrons |

(d) When n = 3 in Na atom that electron is found in s – orbital thus â„“ = 0 and also ml = 0

(e) 1 electrons:

(1^{st}) n = 1, l = 0, m = 0, s = +^{1}/_{2 } (2^{nd}) n = 1, l = 0, m = 0, s = ^{-1}/_{2}

Electrons:

1^{st} n = 1, l = 0, m = 0, s = ^{+1}/_{2} (2^{nd}) n = 1, l = 0, m = 0, s = ^{-1}/_{2 }

2 Px; n = 2, l = 1, m = ^{+}1, s = ^{+}1/2 2Py; n = 2, l = ^{–}1, m = ^{–}1, s = ^{+}1/2

2Pz; n = 2, l = 1, m = 0, s = ^{+}1/2

(f) 2P electrons, there’s

2Px; n = 2, l = +1, m = +1, s = +1/2 2Py: n = 2 , l= ^{–}1, m = ^{–}1, s = ^{+}1/2

2Pz: n = 2 l = 1 m = 0, s = +1/2

(g) (i) l value must be smaller than n value since l = n-1

(ii) When l = 0 value of m is only 0

(iii) For l= 3, m can range from^{ –}3 to ^{+}3. Thus m=^{+}4 is not allowed

(iv) n value cannot be equal to zero.

(v) l value cannot be a negative number.

**Example**

(a) briefly explain the meaning of the following quantum numbers:-

(i) ml

(ii) n

(iii) l

(iv) ms

(b) If n = 2, tabulate the values of 1, _{ml} and ms.

(c) Give the possible values of 1 and m, for an electron with the principal quantum number, n= 3

(d) Briefly explain why an electron cannot have the quantum number n = 2, l = 2 and ml = 2

**Solution**

(a) (i) ml – Magnetic quantum number with values from -l to +l. These are numbers showing the number of sub – level in each energy level.

– Describes orientation

(ii) n – Principal (shell) Quantum number value of 1, 2, 3 etc theses are numbers used to represent the main energy level in which an electrons is found.

(iii) l – Azimuthal or subsidiary or Angular or orbital Quantum number. They specify the shape of orbitals.

(iv) ms – Magnetic spin Quantum number with values ^{+}1/2 and ^{–}1/2 shows the spinning of electrons in orbitals (clockwise and anticlockwise direction).

(a) For n = 2, the scheme below illustrates the value of 1, ml and ms.

l=0, 1

ml=-1, 0, +1, 0

ms= ±1/2, ±1/2, ±1/2, ±1/2

Therefore the total number of electrons = 8

(b) If n = 3, possible values of 1 and m are;

l = 0, 1 and 2

m = 0 for l = 0

m =^{ –}1, 0 and ^{+}1 for l = 1

m = ^{–}2, ^{–}1, 0, ^{+}1 and 2 for l = 2

Note that: l=n-1

(c) An electrons cannot have quantum numbers, n = 2, l = 2 and m = 2. This is because when n = 2, l can be 0 and 1; then ml can be 0, ^{–}1, 0, ^{+}1 different values which can occupy in orbitals (i.e. they have n, l and m values the same but they must spin in opposite directions and hence have different ms values).

**RULES FOR FILLING ELECTRONS IN THE ORBITALS OF AN ATOM**

The orbitals of an atom can be filled with electrons by applying the following rules.

**i)Aufbau Principle**

This is also known as the building up principle. According to this principle “The electrons in an atom are so arranged that they occupy orbitals in the order of their increasing energy.”

Thus, the orbital with the lowest energy will be filled first, then the next higher in energy, and so on. Since, the energy of an orbital in the absence of any magnetic field depends upon the principal quantum number *(n) *and the azimuthal quantum number (l), hence the order of filling orbitals with electrons may be obtained from the following generalizations.

a) The orbitals for which *n + I *is the lowest is filled first.

b) When two orbitals have the same values of n+l, the orbital having the lower value of *n *is filled first.

The order of filling of various orbitals with electrons obtained by this rule is given below:

1s, 2s. 2p, 3s, 3p, 4s, *3d, 4p, *5s, *4d, 5p, *6s, *4f,5d, 6p, 7s, 5f…*

To remember this sequence may be a difficult task. Given a long side is a simple way of working out this order. In this method a series of arrows running from upper right to the lower left gives the order of orbitals with increasing energy.

**ii) Pauli’s Exclusion principle**

In 1925 Wolfgang Pauli discovered what is known as the exclusion principle. This principle is very useful in constructing the electronic configuration of atoms. According to this principle “No two electrons in an atom can have the same values for all the four quantum numbers”.

For example in 1s orbital of helium atom there are two electrons. According to the concept of quantum number and Pauli’s rule, their quantum numbers are:

The + and – sign before *j *refers to the clockwise and anticlockwise spins of the electrons.

Thus, the two electrons having the same values of *n, I *and *m *could have different values

*of s, i.e., *their spins are in the opposite directions. This leads to a very significant observation that

*“Each orbital can accommodate at the maximum two electrons with opposite spins.”*

**Applications of the Pauli’s exclusion principles **

The Pauli’s exclusion principle leads to the following conclusions:

(a) An orbital cannot have more than two electrons.

(b) In any main energy level (shell), the maximum number of electrons is twice the in number of orbitals, or is equal to 2n^{2}, where *n *is the principal quantum number.

**iii) Hund’s Rule of Maximum Multiplicity**

The rules discussed above do not give any idea for filling electrons into the orbitals having equal energies (such states are called *degenerate states). *For example, three p-orbitals, i.e. px, py and *pz, *have equal energy. How should electrons be filled into these orbitals? Let us take an example in which three electrons are to be filled into three p-orbitals. The three electrons can be filled into three *p *orbitals in two different ways as shown below.

Now, which of the two is correct? The answer to this question is given by Hund’s rule, Hund’s rule states that,

“When more than one orbitals of equal energies are available, then the electron*,] first occupy these orbitals separately with parallel spins. The pairing of electrons will | only after all the orbitals of a given sub-level are singly occupied.”

According to the Hund’s rule, the correct way of filling three electrons in three *p *orbitals that in which each orbital is singly occupied, (arrangement II above).

Hund’s rule is also known as the *Hund’s rule of maximum multiplicity.*

**Explanation**

Two electrons with parallel spins, tend to be as far apart as possible to minimize the electrostatic repulsion. Therefore, the electrons prefer to occupy the orbitals singly

as far as possible. When all the orbitals get singly occupied, then the incoming electron has two choices either to pair up with the other electron or to go to the next higher orbital.

When vacant orbital of suitable energy is not available, then the incoming electron will have no choice except to pair up with another electron.

** Example**

(a) State the;

(i) Heisenberg uncertainty principle.

(ii) Hund’s rule.

(iii) Paul’s Principle.

(iv) Aufbau’s Principle.

(b) Brief explain why the following sets of quantum number are NOT allowed in hydrogen atom:

(i) n = 1, l = 1, ml = 0

(ii) n = 1, 1 = 0, ml = 2

(iii) n = 4,1 = 3, ml = 4

(iv) n = 0, 1 = 0, ml = 0 and

(v) n = 2, 1 = ^{–}1, ml = 1

(c) How many orbitals are there in each of the following sublevel?

(i) 1s

(ii) 2p

(iii) 3d

(iv) 4f

(d) How many sublevels are there in each of the following shells?

(i) K

(ii) N

(iii) L

(e) Which of the following electronic configuration is correct for 14p in its ground state?

State the principle violated in each case

**Solution**

(a) (i) The Heisenberg’s uncertainly principle states that; “it is impossible to measure simultaneously the exact position and exact velocity (momentum) of an electron.” Any attempt to measure one quality will distract the measurements of the other quantity

(ii) Hund’s rule states that, “electron pairing is not allowed until all orbitals of the particular energy sub – level are occupied by at least one electron.”

(iii) Paul’s Exclusion Principle, “states that no two electrons may have all the four quantum number the same.” Two electrons may have three quantum numbers of the same; the paired electrons are always in opposite direction at the same time.

(E.g. and).

(iv)Aufbau Principle states that, “In electrons configuration, electrons are filled in order of increasing energy levels.” Thus the lowest energy available must be filled up first. The orbital energy increase in the following order; is 2s, 2p, 3s, 4p, 5p etc.

(b) (i) n = 1, 1 = 1, ml = 0 this is not allowed because l must be smaller than n, in the case l = 1 which is not allowed.

(ii) n = 1, 1 = 0, ml = 2. This is not allowed because 1 = 0 and ml = 0

(iii) n = 4, 1 = 3, ml = 2. This is not allowed because for 1 = 3 can range from ^{–}3, ^{+}3, thus, ^{+}4 is not allowed.

(iv) n = 0, l = 0, ml = 0; this is not allowed because n cannot be zero.

(v) n = 2, l = -1, ml = 1. This is not allowed because l cannot be a negative number and n can never be = 0

(c) (i) 1 s sub energy level has one – orbital (which is 1 s – orbital).

(ii) 2 p sub energy has three p – orbital (which is 1 s – orbital).

(iii) 3 d sub energy level has five d – orbital [which are 3 d (x^{2} – y^{2}), 3dz^{2} 3dxy, 3 and 3dyz].

(iv) 4 f sub energy level has seven f – orbital [which are 4xy, 4fyz, 4fz, 4f(x^{2} – y^{2}),

4f(x^{2}-z^{2}) and 4f (y^{2} – z^{2})].

(d) (i) K has one sub level (which is l s^{2}).

(ii) N has four sub levels (which are 4s^{2}, 4p^{6}, 4d^{10} and 4f^{14}).

(iii) L has two sub levels (which are 2s2 and 2p6).

(e) Is the correct electronic configuration of 15p

This is correct since it obeys principle governing electron in an atom distribution

(i) [(Ne)]

(ii) [(Ne)] aufbau (building up) principle is violated as it states that electrons available as the case here.

(iii) [(Ne)]

(iv) [(Ne)] Hund’s Rule of maximum that multiplicity is violated as it states electrons occupy orbital as singly as possible

**Example**

(a) State the rules used in filling electrons in various orbitals of an atom.

(b) By using 3 different ways for each of the following atoms write their electron configuration

(i) C

(ii) N

(iii) Ag

(c)Write the electronic configuration of Na^{+} and F^{–} then show the other element whose electronic configuration resembles these ions. Show that electronic configurations of Cu and Cr are unusually written. Give reasons.

**Solution**

(a)(i)Paul’s exclusion principle states that; “no two electrons may have all the four quantum number the same i.e. electrons may occupy the same orbital only on conditions that they have their spins in the opposite direction.”

(ii) Hund’s rule of maximum multiplicity states that, “when more than one orbital with equal energies are available, electrons tend occupy those orbital separately first with parallel spins and separately first with parallel spin and pairing of electrons will start only after all the orbitals of a given sub level are singly occupied.”

(iii) Aufbau principle states that, “Electrons in an atom are so arranged that they occupy the orbitals in the order of their increasing energy.”

(b) (i) C Atomic number 6

Using Box method:

1s^{2} 2s^{2} 2p^{2}

Using noble gas structure;

[He] 2p^{2}

(ii) N atomic number 7

Using box method

Using sub levels only;

1s^{2} 2s^{2} 2p^{3}

Using noble gas structure;

[He] 2p^{3}

(iii) Ag atomic number 47

Using box method;

Note: 4d is filled first to maintain stability of full filled d – orbital before 5s is filled.

Using sub levels only;

1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{6} 4d^{10 }5s^{1}

Using noble gas structure;

[Kr] 4d^{10} 5s^{1}

(c)

(i) Na^{+} number of electrons also 10

1s^{2} 2s^{2} 2p^{6} resembles Ne

(ii) F – number of electrons also 10

1s^{2} 2s^{2} 2p^{6} resembling Ne

(d) Cu Atomic number 29

e.g. [Ar] 4s^{1} 3d^{10}

Cr Atomic number 24

e.g. [Ar] 4s^{1} 3d^{5} full filled and half filled orbitals are very stable electronic structures. The stability gained is the cause of 4s electrons to be unpaired and make 3d orbitals either full filled or half filled.

Note that: In writing the electronic configuration the atoms or orbitals of the same value are usually written together irrespective of their relation energy levels.

E.g. Cu 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 3d^{10} 4s^{1} not 1s^{2} 2s^{2 }2p^{6} 3s^{2} 2p^{6} 4s^{1} 3d^{10}. To save time and space noble gases are often utilized e.g. sodium [Ne] 3s^{1} and Iron [Ar] 3d^{6} 4s^{2}

Wave mechanics deal with electron occupying space. The electrons revolve the nucleus through orbit and occur in the orbital.

Orbital:Is a region within an atomic sublevel that can be occupied by maximum two electrons that have opposite spin. Any orbital can accommodate a maximum of two electrons.

The energy level of atoms is a specific region around the nucleus electrons can occupy in atoms. A shell is defined as a complete group of orbital possessing the quantum number. The

electrons arranged in the atom starting from orbit from nucleus toward the orbit of highest energy level. The following include position of locating electrons in the atom.

i. Principal quantum number (n)

The atom shells called principal quantum number (main energy level). The electrons filled in atom in order of increasing the energy from the nucleus. Each shell have specific constant number or electrons obtained taking

e = 2n^{2}

Old name | 1 | 2 | 3 | 4 |

New name | K | L | M | N |

** (ii)Subsidiary quantum number (L)**

Subsidiary quantum number is a sub energy level. The sub energy level known as azimuthal quantum number, sub shell or sub energy level. The shell divides to form sub shell. The total number of sub shells in each shell is equal to the number of shell from the nucleus. This describes shapes of orbitals (sub-energy level).Values of subsidiary quantum number start from 0——-n-1

The names of sub shell include the following;

iii. Magnetic quantum number (m)

This is a quantum number which describes the orbital of each sub shell. This sub shell divided and form orbital. Each sub shells have specific number of orbital.

(iv) Spin Quantum number (s)

These quantum numbers describe the spinning of electrons in the orbital. Maximum number of electrons in the orbital in only two which spines in opposite direction. Since each orbital has two maximum electrons result the sub shell to have constant total number of electrons which accommodated in all shells S^{2}, p^{6}, d^{10}, f^{14}, g^{18}, h^{22} and i^{26}. In order to write electrons of element consider the knowledge of all four quantum numbers.

The following series of sub shell used to write the electrons structure.

This arrangement used to write the electronic configuration.

Arrangement used to write the electronic configuration has some irregularities or abnormality. This irregularity occurs because there is overlapping of orbital. The orbital of low energy jump toward highest energy level and orbital of high energy drop down towards lowest energy level. Example 4s arranged first before 3d orbital.

RULES WHICH GOVERN ARRANGEMENT OF ELECTRONS IN ORBITALS

1. Aufbau’s principle – State that “in the ground state of orbital the lowest energy is filled by electrons before filling orbital of the highest energy”.

This means that electron filled in order of increasing energy level. The orbital of lowest energy when is full of electrons then the electron filled on the next orbital of highest energy level. Consider the following arrangement

2. Hund’s rule of maximum multiplicity

State that, “The electrons pair in the orbital of the same sub shell is allowed if all orbital of the same energy are occupied by electrons of maximum multiplicity. This means that the degenerate orbital are not allowed to pair up unless each orbital is singly occupied. The incoming pairing is done in an opposite spin. Consider a case with 5 electrons in valency shell.

3. Paul’s exclusive principle

States that, “Two electrons in an atom can never have exactly similar set of all quantum numbers.” This means that the maximum number of electrons which is occupied in the orbital is only two. Two electrons may occupy the same orbital only on condition that they have their spin in the opposite direction. Two electrons may have three quantum numbers the same; n, l and ml but the fourth, ms must be different.

Sometimes the electrons structure or electronic diagram of ions is written according to what kind of ions given.

There are two kinds of ions which include the following;-

Cation or metallic ion is formed through losing of electrons. The electrons which are lost occur in the orbital out the of the noble gas structure. The electron removed starting from orbital of low energy toward the orbital of highest energy.

Anion or non-metallic ion – is formed by gaining electrons. Electrons which gain is added in the orbital having unpaired electrons starting from orbital of lowest energy to the orbital of highest energy level

QUANTUM NUMBER

Is a number which describe the characteristics of electrons or is a number which is used to characterize electrons as they occupy orbitals in different energy levels. The quantum number is a number which describe main energy level, orientation of orbitals and spinning of electrons.

**There are four type of quantum number which includes the following:-**

**1.Principle quantum number (n)**: Is a number which specifies the location and the energy of electron. This is a number which describe the main energy level of electrons. The principle quantum number called shell orbit, main energy level and stationary state. The principle quantum number have specific name from the nucleus. K, L, M, N etc.

**2. Subsidiary principle quantum number (L)**: Is a number which specifies the shape of orbitals. Subsidiary principle quantum number is a sub energy level. The sub energy level is known as azimuthal number sub shell or sub energy level. The shell is divided to form sub shell. The total number of sub shells in each shell is equal to the number of shells from the nucleus. The names of sub shells include the following; s, p, d, f, g, h, i. The value of sub energy level i.e. 0, 1, 2, …L = n – 1.The value of L must be smaller that n because L = n – 1

n | Value of L | Sub shell |

1 | 0 | 1s |

2 | 0, 1 | 2s2p |

3 | 0, 1, 2 | 3sp3d |

4 | 0, 1, 2, 3 | 4s4p4d4f |

5 | 0, 1, 2, 3, 4 | 5s5p5d5f5f |

** 3. Magnetic quantum number**: Is a number which specifies the number of orbital present in a given value of subsidiary quantum number. This is a quantum number which describes the orientation of orbital. The magnetic quantum number is a number of orbital. When the sub shell divides form orbital, these orbitals are called Magnetic quantum number.

1s – Orbital S = X

**4. Spin quantum number (s)** – This is a quantum number which describes the direction in which the electrons spin. There are two maximum electrons, these electrons are moving in opposite direction. They revolve clockwise + ½ and another revolve anti – clock wise – ½

**APPLICATION OF QUANTUM NUMBER**

Quantum number is applied to find the number of sub shell (L), number of orbital (m) and total number of electrons (s) of the given quantum number (n). There are two methods which are applied to find total sub shells, orbital and electrons.

Find total number of electrons in the principle quantum number 2.

(By using tree diagram)

Total shells = 2 (s, p)

Orbitals = 4 (0, 0, +1, -1)

Electrons = 8

Also: by using tabular form

(b) Find the total number of electrons in the principle quantum number 3

Soln: (by using tree diagram)

**Example**

(a) The modern theory of electron behavior is based on two assumptions. State them

(b) Define the following terms

i. Orbital.

ii. Energy levels of atoms.

iii A shell.

iv. Quantum.

**Solution:**

(a) The modern theory of electron behavior is based on two assumptions.

(i) An electron has a dual nature i.e. it behaves both as a particle and wave.

(ii) It is practically impossible to determine simultaneously both the position and momentum of an electron with any degree of precision.

(b) (i) An Orbital is a region within an atomic sublevel that can be occupied by a maximum of two electrons that have opposite spin. There are s,p,d, and f orbitals is a volume of space surrounding the nucleus within which there is a more than 95% chances of finding electrons. Any orbital can accommodate a maximum of two electrons.

(ii) Energy level of atoms in specific region around the nucleus that electrons can occupy in atoms.

(iii) A shell is defined as a complete group of orbital possessing the same quantum number.

(iv) Quantum is the smallest countable, discrete packet or increment of radiant energy that can be absorbed or emitted.

**Example**

Explain the meaning of the following terms:

(a) Quantization of energy and radiation.

(b) Wave particles duality of matter.

(c) Quantum number.

**Solution**

(a)Light and other forms of electromagnetic radiations are not emitted continuously but in discrete “packets” called photons. Energy of an electromagnetic radiation is also not emitted continuously but in “packets” called quanta. A photon of radiation carries a quantum of energy. Energy and radiation are quantized in that electron shall fall from one energy level to another and not anywhere else in between.

(b) According to de Broglie, every sub – atomic particles has both wave and particle properties. Photons of light and other electromagnetic radiations, are regarded and wavelengths. They are regarded as waves because they have specific frequencies. Although electrons have extremely small mass, they have both velocity and momentum as particles.

(c) Quantum numbers are used to characterize electrons as they occupy orbitals in different energy levels. Each electron is characterized by four quantum numbers:-

The principal quantum number, n, which specifies the location and energy of the electron.

The azimuthal, subsidiary or angular momentum quantum number, l, which specifies the shapes of the orbitals.

The magnetic quantum number, m, which specifies the orbital orientation in space.

The spin quantum number, s, which specifies the direction in which the electron is spinning. These quantum numbers completely describe the stationary stage of the electron.

**Example**

(a) Distinguish between the following terms:-

(i) An orbital and orbit.

(ii) S – Orbital and P – orbital.

(iii) A quantum of light and quantization of energy.

(iv) Quantum shell and quantum number.

(b) Explain briefly the meaning of the following quantum numbers:-

(i) n

(ii) l

(iii) m

(iv) ms

(c) In a tabular form specify all the four quantum number for each electron in an atom whose n value is 2. Given all the orbitals are full of electrons.

(d) Given the value of the quantum numbers n, l and m for the electron with the highest energy in sodium atom in the ground state.

(e) Write down all the quantum number of all the electrons in the ground state of nitrogen atom.

(f) Give the values of all the four quantum number for 2p electrons in Nitrogen.

(g) Briefly explain why the following quantum numbers are not allowed.

i) n = 1, l = 1, m = 0

ii) n = 1, l = 0, m = 2

iii) n = 4, l = 3, m = 4

iv) n = 0, l = -1, m = 1

v) n = 2, l = -1, m = 1

**Solution**

(a) (i) An orbit is a well define circular path in which electrons were assumed to revolve around the nucleus. Whereas an orbital is a three dimensional region of space around the nucleus whereby there is high probability of finding an electron.

(ii) S–Orbital is spherical and hence non directional whereas p-orbital is dumbbell in shape and it is directional.

(iii) A quantum of light is a photon of radiation emitted when an electron jumps from one energy level to another whereas quantization of energy means energy emitted or absorbed following electron transition between one energy level and another is not continuous rather it is in form of small packet called quanta.

(iv) A quantum shell is the energy level of an electron in a given atom whereas quantum number refers to specifies way of defining an electron in a given atom whereas quantum number refers to specified way of defining an electron in a given orbital of an atom using n, l, m and ms values.

(b) (i) n – represents the principle quantum number;

It specifies the location and the energy of an electron.

It is a measure of the volume of the electrons cloud.

As the value of n increases or becomes less negative it means the energy levels of the electrons get further away from the nucleus n can have only integers 1, 2, 3, 4 to infinite represented by K,L,M,N etc. Also called azimuthal quantum number or subsidiary quantum number, specifies the shapes of the orbital, when n = 1 there’s only l s orbital where n = 2, there’s 2s and 2p orbital, thus for a given value of n, l, are all from 0, 1, 2…up to l = n – 1. Thus when n = 4 values are 0, 1, 2 and 3. Which represents 4s, 4p, 4d and 4, â„“ sub–levels m Magnetic quantum number specifies the number of orbital present in a given value of â„“.

It specifies the orientating i.e. the direction of the orbital to magnetic field in which it’s placed.

It accounts for the splitting of the spectral lines observed when an atom emitting radiation is placed in a magnetic field, ms – Refers to spin quantum number.

(i) It indicates the direction in which the electron spins.

(ii) Only two values are allowed for an electrons i.e. electrons may spin clockwise, shown as +1/2 or may spin anticlockwise as shown as ½ or

(c)

Principle (n) | Azimuthal | Magnetic quantum | Spins ms |

n = 1 | â„“ = 0 | m = 0 | ^{+1}/_{2 }, ^{–1}/_{2} |

n = 2 | â„“ = 0
â„“ = 1 |
m = 0
m = -1 m = 0 m = 1 |
^{+1}/_{2} , ^{–1}/_{2}
Total of 10 electrons |

(d) When n = 3 in Na atom that electron is found in s – orbital thus â„“ = 0 and also ml = 0

(e) 1 electrons:

(1^{st}) n = 1, l = 0, m = 0, s = +^{1}/_{2 } (2^{nd}) n = 1, l = 0, m = 0, s = ^{-1}/_{2}

Electrons:

1^{st} n = 1, l = 0, m = 0, s = ^{+1}/_{2} (2^{nd}) n = 1, l = 0, m = 0, s = ^{-1}/_{2 }

2 Px; n = 2, l = 1, m = ^{+}1, s = ^{+}1/2 2Py; n = 2, l = ^{–}1, m = ^{–}1, s = ^{+}1/2

2Pz; n = 2, l = 1, m = 0, s = ^{+}1/2

(f) 2P electrons, there’s

2Px; n = 2, l = +1, m = +1, s = +1/2 2Py: n = 2 , l= ^{–}1, m = ^{–}1, s = ^{+}1/2

2Pz: n = 2 l = 1 m = 0, s = +1/2

(g) (i) l value must be smaller than n value since l = n-1

(ii) When l = 0 value of m is only 0

(iii) For l= 3, m can range from^{ –}3 to ^{+}3. Thus m=^{+}4 is not allowed

(iv) n value cannot be equal to zero.

(v) l value cannot be a negative number.

**Example**

(a) briefly explain the meaning of the following quantum numbers:-

(i) ml

(ii) n

(iii) l

(iv) ms

(b) If n = 2, tabulate the values of 1, _{ml} and ms.

(c) Give the possible values of 1 and m, for an electron with the principal quantum number, n= 3

(d) Briefly explain why an electron cannot have the quantum number n = 2, l = 2 and ml = 2

**Solution**

(a) (i) ml – Magnetic quantum number with values from -l to +l. These are numbers showing the number of sub – level in each energy level.

– Describes orientation

(ii) n – Principal (shell) Quantum number value of 1, 2, 3 etc theses are numbers used to represent the main energy level in which an electrons is found.

(iii) l – Azimuthal or subsidiary or Angular or orbital Quantum number. They specify the shape of orbitals.

(iv) ms – Magnetic spin Quantum number with values ^{+}1/2 and ^{–}1/2 shows the spinning of electrons in orbitals (clockwise and anticlockwise direction).

(a) For n = 2, the scheme below illustrates the value of 1, ml and ms.

l=0, 1

ml=-1, 0, +1, 0

ms= ±1/2, ±1/2, ±1/2, ±1/2

Therefore the total number of electrons = 8

(b) If n = 3, possible values of 1 and m are;

l = 0, 1 and 2

m = 0 for l = 0

m =^{ –}1, 0 and ^{+}1 for l = 1

m = ^{–}2, ^{–}1, 0, ^{+}1 and 2 for l = 2

Note that: l=n-1

(c) An electrons cannot have quantum numbers, n = 2, l = 2 and m = 2. This is because when n = 2, l can be 0 and 1; then ml can be 0, ^{–}1, 0, ^{+}1 different values which can occupy in orbitals (i.e. they have n, l and m values the same but they must spin in opposite directions and hence have different ms values).

**RULES FOR FILLING ELECTRONS IN THE ORBITALS OF AN ATOM**

The orbitals of an atom can be filled with electrons by applying the following rules.

**i)Aufbau Principle**

This is also known as the building up principle. According to this principle “The electrons in an atom are so arranged that they occupy orbitals in the order of their increasing energy.”

Thus, the orbital with the lowest energy will be filled first, then the next higher in energy, and so on. Since, the energy of an orbital in the absence of any magnetic field depends upon the principal quantum number *(n) *and the azimuthal quantum number (l), hence the order of filling orbitals with electrons may be obtained from the following generalizations.

a) The orbitals for which *n + I *is the lowest is filled first.

b) When two orbitals have the same values of n+l, the orbital having the lower value of *n *is filled first.

The order of filling of various orbitals with electrons obtained by this rule is given below:

1s, 2s. 2p, 3s, 3p, 4s, *3d, 4p, *5s, *4d, 5p, *6s, *4f,5d, 6p, 7s, 5f…*

To remember this sequence may be a difficult task. Given a long side is a simple way of working out this order. In this method a series of arrows running from upper right to the lower left gives the order of orbitals with increasing energy.

**ii) Pauli’s Exclusion principle**

In 1925 Wolfgang Pauli discovered what is known as the exclusion principle. This principle is very useful in constructing the electronic configuration of atoms. According to this principle “No two electrons in an atom can have the same values for all the four quantum numbers”.

For example in 1s orbital of helium atom there are two electrons. According to the concept of quantum number and Pauli’s rule, their quantum numbers are:

The + and – sign before *j *refers to the clockwise and anticlockwise spins of the electrons.

Thus, the two electrons having the same values of *n, I *and *m *could have different values

*of s, i.e., *their spins are in the opposite directions. This leads to a very significant observation that

*“Each orbital can accommodate at the maximum two electrons with opposite spins.”*

**Applications of the Pauli’s exclusion principles **

The Pauli’s exclusion principle leads to the following conclusions:

(a) An orbital cannot have more than two electrons.

(b) In any main energy level (shell), the maximum number of electrons is twice the in number of orbitals, or is equal to 2n^{2}, where *n *is the principal quantum number.

**iii) Hund’s Rule of Maximum Multiplicity**

The rules discussed above do not give any idea for filling electrons into the orbitals having equal energies (such states are called *degenerate states). *For example, three p-orbitals, i.e. px, py and *pz, *have equal energy. How should electrons be filled into these orbitals? Let us take an example in which three electrons are to be filled into three p-orbitals. The three electrons can be filled into three *p *orbitals in two different ways as shown below.

Now, which of the two is correct? The answer to this question is given by Hund’s rule, Hund’s rule states that,

“When more than one orbitals of equal energies are available, then the electron*,] first occupy these orbitals separately with parallel spins. The pairing of electrons will | only after all the orbitals of a given sub-level are singly occupied.”

According to the Hund’s rule, the correct way of filling three electrons in three *p *orbitals that in which each orbital is singly occupied, (arrangement II above).

Hund’s rule is also known as the *Hund’s rule of maximum multiplicity.*

**Explanation**

Two electrons with parallel spins, tend to be as far apart as possible to minimize the electrostatic repulsion. Therefore, the electrons prefer to occupy the orbitals singly

as far as possible. When all the orbitals get singly occupied, then the incoming electron has two choices either to pair up with the other electron or to go to the next higher orbital.

When vacant orbital of suitable energy is not available, then the incoming electron will have no choice except to pair up with another electron.

** Example**

(a) State the;

(i) Heisenberg uncertainty principle.

(ii) Hund’s rule.

(iii) Paul’s Principle.

(iv) Aufbau’s Principle.

(b) Brief explain why the following sets of quantum number are NOT allowed in hydrogen atom:

(i) n = 1, l = 1, ml = 0

(ii) n = 1, 1 = 0, ml = 2

(iii) n = 4,1 = 3, ml = 4

(iv) n = 0, 1 = 0, ml = 0 and

(v) n = 2, 1 = ^{–}1, ml = 1

(c) How many orbitals are there in each of the following sublevel?

(i) 1s

(ii) 2p

(iii) 3d

(iv) 4f

(d) How many sublevels are there in each of the following shells?

(i) K

(ii) N

(iii) L

(e) Which of the following electronic configuration is correct for 14p in its ground state?

State the principle violated in each case

**Solution**

(a) (i) The Heisenberg’s uncertainly principle states that; “it is impossible to measure simultaneously the exact position and exact velocity (momentum) of an electron.” Any attempt to measure one quality will distract the measurements of the other quantity

(ii) Hund’s rule states that, “electron pairing is not allowed until all orbitals of the particular energy sub – level are occupied by at least one electron.”

(iii) Paul’s Exclusion Principle, “states that no two electrons may have all the four quantum number the same.” Two electrons may have three quantum numbers of the same; the paired electrons are always in opposite direction at the same time.

(E.g. and).

(iv)Aufbau Principle states that, “In electrons configuration, electrons are filled in order of increasing energy levels.” Thus the lowest energy available must be filled up first. The orbital energy increase in the following order; is 2s, 2p, 3s, 4p, 5p etc.

(b) (i) n = 1, 1 = 1, ml = 0 this is not allowed because l must be smaller than n, in the case l = 1 which is not allowed.

(ii) n = 1, 1 = 0, ml = 2. This is not allowed because 1 = 0 and ml = 0

(iii) n = 4, 1 = 3, ml = 2. This is not allowed because for 1 = 3 can range from ^{–}3, ^{+}3, thus, ^{+}4 is not allowed.

(iv) n = 0, l = 0, ml = 0; this is not allowed because n cannot be zero.

(v) n = 2, l = -1, ml = 1. This is not allowed because l cannot be a negative number and n can never be = 0

(c) (i) 1 s sub energy level has one – orbital (which is 1 s – orbital).

(ii) 2 p sub energy has three p – orbital (which is 1 s – orbital).

(iii) 3 d sub energy level has five d – orbital [which are 3 d (x^{2} – y^{2}), 3dz^{2} 3dxy, 3 and 3dyz].

(iv) 4 f sub energy level has seven f – orbital [which are 4xy, 4fyz, 4fz, 4f(x^{2} – y^{2}),

4f(x^{2}-z^{2}) and 4f (y^{2} – z^{2})].

(d) (i) K has one sub level (which is l s^{2}).

(ii) N has four sub levels (which are 4s^{2}, 4p^{6}, 4d^{10} and 4f^{14}).

(iii) L has two sub levels (which are 2s2 and 2p6).

(e) Is the correct electronic configuration of 15p

This is correct since it obeys principle governing electron in an atom distribution

(i) [(Ne)]

(ii) [(Ne)] aufbau (building up) principle is violated as it states that electrons available as the case here.

(iii) [(Ne)]

(iv) [(Ne)] Hund’s Rule of maximum that multiplicity is violated as it states electrons occupy orbital as singly as possible

**Example**

(a) State the rules used in filling electrons in various orbitals of an atom.

(b) By using 3 different ways for each of the following atoms write their electron configuration

(i) C

(ii) N

(iii) Ag

(c)Write the electronic configuration of Na^{+} and F^{–} then show the other element whose electronic configuration resembles these ions. Show that electronic configurations of Cu and Cr are unusually written. Give reasons.

**Solution**

(a)(i)Paul’s exclusion principle states that; “no two electrons may have all the four quantum number the same i.e. electrons may occupy the same orbital only on conditions that they have their spins in the opposite direction.”

(ii) Hund’s rule of maximum multiplicity states that, “when more than one orbital with equal energies are available, electrons tend occupy those orbital separately first with parallel spins and separately first with parallel spin and pairing of electrons will start only after all the orbitals of a given sub level are singly occupied.”

(iii) Aufbau principle states that, “Electrons in an atom are so arranged that they occupy the orbitals in the order of their increasing energy.”

(b) (i) C Atomic number 6

Using Box method:

1s^{2} 2s^{2} 2p^{2}

Using noble gas structure;

[He] 2p^{2}

(ii) N atomic number 7

Using box method

Using sub levels only;

1s^{2} 2s^{2} 2p^{3}

Using noble gas structure;

[He] 2p^{3}

(iii) Ag atomic number 47

Using box method;

Note: 4d is filled first to maintain stability of full filled d – orbital before 5s is filled.

Using sub levels only;

1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{2} 3d^{10} 4p^{6} 4d^{10 }5s^{1}

Using noble gas structure;

[Kr] 4d^{10} 5s^{1}

(c)

(i) Na^{+} number of electrons also 10

1s^{2} 2s^{2} 2p^{6} resembles Ne

(ii) F – number of electrons also 10

1s^{2} 2s^{2} 2p^{6} resembling Ne

(d) Cu Atomic number 29

e.g. [Ar] 4s^{1} 3d^{10}

Cr Atomic number 24

e.g. [Ar] 4s^{1} 3d^{5} full filled and half filled orbitals are very stable electronic structures. The stability gained is the cause of 4s electrons to be unpaired and make 3d orbitals either full filled or half filled.

Note that: In writing the electronic configuration the atoms or orbitals of the same value are usually written together irrespective of their relation energy levels.

E.g. Cu 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 3d^{10} 4s^{1} not 1s^{2} 2s^{2 }2p^{6} 3s^{2} 2p^{6} 4s^{1} 3d^{10}. To save time and space noble gases are often utilized e.g. sodium [Ne] 3s^{1} and Iron [Ar] 3d^{6} 4s^{2}

Recommended;

FORM THREE PHYSICS STUDY NOTES TOPIC 3:LIGHT & TOPIC 4: OPTICAL INSTRUMENT