B.Sc. Semester I Chemistry

B.Sc. Semester I (Generic) Chemistry (Section-A Inorganic Chemistry-1) DCH-101 Unit-1 (Atomic Structure) BOHR’S ATOMIC MODEL

Bohr proposed a quantum mechanical model of the atom, to overcome the objections of Rutherford’s model and to explain the hydrogen spectrum. This model was based on the quantum theory of radiation and the classical laws of physics. The important postulates on which Bohr’s model is based are the following:

(a) The atom has a nucleus where all the protons and neutrons are present. The size of the nucleus is very small. It is present at the center of the atom.
(b) Negatively charged electrons are revolving around the nucleus in a similar way as the Planets are revolving around the Sun. The path of the electron is circular. The force of attraction between the nucleus and the electron is equal to centrifugal force of the moving electron.
Force of attraction toward nucleus = Centrifugal force
(c) Out of infinite number of possible circular orbits around the nucleus, the electron can revolve only on those orbits whose angular momentum is an integral multiple of $\frac{h}{2\pi}$, that is, $mvr = \frac{nh}{2\pi}$ where $m$ = Mass of the electron, $v$ = Velocity of electron, $r$ = Radius of the orbit and $n = 1, 2, 3,\dots$ number of the orbit. The angular momentum can have values such as $\frac{h}{2\pi}$, $\frac{2h}{2\pi}$, $\frac{3h}{2\pi}$, etc., but it cannot have fractional value. Thus, the angular momentum is quantized. The specified or circular orbits (quantized) are called stationary orbits.

(d) When the electron remains in any one of the stationary orbits, it does not lose energy. Such a state is called ground or normal. In the ground state potential energy of electron will be minimum; hence it will be the most stable state.
(e) Each stationary orbit is associated with a definite amount of energy. The greater is the distance of the orbit from the nucleus, more shall be the energy associated with it. These orbits are also called energy levels and are numbered as 1, 2, 3, 4,... or K, L, M, N,... from nucleus outward, i.e. $E_1 < E_2 < E_3 < E_4 \dots (E_2 - E_1) > (E_3 - E_2) > (E_4 - E_3)$.
(f) The emission or absorption of energy in the form of radiation can occur only when an electron jumps from one stationary orbit to another. $\Delta E = E_{\text{high}} - E_{\text{low}} = h\nu$; energy is absorbed when the electron jumps from an inner to an outer orbit and it is emitted when the electron moves from an outer to an inner orbit. When the electron moves from an inner to an outer orbit by absorbing definite amount of energy, the new state of the electron is said to be excited state.

Limitations of Bohr’s Model (a) It does not explain the spectra of multi-electron atoms.
(b) By using a high resolving power spectroscope it is observed that a spectral line in the hydrogen spectrum is not a simple line but a collection of several lines which are very close to one another. This is known as fine spectrum. Bohr’s theory does not explain the fine spectra of even the hydrogen atom.
(c) Spectral lines split into a group of inner lines under the influence of magnetic field (Zeeman effect) and electric field (Stark effect); but, Bohr’s theory does not explain this.
(d) Bohr’s theory is not in agreement with Heisenberg’s uncertainty principle.

PARTICLE AND WAVE NATURE OF ELECTRON

In 1924, Louis de Broglie proposed that an electron, like light behaves both as a material particle and as wave. This proposal gave birth to a new theory as wave mechanical theory of matter. According to this theory, the electrons, protons and even atoms, when in motion, possess wave properties.

De Broglie derived an expression for calculating the wavelengths of the wave associated with the electron. Using Planck’s equation,
$E = h\nu = \frac{hc}{\lambda}$ ..................(i)

On the basis of Einstein’s mass-energy relationship the energy of a photon is
$E = mc^2$ ..................(ii)
where $c$ is the velocity of the electron.

Equating both the equations, we get $\frac{c}{\lambda} = mc^2$; $\lambda = \frac{h}{mc} = \frac{h}{p}$

Momentum of the moving electron is inversely proportional to its wavelengths.

Let kinetic energy of the particle of mass '$m$' is $E$. $E = \frac{1}{2}mv^2$; $2Em = m^2v^2$
$\sqrt{2Em} = mv = p \text{ (momentum)}$; $\lambda = \frac{h}{p} = \frac{h}{\sqrt{2Em}}$

Bohr’s Theory vs. de Broglie Equation Bohr’s theory postulates that angular momentum of an electron is an integral multiple of $\frac{h}{2\pi}$. This postulate can be derived with the help of de Broglie concept of wave nature of electron.

Consider an electron moving in a circular orbit around nucleus. The wave train would be associated with the circular orbit as shown in the figure. If the two ends of an electron wave meet to give a regular series of crests and troughs, the electron wave is said to be in phase, i.e. the circumference of Bohr orbit is equal to whole number multiple of the wavelength of the electron wave.

So, $2\pi r = n\lambda$ Or $\lambda = \frac{2\pi r}{n}$ ..................(i)
From de Broglie equation, $\lambda = \frac{h}{mv}$ ..................(ii)
Thus, $\frac{h}{mv} = \frac{2\pi r}{n}$ or $mvr = \frac{nh}{2\pi}$ ($v$ = velocity of electron and $r$ = radii of the orbit)
i.e. Angular momentum = $\frac{nh}{2\pi}$ ..................(iii)

This proves that the concepts of de Broglie and Bohr are in perfect agreement with each other.

HEISENBERG’S UNCERTAINTY PRINCIPLE

We see around us all moving particles, e.g. a car, a ball thrown in the air etc. move along definite paths. Hence their position and velocity can be measured accurately at any instant of time. Likewise is it possible to measure the position and velocity for the subatomic particle also?

Heisenberg, in 1927 gave a principle about the uncertainties in simultaneous measurement of position and momentum (mass $\times$ velocity) of small particles. This principle is due to the consequence of dual nature of matter.

This Principle States: ‘It is impossible to measure simultaneously the position and momentum of a small microscopic moving particle with absolute accuracy or certainty’, i.e. if an attempt is made to measure any one of these two quantities with higher accuracy, the other becomes less accurate. The product of the uncertainty in position ($\Delta x$) and the uncertainty in the momentum ($\Delta p = m \Delta v$ where $m$ is the mass of the particle and $\Delta v$ is the uncertainty in velocity) is equal to or greater than $\frac{h}{4\pi}$ where $h$ is the Planck’s constant. Thus, the mathematical expression for the Heisenberg’s uncertainty principle is readily written as $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$

Explanation of Heisenberg’s uncertainty principle: Let us attempt to measure both the position and momentum of an electron; to pinpoint the position of the electron we have to use light so that the photon of light strikes the electron and the reflected photon is seen in the microscope. As a result of the hitting, both the position and the velocity of the electron are disturbed. The accuracy with which the position of the particle can be measured depends upon the wavelength of the light used.

The uncertainty in position is $\pm \lambda$. The shorter the wavelength, the greater is the accuracy. But shorter wavelength means higher frequency and hence higher energy. This high energy photon on striking the electron changes its speed as well as direction. But this is not true for a moving macroscopic particle. Hence Heisenberg’s uncertainty principle does not apply to macroscopic particles.

HYDROGEN SPECTRUM

Hydrogen spectrum is an example of atomic or line emission spectrum. Whenever an electric discharge is passed to hydrogen gas at low pressure, a blue light is emitted. The light shows discontinuous line spectrum of several isolated sharp lines through prism. All these lines of H-spectrum have Lyman, Balmer, Paschen, Brackett, Pfund and Humphrey series. Wavelength of various H-lines Rydberg introduced the following expression,

$\bar{\nu} = \frac{1}{\lambda} = \frac{\nu}{c} = R \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right]$

$R$ is a Rydberg’s constant its value is 109,67800 m⁻¹.

$n$ is an integral; $n = 3, 4, 5$ and $6$ for $H_\alpha$, $H_\beta$, $H_\gamma$ and $H_\delta$ lines respectively in the hydrogen spectrum.

Total Number of spectral Lines = $\frac{(n_2-n_1)(n_2-n_1+1)}{2}$

Series	Region of spectrum	Equation for wavenumber($\nu$)
Lyman series	Ultraviolet	$\nu = R_H \left(\frac{1}{1^2} - \frac{1}{n^2}\right)$, $n = 2,3,4,5\dots$
Balmer series	Visible/ultraviolet	$\nu = R_H \left(\frac{1}{2^2} - \frac{1}{n^2}\right)$, $n = 3,4,5,6\dots$
Paschen series	Infrared	$\nu = R_H \left(\frac{1}{3^2} - \frac{1}{n^2}\right)$, $n = 4,5,6,7\dots$
Brackett series	Infrared	$\nu = R_H \left(\frac{1}{4^2} - \frac{1}{n^2}\right)$, $n = 5,6,7,8\dots$
Pfund series	Infrared	$\nu = R_H \left(\frac{1}{5^2} - \frac{1}{n^2}\right)$, $n = 6,7,8\dots$

When a hydrogen atom absorbs a photon, it causes the electron to experience a transition to a higher energy level, for example, $n=1$, $n=2$. When a photon is emitted through a hydrogen atom, the electron undergoes a transition from a higher energy level to a lower, for example, $n=3$, $n=2$. During this transition from a higher level to a lower level, there is the transmission of light occurs. The quantized energy levels of the atoms cause the spectrum to comprise wavelengths that reflect the differences in these energy levels. For example, the line at 656 nm corresponds to the transition $n=3$ to $n=2$.

The Balmer series is basically the part of the hydrogen emission spectrum responsible for the excitation of an electron from the second shell to any other shell. Similarly, other transitions also have their own series names. Some of them are listed below:
• The transition from the first shell to any other shell – Lyman series
• The transition from the second shell to any other shell – Balmer series
• The Transition from the third shell to any other shell – Paschen series
• The transition from the fourth shell to any other shell – Brackett series
• The transition from the fifth shell to any other shell – Pfund series

Need of a new approach to atomic structure The need for a new approach to atomic structure arises from the limitations of the current understanding of atomic structure. The current model, known as the Bohr model, was developed over a century ago and is based on classical mechanics. While it explains many aspects of atomic behavior, it is not able to fully account for the behavior of electrons in atoms, particularly in the case of highly excited states or in the presence of strong electromagnetic fields. A new approach, such as quantum mechanics, is needed to fully understand atomic behavior and make predictions about the behavior of atoms in various conditions.

What is Quantum Mechanics? Quantum mechanics is the branch of physics that deals with the behavior of matter and light on a subatomic and atomic level. It attempts to explain the properties of atoms and molecules and their fundamental particles like protons, neutrons, electrons, gluons, and quarks. The properties of particles include their interactions with each other and with electromagnetic radiation. So below mentioned are those two pointers one should know necessarily before tackling quantum mechanics.

Quantum Mechanics Formulas | Quantity | Formula |
|---------------------------------|------------------------------------------------------|
| Wavefunction probability density| $\rho = |\Psi|^2 = \Psi^* \Psi$ |
| Photoelectric equation | $K_{\text{max}} = hf + \Phi$ |
| Hydrogen atom spectrum | $\frac{1}{\lambda_i} = R\left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right)$ $n_i > n_f$ |
| Dipole moment potential | $U = -\mu B = -\mu_z B$ |

WAVE MECHANICAL MODEL OF ATOMS Erwin Schrodinger in 1926 put forward this model by taking into account the de Broglie concept of dual nature of matter and Heisenberg’s uncertainty principle. In this model, the discrete energy levels or orbits proposed by Bohr’s model are replaced by mathematical function $\Psi$ (psi) which is related with probability of finding electrons around the nucleus. The wave equation for an electron wave propagating in 3-D space is:

$\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} + \frac{8\pi^2 m}{h^2}(E - V)\psi = 0$

where $\psi$ is the amplitude of the electron wave at point with coordinates $x$, $y$, $z$, $m$ = mass of the electron, $h$ = Planck’s constant, $E$ = total energy and $V$ = potential energy of the electron; $\Psi$ is also called wave function and $\Psi^2$ gives the probability of finding the electron at ($x$, $y$, $z$). The acceptable solutions of the above equation for the energy $E$ are called Eigen values and the corresponding wave functions are called Eigen functions.

Every function is not an Eigen function. An acceptable solution for Schrödinger wave equation must satisfy the following conditions:

1. The function should be finite.
2. It should always bear a single value at a particular point in space.
3. It should be a continuous function.

Schrödinger wave equation can be written as
$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)\psi + \frac{8\pi^2 m}{\hbar^2}(E - V)\psi = 0$;
$\nabla^2 \psi + \frac{8\pi^2 m}{\hbar^2}(E - V)\psi = 0$

Where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ is called Laplacian operator. This equation can be rewritten as $\nabla^2 \psi = -\frac{8\pi^2 m}{\hbar^2}(E - V)\psi$;
Or $\left(-\frac{\hbar^2}{8\pi^2 m}\nabla^2 + V\right)\psi = E\psi$; Or $\hat{H}\psi = E\psi$, where $\hat{H} = -\frac{\hbar^2}{8\pi^2 m}\nabla^2 + V$ is called Hamiltonian operator.

In this operator, the first term represents kinetic energy operator ($T$) and the second term represents potential energy operator ($V$).

Significance of $\psi$: It represents the amplitude of an electron wave. It can be positive or negative. It has no physical value.
Significance of $\psi^2$: It is a probability function. It determines the probability of finding an electron within a smaller region of space around nucleus. The space in which there is maximum probability of finding an electron is termed as orbital.

Schrodinger wave equation for hydrogen atom In case of hydrogen atom, since a single electron of charge $e$ is revolving round the nucleus of charge $+e$, potential energy $PE$ is equal to $-\frac{e^2}{r}$. thus putting the $PE = -\frac{e^2}{r}$ in Schrodinger wave equation, we get
$\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} + \frac{8\pi^2 m}{\hbar^2}\left(E - \frac{e^2}{r}\right)\psi = 0$

This equation is the Schrödinger wave equation of hydrogen atom.

Radial and Angular functions Since Schrödinger wave equation is given as
$\nabla^2 \psi = -\frac{8\pi^2 m}{h^2}(E - V)\psi$ or $\nabla^2 \psi - \frac{8\pi^2 m}{h^2}(E - V)\psi = 0$

Where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$

Changing the polar coordinates, $\nabla^2 \psi$ becomes
$\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \psi}{\partial \phi^2} + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)$

The solutions of Schrödinger wave equation is obtained by separating the variables so that the wave function is represented by the product
$\Psi_{\text{cartesian}}(x,y,z) = \Psi_{\text{radial}}(r) \Psi_{\text{angular}}(\theta,\phi)$
$\Psi(r, \theta, \phi) = R(r) A(\theta, \phi)$ or $\Psi = R_{nl}(r) \cdot A_{ml}(\theta, \phi)$

where $R(r)$ and $A(\theta, \phi)$ are radial and angular wave functions respectively.

This splits the wave function into two parts which can be solved separately:

1. $R(r)$ the radial function, which depends on the quantum numbers $n$ and $l$.
2. $A_{ml}$ the total angular wave function which depends on the quantum number $m$ and $l$.

The radial function $R$ has no physical meaning, but $R^2$ gives the probability of finding the electron in a small volume $dv$ near the point at which $R$ is measured. For a given value of $r$ the number of small volumes is $r^2$ so, the probability of the electron being at a distance $r$ from the nucleus is $r^2 R^2$. This is called the radial distribution function. Graphs of the radial distribution function for hydrogen plotted against $r$ are shown below.

Figure. (a) The electronic radial wave function $R(r)$ for a hydrogen atom. (b) The probability density for finding the electrons, $r^2 R^2(r)$ for a hydrogen atom.

These diagram shows that the probability is zero at the nucleus (as $r=0$) and by examining the plots for $1s$, $2s$ and $3s$ that the most probable distance increases markedly as the principal quantum number increases. Furthermore by comparing the plots for $2s$ and $2p$ or $3s$, $3p$ and $3d$ it can be seen that the most probable radius decreases slightly as the subsidiary quantum number increases. All $s$ orbitals except the first one ($1s$) have a shell-like structure, rather like an onion or a hailstone, consisting of concentric layers of electron density. Similarly all but the first $p$ orbitals ($2p$) and the first $d orbitals ($3d$) have a shell structure.

The angular function $A$ depends only on the direction, and is independent of the distance from the nucleus ($r$). Thus $A^2$ is the probability of finding an electron at a given direction $\theta, \phi$ at any distance from the nucleus to infinity. The angular functions $A$ are plotted as polar diagrams. It must be emphasized that these polar diagrams do not represent the total wave function $\psi$, but only the angular part of the wave function. The total wave function is made up from contributions from both the radial and the angular functions.
$\Psi = R(r) \cdot A$

Thus the probability of finding an electron simultaneously at a distance $r$ and in a given direction $\theta, \phi$ is $\psi^2(r,\theta,\phi)$.

Drawings of the angular part of the wave function are commonly used to illustrate the overlap of orbitals giving bonding between atoms. The symmetry of the angular wave function is shown in $+$ and $-$ signs and the like signs must overlap for bonding.

1. These drawing show the symmetry for the $1s, 2p, 3d$ orbitals. However in the others $2s, 3s, 4s, \dots 3p, 4p, 5p, \dots 4d, 5d, \dots$ changes inside the boundary surface of the orbitals. This is readily seen as nodes in the graphs of the radial functions.
2. For $p$ orbitals the electron density is zero at the nucleus.
3. The probability of finding an electron at a direction $\theta, \phi$ is the wave function squared $A^2$ or more precisely $\psi^2_\theta \psi^2_\phi$. Squaring does not change the shape of an $s$ orbital, but it elongates the lobe of $p$ orbitals.
4. A full representation of the probability of finding an electron requires the total wave function squared and includes both the radial and angular probabilities squared. It needs three dimensional models to display the probability and show the shapes of the orbitals.

Shapes of Orbitals The electron cloud represents the shape of the orbital. It is not uniform but it is dense where the probability for finding the electron is maximum.
(a) $s$-orbitals do not vary with angles, i.e. they do not have directional dependence. Thus, all $s$-orbitals are called spherically symmetrical. Their size increases with increases in the value of $n$. $1s$-orbital has no nodal plane (the plane at which zero electron density is noticed). $2s$-orbital has one nodal plane; $3s$-orbital has two nodal planes. Thus it is evident that the number of nodal planes increases with increasing value of principal quantum number.
(b) All orbitals with $l \neq 0$ have angular dependence. Therefore, $p$ and $d$ and other higher angular momentum orbitals are not spherically symmetrical. $p$-orbitals consist of two lobes to form dumbbell shaped structure. The three $p$-orbitals along $x, y, z$-axes named as $p_x$, $p_y$, $p_z$ orbitals are perpendicular to each other. All the three $p$-orbitals of a sub-shell have the same size and shape but differ from each other in orientation. The subscripts $x, y$ and $z$ indicate the axis along which the orbitals are oriented and possess maximum electron density. Also, the orbitals of a sub-shell having same energy are referred as degenerate orbitals.
(c) The probability density is the square of the wave function and is positive everywhere. The lobes on the positive or negative side of both the axes are assigned ($+$) sign and those on positives side of one axis and negative side of the other or vice versa are assigned ($-$)sign.

The characteristics of the $d$-orbitals may be summarised as follows:
(i) $d_{xy}$, $d_{yz}$ and $d_{zx}$ (or $d_{xz}$) as well as $d_{x^2-y^2}$ orbitals are double dumb-bell shaped and contain four lobes. The lobes of the first three orbitals are concentrated between $xy$, $yz$ and $zx$ planes, respectively and lie between their coordinate axes. The lobes of $d_{x^2-y^2}$ orbital are concentrated along $x$ and $y$ axes. $d_{z^2}$ orbital has a dumb-bell shape with two lobes along $z$-axis with ($+$) sign and a concentric collar or ring around the nucleus in $xy$ plane with ($-$)sign.
(ii) The $d$-orbitals belonging to same energy shell are degenerate, i.e. have the same energy in a free atom.
(iii) The $d$-orbitals belonging to all main energy shells have similar shape but their size goes on increasing as the value of $n$ and number of nodal points increase. For example, the size of $5d$ orbital (number of nodal points = $5 - 2 - 1 = 2$) is larger than that of $4d$-orbital (number of nodal points = $4 - 2 - 1 = 1$).

ORBITAL NODES Orbital nodes refer to places where the quantum mechanical wave function $\Psi$ and its square $\Psi^2$ change phase. Since the phase is either moving from positive to negative or vice versa, both $\Psi$ and $\Psi^2$ are zero at nodes. Where $\Psi^2$ is zero, the electron density is zero. Hence, at a node, the electron density is zero.

The regions or spaces around the nucleus where the probability of finding an electron is zero are called nodes.

The atomic orbitals or orbital wave functions can be represented by the product of two wave functions, radial and angular wave function. A node is a point where a wave function passes through zero. The nodes are classified into two types (i) Radial nodes and (ii) Angular nodes.

The spherical surfaces around the nucleus where the probability of finding an electron is zero are called radial nodes. The planes or planar areas around the nucleus where the probability of finding an electron is zero are called angular nodes.

QUANTUM NUMBERS An atom contains large number of shells and sub-shells. These are distinguished from one another on the basis of their size, shape and orientation (direction) in space. The parameters are given in terms of different numbers called quantum numbers.

Quantum numbers may be defined as a set of four numbers with the help of which we can get complete information about all the electrons in an atom. It tells us the address of the electron i.e. location, energy, the type of orbital occupied and orientation of that orbital.

Principal Quantum Number (a) This is denoted by $n$, an integer.
(b) The values of $n$ are from $1$ to $n$. $n = 1$ K shell; $n = 2$ L shell; $n = 3$ M shell; $n = 4$ N shell
(c) ‘$n$’ represents the major energy shell to which an electron belongs.
(d) The values of ‘$n$’ signify the size and energy level of major energy shells.
(e) As the value of ‘$n$’ increases, the energy of the electron increases and thus, the electron is less tightly held with nucleus.
(f) Angular momentum can be calculated using principal quantum number: $mvr = \frac{nh}{2\pi}$

Azimuthal Quantum Number This is denoted by $l$.
(a) The values of $l$ are from $0$ to $(n - 1)$
(b) $l = 0$, $s$-sub-shell, spherical (The representation is independent of the value of $n$); $l = 1$, $p$-sub-shell, dumbbell; $l = 2$, $d$-sub-shell, double dumbbell or like leaf
(c) The letters $s$, $p$, $d$, $f$ designate old spectral terms. Sharp ($s$), principal ($p$), diffuse ($d$), fundamental ($f$)
(d) For a given value of $n$, total values of ‘$l$’ are $n$.
(e) The values of $l$ signify the shape and energy level of sub-shells in a major energy shell.
(f) The angular momentum of an electron in an orbital is given by $\frac{nh}{2\pi}$.
(g) The energy level for sub-shells of a shell shows the order: $s < p < d < f$

Magnetic Quantum Number (a) Denoted by $m_l$, an integer.
(b) Zeeman effect: Zeeman studied the fine spectrum of H using a spectroscope of high resolving power as well as putting the source under the influence of magnetic field. He noticed that the spectral line splits up to more than one component.
(c) Each frequency of radiation emitted by the atom in the presence of magnetic field splits up into components if the angular momentum of the electron along the magnetic field are restricted to the value, $m_l = \frac{h}{2\pi}$.
(d) The values of $m_l$ lie from $\pm l$ through zero.
(e) The positive values of magnetic quantum number $m_l$ represent the angular momentum component of the orbital in the direction of the applied magnetic field whereas the negative values of $m_l$ account for the angular momentum component of orbital in the opposite direction of applied magnetic field.
(f) Total values of $m_l$ for a given value of $n = n^2$.
(g) Total values of $m_l$ for a given value of $l = (2l + 1)$
(h) The values of $m_l$ signify the possible numbers of orientations of a sub-shell.
(i) In the absence of magnetic field, the three $p$-orbitals are equivalent in energy and are said to be threefold degenerate, i.e. sub sub-shell (orbitals) having same energy level are known as degenerate orbitals.

Spin Quantum Number (a) Wave mechanical treatment required no more than three quantum number $n$, $l$ and $m$. The existence of multiple, i.e. doublet structure led to the introduction of a spin quantum number $m_s$.
(b) The values of $m_s$ are $+1/2$ and $-1/2$. This is due to the fact of the doublet structures of spectral lines which can be explained by proposing only two directions of spin of electron along its own axis.
(c) The values of $m_s$ signify the direction of rotation or spin of an electron in its axis during its motion.
(d) Spin angular momentum is given by $\frac{h}{2\pi} \sqrt{m_s(m_s + 1)} = \frac{h}{2\pi} \sqrt{\frac{1}{2} \left(\frac{1}{2} + 1\right)} = \frac{\sqrt{3}}{4} \frac{h}{\pi} = \frac{\sqrt{3}}{2} h$.
(e) The spin may be clockwise or anticlockwise ($+1/2$ or $-1/2$) or anticlockwise ($-1/2$ or $+1/2$)
(f) Spin multiplicity of an atom = $\sqrt{s(s + 1)}$.
(g) These values i.e. ($+1/2$) and ($-1/2$) are also represented as $\uparrow$ (upward arrow) and $\downarrow$ (downward arrow). Being a charged particle, a spinning electron generates a so called spin magnetic moment which can be oriented either up or downward. The value of $s$ for an electron in an orbital does not affect the energy, shape, size or orientation of an orbital but shows only how the electrons are arranged in that orbital.

Table: Quantum number and orbital chart

Shell	Principal quantum number ($n$)	maximum number of electron in a shell (orbital) $2n^2$	Azimuthal quantum number ($l$) = $0,1,\dots, (n-1)$	Maximum no. of electron in a orbital $2(2l+1)$	Magnetic quantum number ($m$) different possible orientation of orbital	Designation of orbitals in a given shell
K	$1$	$2(1)^2 = 2$	$0$	$2[2(0)+1] = 2$	$0$	$1s$
L	$2$	$2(2)^2 = 8$	$0$	$2$	$0$	$2s$
			$1$	$2[2(1)+1] = 6$	$-1, 0, +1$	$2p_y, 2p_z, 2p_x$
M	$3$	$2(3)^2 = 18$	$0$	$2$	$0$	$3s$
			$1$	$6$	$-1,0,+1$	$3p_x,3p_y,3p_z$
			$2$	$2[2(2)+1] = 10$	$-2,-1,0,+1,+2$	$3d_{x^2-y^2}, 3d_{yz}, 3d_{z^2}, 3d_{xz}, 3d_{xy}$
N	$4$	$2(4)^2 = 32$	$0$	$2$	$0$	$4s$
			$1$	$6$	$-1,0,+1$	$4p_x,4p_y,4p_z$
			$2$	$10$	$-2,-1,0,+1,+2$	$4d_{x^2-y^2},4d_{yz},4d_{z^2},4d_{xz},4d_{xy}$
			$3$	$2[2(3)+1] = 14$	$-3,-2,-1,0,1,+2,+3$	$f$ orbitals (various designations)

The labels on the orbitals, such as $p_x$, $d_{z^2}$, $f_{xyz}$, etc. are not associated with specific '$m$' values.

RULES FOR FILLING OF ELECTRONS IN VARIOUS ORBITALS

PAULI’S EXCLUSION PRINCIPLE This principle was proposed by Pauli in 1924 and as an important rule, governs the quantum numbers allowed for an electron in an atom and determines the electronic configuration of poly electron atoms. In a general form, this principal states that “In an atom, any two electrons cannot have the same values of four quantum numbers”. Alternatively, this can be put in the form “any two electrons in an atom cannot exist in the same quantum state”. Consequently, it can be said that any two electrons in an atom can have same values of any three quantum numbers but the fourth (may be $n$ or $l$ or $m$ or $s$) will definitely have different values for them.

This principle has been used to calculate the maximum number of electrons that can be accommodated in an orbital, a subshell and in a main shell. ... (full explanation as in document, including examples for K, L shells and table of quantum number combinations).

HUND’S RULE OF MAXIMUM MULTIPLICITY This rule states that “electron pairing in the orbitals of a subshell will not take place until each orbital is filled with single electron” (due to same energy of orbitals of a subshell). ... (full explanation with examples of $p^3$, $d^5$, $f^7$ vs $p^4$, $d^6$, $f^8$ and diagrams of parallel spins).

THE AUFBAU PRINCIPLE Aufbau is a German word which means building up or construction. ... The principle can be stated as “in the ground state of poly electronic atoms, the electrons are filled in various subshells in the increasing order of their energy”. ... (rules, $(n+1)$ rule, order $1s < 2s < 2p < 3s < 3p < 4s < 3d < \dots$ and exceptions for La, Ac).

Relative Energies of Orbitals The energy which is essential to take an electron present in that orbital to infinity or the release of energy when an electron from an infinity it is added to that orbital... (factors affecting orbital energy, energy level diagram).

ELECTRONIC CONFIGURATION OF ELEMENTS ... (table up to Ca, then Sc to Zn with configurations, including Cr and Cu anomalies).

ANOMALOUS ELECTRON CONFIGURATION OF ELEMENTS It is observed that in case of chromium (Cr, $Z = 24$) and copper (Cu, $Z = 29$)... (explanation of half-filled and fully-filled stability, table of predicted vs actual for Cu, Ag, Au, Pd, Cr, Mo).

HALF FILLED AND FULLY FILLED ORBITALS Half-filled orbitals... greater stability... explained by symmetrical distribution and exchange energy.

Exchange energy (a) ... (full details with counting of exchanges in $d^4$ and $d^5$, symmetry examples for $p$ and $d$ orbitals, basis of Hund’s rule).

• Relatively small shielding
• Larger exchange energies
• Low coulombic repulsion energy

The exchange energy is the basis of Hund’s rule...

Examples:
• Chromium ... $1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 4s^1$
• Copper ... $1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^1$

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