Foundations of Inorganic & Polymer Chemistry PDF

FOUNDATIONS OF INORGANIC & POLYMER CHEMISTRY MODULE I ATOMIC STRUCTURE & PERIODICITY UNIT 1 Atomic structure – Introduction Dalton's atomic theory Dalton’s atomic theory was a scientific theory on the nature of matter put forward by the English physicist and chemist...

FOUNDATIONS OF INORGANIC & POLYMER CHEMISTRY MODULE I ATOMIC STRUCTURE & PERIODICITY UNIT 1 Atomic structure – Introduction Dalton's atomic theory Dalton’s atomic theory was a scientific theory on the nature of matter put forward by the English physicist and chemist John Dalton in the year 1808. It stated that all matter was made up of small, indivisible particles known as ‘atoms’. All substances, according to Dalton’s atomic theory, are made up of atoms, which are indivisible and indestructible building units. While an element’s atoms were all the same size and mass, various elements possessed atoms of varying sizes and masses. Postulates of Dalton’s Atomic Theory All matter is made up of tiny, indivisible particles called atoms. All atoms of a specific element are identical in mass, size, and other properties. However, atoms of different element exhibit different properties and vary in mass and size. Atoms can neither be created nor destroyed. Furthermore, atoms cannot be divided into smaller particles. Atoms of different elements can combine with each other in fixed whole-number ratios in order to form compounds. Atoms can be rearranged, combined, or separated in chemical reactions. Limitations of Dalton’s Atomic Theory It does not account for subatomic particles: Dalton’s atomic theory stated that atoms were indivisible. However, the discovery of subatomic particles (such as protons, electrons, and neutrons) disproved this postulate. It does not account for isotopes: As per Dalton’s atomic theory, all atoms of an element have identical masses and densities. However, different isotopes of elements have different atomic masses (Example: hydrogen, deuterium, and tritium). It does not account for isobars: This theory states that the masses of the atoms of two different elements must differ. However, it is possible for two different elements to share the same mass number. Such atoms are called isobars (Example: 40Ar and 40Ca). 1|Page  Elements need not combine in simple, whole-number ratios to form compounds: Certain complex organic compounds do not feature simple ratios of constituent atoms. Example: sugar/sucrose (C11H22O11). The theory does not account for allotropes: The differences in the properties of diamond and graphite, both of which contain only carbon, cannot be explained by Dalton’s atomic theory. An atom is a complex arrangement of negatively charged electrons arranged in defined shells about a positively charged nucleus. This nucleus contains most of the atom's mass and is composed of protons and neutrons (except for common hydrogen which has only one proton). All atoms are roughly the same size. A convenient unit of length for measuring atomic sizes is the angstrom (Å), which is defined as 1 × 10-10 meters. The diameter of an atom is approximately 2-3 Å. In 1897, J. J. Thomson discovered the existence of the electron, marking the beginning of modern atomic physics. The negatively charged electrons follow a random pattern within defined energy shells around the nucleus. Most properties of atoms are based on the number and arrangement of their electrons. The mass of an electron is 9.1 × 10-31 kilograms. One of the two types of particles found in the nucleus is the proton. The existence of a positively charged particle, a proton, in the nucleus was proved by Sir Ernest Rutherford in 1919. The proton's charge is equal but opposite to the negative charge of the electron. The number of protons in the nucleus of an atom determines what kind of chemical element it is. A proton has a mass of 1.67 × 10-27 kilograms. The neutron is the other type of particle found in the nucleus. It was discovered by a British physicist, Sir James Chadwick. The neutron carries no electrical charge and has the same mass as the proton. With a lack of electrical charge, the neutron is not repelled by the cloud of electrons or by the nucleus, making it a useful tool for probing the structure of the atom. Atomic spectrum of Hydrogen Different series When an electron is excited to an outer orbit, it uses different routes to get to its original or lower orbits. These transitions from higher orbit to lower orbit lead to the emission of many lines in the hydrogen spectrum. These lines are mainly five groups in the hydrogen atomic spectrum. These are: 2|Page  Lyman series: When an electron transition takes place from a higher-level orbit, i.e. n2=2,3,4,5,…. to the first orbit (n1=1), the produced lines on the spectrum correspond to the Lyman series. The wavelength of the produced photons lies in the ultraviolet region. Theodore Lyman discovered this series. Balmer series: This series was discovered by Johann Balmer. When an electron transition takes place from a higher-level orbit, i.e. n2=3,4,5,6 …, to the second orbit (n2=2), the produced lines on the spectrum correspond to the Balmer series. These Balmer series lines lie in the visible region and have a wavelength range of 700 nm-400 nm. The Paschen series consists of lines corresponding to the transition of an electron from n2=4,5,6,7… and so on to n1=3. This series was named after Friedrich Paschen and the wavelength lies in the infrared band. The wavelength range of this series is 820 nm-1875 nm. Brackett series: When an electron transition takes place from a higher-level orbit, i.e. n2=5,6,7,8…. to the first orbit (n1=4), the produced lines on the spectrum correspond to the Brackett series. The wavelength of the producer, Frederick Summer Brackett. Pfund series: This series was discovered by August Herman Pfund. When an electron transition takes place from a higher-level orbit, i.e. n2=6,7,8,9 … to the second orbit (n2=5), the produced lines on the spectrum correspond to the Pfund series. These produced lines lie in the visible region with a wavelength range of 2279 nm-7460 nm. Rydberg equation The Rydberg formula is the mathematical formula to determine the wavelength of light emitted by an electron moving between the energy levels of an atom. When an electron transfers from one atomic orbital to another, it’s energy changes. When an electron shifts from an orbital with high 3|Page energy to a lower energy state, a photon of light is generated. A photon of light gets absorbed by the atom when the electron moves from low energy to a higher energy state. The Rydberg Formula applicable to the spectra of the different elements and is it is expressed as where, n1 and n2 are integers and n2 is always greater than n1. R is constant, called Rydberg constant and formula is usually written as The modern value of Rydberg constant is known as 109677.57 cm-1 and it is the most accurate physical constant. Bohr theory According to the Bohr Atomic model, a small positively charged nucleus is surrounded by revolving negatively charged electrons in fixed orbits. He concluded that electron will have more energy if it is located away from the nucleus whereas electrons will have less energy if it located near the nucleus. Postulates Electrons revolve around the nucleus in a fixed circular path termed “orbits” or “shells” or “energy level.” The orbits are termed as “stationary orbit.” Every circular orbit will have a certain amount of fixed energy and these circular orbits were termed orbital shells. The electrons will not radiate energy as long as they continue to revolve around the nucleus in the fixed orbital shells. The different energy levels are denoted by integers such as n=1 or n=2 or n=3 and so on. These are called quantum numbers. The range of quantum numbers may vary and begin from the lowest energy level (nucleus side n=1) to the highest energy level. The different energy levels or orbits are represented in two ways such as 1, 2, 3, 4… or K, L, M, N….. shells. The lowest energy level of the electron is called the ground state. 4|Page  The change in energy occurs when the electrons jump from one energy level to other. In an atom, the electrons move from lower to higher energy level by acquiring the required energy. However, when an electron loses energy it moves from higher to lower energy level. Statement of Bohr energy equation 1. An atom has a positively charged nucleus which is responsible for almost the entire mass of the atom. 2. Electrons revolve around this nucleus in circular orbits of fixed radius. 3. While moving in stable orbits, two forces act on the electron. One is directed towards the nucleus, which is called Coulomb's electrostatic attraction force. This is the centripetal force. Whereas the second force is exerted outside the nucleus, which is called centrifugal force. These two forces are equal and opposite to each other, so that the electrons remain in their orbit. 4. There are many circular orbits around the nucleus, but electrons move in only those orbits for which the value of angular momentum is an absolute multiple of h/2π. If an electron of mass m moves in an orbit of radius r with a velocity v, the angular momentum of the electron is given by: L = mvr = nh/2π Where n=1,2,3,4,… 5. When an electron jumps from one stable orbit or energy level to another stable energy orbital, the difference in energy (ΔE) of both the energy levels is absorbed or emitted as discrete radiation. The value of the frequency or wavelength of this radiation is given in the following equation known from: E2 − E1 = ΔE = hv = hc/λ Where E2 = energy of higher energy level, E1 = energy of lower energy level. 6. The greater the distance of the energy level from the nucleus, the higher its energy; the energy of the orbit closest to the nucleus is the lowest. It is denoted by n=1 or by the sign K. Similarly, n=2,3,4 correspond to the signs L, M, N, etc. 7. When the electron moves in these orbits, there is no change in their energy. After emitting energy, the electron falls into the lower orbit. The absorption or emission of energy occurs in absolute quantum (hv) or quantum coefficients (hv, 2hv, 3hv, etc.) of energy. 8. As a result of energy emission, lines of fixed frequency are produced in the obtained emission spectrum; thus, this model explains the linear spectrum of the atom. 5|Page 6|Page 7|Page Limitations of Bohr model The Bohr Model was an important step in the development of atomic theory. However, it has several limitations. It is in violation of the Heisenberg Uncertainty Principle. The Bohr Model considers electrons to have both a known radius and orbit, which is impossible according to Heisenberg. The Bohr Model is very limited in terms of size. Poor spectral predictions are obtained when larger atoms are in question. It cannot predict the relative intensities of spectral lines. It does not explain the Zeeman Effect, when the spectral line is split into several components in the presence of a magnetic field. The Bohr Model does not account for the fact that accelerating electrons do not emit electromagnetic radiation. 8|Page  UNIT 2 Dual nature of matter and radiation This is based on the wave mechanical concept of an electron in an atom. Albert Einstein proposed the dual character of electromagnetic radiation in 1905, viz. wave character based on Maxwell’s concept, evidenced by diffraction, interference, and polarization phenomena, and particle character based on Planck’s quantum theory, witnessed by the quantization of energy and hence the photoelectric effect, i.e., the ejection of photoelectrons from a metal surface upon striking electromagnetic radiation. On the basis of the above analogy, French physicist Louis de Broglie (1924) postulated that not only light but all material objects (both micro and macroscopic) in motion, such as electrons, protons, atoms, molecules, etc., possess both wave and particle properties and thus have a dual character, i.e., wave character and particle (corpuscular) character. He called the waves associated with material particles matter waves, now named de Broglie’s waves. These waves differ from electromagnetic or light waves in the sense that they are unable to travel through empty space, and their speed is different from that of light waves. Photoelectric effect The photoelectric effect is the process of emitting electrons from a metal surface when the metal surface is exposed to electromagnetic radiation of sufficiently high frequency. For example, ultraviolet light is required in the case of ejection of electrons from an alkali metal. Apparatus Description An evacuated tube has two electrodes connected to an external circuit. The metal plate whose surface is to be irradiated acts as the anode. 9|Page  Some photoelectrons that emerge from the radiated surface have sufficient energy to reach the cathode despite its negative polarity, constituting the current. As the retarding potential is increased, fewer electrons are able to reach the cathode, causing the current to drop. When V exceeds a certain value V0, no further electrons are able to strike the cathode, and the current drops to zero. Laws of Photoelectric Emission There is no time lag between the irradiation of the surface and the ejection of the electrons. At a particular fixed frequency of incident radiation, the rate of the emission of photoelectrons (i.e., the photocurrent) increases with an increase in the intensity of the incident light. The photoelectric effect does not occur at frequencies less than the threshold frequency. At frequencies above the threshold frequency, the kinetic energy of the ejected electrons depends only on the frequency of the exposed radiation and not on its intensity. Explanation of Photoelectric Effect The photoelectric effect cannot be explained on the basis of electromagnetic theory. In 1905, Einstein proposed that the photoelectric effect could be understood through the idea introduced by the German theoretical physicist Max Planck in 1900. Planck was seeking to explain the characteristics of the radiation emitted by hot bodies. Planck assumed that while radiation is emitted continuously, it occurs in little bursts of energy called quanta but propagates continuously in space as electromagnetic waves. Einstein proposed that light is emitted as quanta and also propagates as individual quanta, which are sufficiently small to be absorbed by electrons. Planck found that the quantity associated with a particular frequency (ν) of light all had the same energy, which was proportional to ν; that is, E=hν. The photoelectric effect can be explained by the following equation: E(=hν) = hν0 + Tmax Here, E is the total energy of the photon incident on the metallic surface. ν is the frequency of the incident radiation. ν0 is the threshold frequency of the metal. Tmax is the maximum kinetic energy with which the electron moves after ejection from the surface. The photoelectric effect describes the emission of electrons from a metal surface when it is exposed to electromagnetic radiation of sufficient frequency. The total energy of the incoming photon, denoted as Ephoton , is crucial in this process. This energy must equal the kinetic energy of the ejected electron, referred to as KEelectron, plus the energy required to liberate the electron from the 10 | P a g e metal surface, which is known as the work function and is represented by the symbol Φ. This relationship can be summarized by the equation: Ephoton = KEelectron + Φ. The work function Φ varies depending on the type of metal and is measured in joules (J). Along with the threshold frequency ν0, it plays a crucial role in determining the conditions under which the photoelectric effect occurs. In order to relate the energy of the incoming photon to the frequency of the incident light, we can use Planck's equation, which states: Ephoton = hν, where h is Planck's constant (6.626×10−34 J s) and ν is the frequency of the incident radiation. By substituting this expression into our earlier equation, we can rewrite it as: hν = KEelectron + Φ Rearranging this equation allows us to solve for the kinetic energy of the emitted electron: KEelectron = hν − Φ This equation indicates that the kinetic energy of the photoelectron increases linearly with the frequency ν of the incident light, provided that the photon energy exceeds the work function Φ. This linear relationship can be illustrated graphically, demonstrating how kinetic energy varies with frequency. Furthermore, we can express the kinetic energy in terms of the velocity v of the ejected electron using the equation: KEelectron = hν – Φ = 1/2mev2 where me is the rest mass of the electron, approximately 9.1094×10−31 kg. This equation indicates that by knowing the frequency of the incident light and the work function of the metal, we can determine both the kinetic energy and the velocity of the emitted photoelectron. de Broglie equation de Broglie deduced a fundamental relation between the wavelength of a moving particle and its momentum by making use of Einstein’s mass-energy relationship and Planck’s quantum theory. The material particle, as a wave, satisfies Planck’s relation for a photon, i.e., E = hν where h is Planck’s constant and  is the frequency of the wave. The frequency for a light wave is ν = c/λ, and for a particle wave, ν = v/λ (where c is the speed of the light wave and v is the speed of the particle wave). At the same time, Einstein’s mass-energy relationship is applicable to it, i.e., 11 | P a g e  E = mc2 (for a photon) or or E = mν 2 (for a particle where ν ≠ c) where m is the mass and ν the speed/velocity of the particle. From equations above, we have: hv = mv 2 or = mv 2 Here, λ corresponds to the wave character of matter and p its particle character. This is known as de Broglie’s relation. From this relationship, it is concluded that 'the momentum of a moving particle is inversely proportional to the wavelength of the wave associated with it.' It is important to note from the above discussion that de Broglie’s relation is applicable to material particles of all sizes and dimensions, but the wave character is significant only for micro objects like electrons and is negligible for macro objects, hence it cannot be measured properly. This infers that de Broglie’s relation is more useful for smaller particles. de Broglie’s relation has been applied to a moving electron around a nucleus in a circular path in an atom to justify Bohr’s postulate, which states that electrons can move only in those orbits for which the angular momentum is equal to an integral multiple of h/2, i.e., mvr = nh/2 This moving electron is considered as a standing wave extended around the nucleus in a circular path and not as a mass particle. If the circumference of the orbit is an integral multiple of the wavelength λ, Heisenberg’s uncertainty principle According to classical mechanics, a moving electron behaves as a particle whose position and momentum could be determined with accuracy. But according to de Broglie, a moving electron has wave as well as particle character, and its precise position cannot be located because a wave is not situated at a particular point but rather extends in space. To describe the character of a subatomic particle that behaves like a wave, Werner Heisenberg, in 1927, formulated a principle known as Heisenberg’s Uncertainty Principle. According to the principle, 'it is impossible to determine simultaneously both the position and the momentum (or velocity) of a moving particle with certainty (or accuracy).' He also proposed a mathematical relationship for the uncertainty principle by relating the uncertainty in position with the uncertainty in momentum, which is given below: ℎ ∆𝑥. ∆𝑝 ≥ 4𝜋 12 | P a g e  ℎ ∆𝑥. 𝑚(∆𝜗) ≥ 4𝜋 (since p = mv and Δp = mΔv) Where x is the uncertainty or error in the position of the particle, p and v are the uncertainties in its momentum and velocity, and h is Planck’s constant. This equation states that the product of x and p can either be greater than or equal to h/2, a constant. If x is measured more precisely (i.e., x is small), then there is a large uncertainty or error in the measurement of momentum (p is large), and vice versa." 13 | P a g e  UNIT 3 Concept of orbital and Shapes of orbitals (s, p, d) An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term may also refer to the physical region where the electron can be calculated to be, as defined by the particular mathematical form of the orbital. Atomic orbitals are typically categorized by n, l and m quantum numbers, which correspond to the electron's energy, angular momentum, and an angular momentum vector component, respectively. Each orbital is defined by a different set of quantum numbers (n, l, and m), and contains a maximum of two electrons, each with their own spin quantum number. The simple names s orbital, p orbital, d orbital, and f orbital refer to orbitals with angular momentum quantum number l = 0, 1, 2 and 3 respectively. These names indicate the orbital shape and are used to describe the electron configurations. They are derived from the characteristics of their spectroscopic lines: sharp, principal, diffuse, and fundamental, the rest being named in alphabetical order (omitting j). Atomic orbitals are the basic building blocks of the atomic orbital model (alternatively known as the electron cloud or wave mechanics model), a modern framework for visualizing the microscopic behavior of electrons in matter. In this model, the electron cloud of a multi-electron atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating periodicity of the blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons which occupy a complete set of s, p, d, and f atomic orbitals, respectively. The shape of s orbitals Orbital is defined as the region in which a given electron can be found at a given time. s orbitals are the spherical region around the nucleus, in which the electron density is uniform like that of the surface of a sphere. The s orbital boundary surface diagram resembles a sphere with the nucleus at its center, which can be shown in two dimensions as a circle. s-orbitals are spherically symmetric, which means that the probability of finding an electron at a given distance is the same in all directions. The size of the s orbital is likewise shown to increase as the value of the primary quantum number (n) increases; hence, 4s > 3s > 2s > 1s. 14 | P a g e  The nodal point is a location where there is no chance of locating the electron. Nodes are classified into two types: radial nodes and angular nodes. The distance from the nucleus is calculated by the radial nodes, while the orientation is determined by the angular nodes. Totally filled subshells have spherical symmetry in two dimensions, hence the name s-orbital (s stands for spherical). The radial distribution of probability density is equal to that of a point charge. The shape of p orbitals p-orbitals are a type of molecular orbital used in molecular orbital theory. p orbitals, or pi-orbitals, are the three equivalent orbitals with two lobes pointing along the x axis, y axis and z axis respectively. These orbitals may be thought of as resulting from the overlap of three atomic orbitals when an electron is removed from a neutral atom. The p orbitals are formed like dumbbells. The p orbital node is located at the nucleus’s center. Because of the presence of three orbitals, the p orbital can occupy a maximum of six electrons. Each p orbital is made up of two parts known as lobes that are located on either side of the plane that runs across the nucleus. Each p orbital has parts known as lobes on either side of the plane that runs across the nucleus. At the plane where the two lobes intersect, the likelihood of finding an electron is nil. The three orbitals are known as degenerate orbitals because they have the same size, shape, and energy. The sole difference between the orbitals is the orientation of the lobes. Because the lobes are orientated along the x, y, or z-axis, they are given the names 2px, 2py, and 2pz. The formula n –2 is used to calculate the number of nodes. Similarly to s orbitals, the size and energy of p orbitals rise as the primary quantum number increases (4p > 3p > 2p). 15 | P a g e The Pauli exclusion principle requires that only two electrons can occupy each p-orbital. The shape of d orbitals The quantum number of a d orbital is given as (-2, -1, 0, 1, 2). Hence, we can say that there are five d-orbitals. These orbitals are dxy, dyz, dxz, dx2–y2, and dz2. Out of these five d orbitals, the shapes of the first four d-orbitals are similar, whereas the fifth d-orbital is different from the others. Also, the energy of all five d orbitals is the same. For d orbitals, the magnetic orbital quantum number is given as (-2, -1, 0, 1, 2). As a result, we can claim there are five d-orbitals. These orbitals are denoted by the symbols dxy, dyz, dxz, dx2–y2, and dz2. The forms of the first four d orbitals are similar to each other, which differs from the dz2 orbital, but the energy of all five d orbitals is the same. 16 | P a g e Understanding the shape of these orbitals will help you to visualise how these orbitals are formed. These five shapes are not identical. The first four orbitals (dxy, dyz, dxz and dx2–y2 ) have a dumbbell shape or two dumbbells connected to each other. But the fifth d-orbital (dz2) is really different from the others. We can say that it looks like a sphere while the other four look like dumbbells. Quantum numbers These are the integral numbers and most of them (i.e. first three) have been derived from the mathematical solution of Schrodinger’s wave equation for ψ. These numbers serve as the address of the electrons in an atom and hence are also known as identification numbers. These describe the energy of an electron in a shell, radius of that shell (i.e. distance of electron from the nucleus), shape and orientation of the electron cloud (or orbital) and the direction of the spinning of the electron on its own axis. There are four quantum numbers viz. principal quantum number (n), azimuthal or subsidiary quantum number (l), magnetic quantum number (m) and spin quantum number (s). (i) Principal or Radial Quantum Number This quantum number represents the number of shell or main energy level to which the electron belongs round the nucleus. It is denoted by the letter n. It arises from the solution of the radial part of ψ. This quantum number can have integral values 1, 2, 3, 4, etc., which are designated by the letters K, L, M, N, etc., as follows (proposed by Bohr): Value of n Designation 1 K 2 L 3 M 17 | P a g e  Value of n Designation 4 N It can be concluded that the principal quantum number (n) gives an idea of: (i) The shell or main energy level which the electron belongs to. (ii) The distance (r) of the electron from the nucleus, i.e., the radius of the shell. (iii) The energy associated with the electron. (iv) The maximum number of electrons that may be accommodated in a given shell. According to the Bohr-Berry scheme, the maximum number of electrons in the nth shell = 2n². Thus, the first shell (n = 1) can accommodate (2 x 1² = 2) two electrons, while the second, third, and fourth shells with n = 2, 3, and 4 can accommodate eight (2 x 2² = 8), eighteen (2 x 3² = 18), and thirty-two (2 x 4² = 32) electrons, respectively. (ii) Azimuthal or Subsidiary Quantum Number (l) This quantum number is also known as the orbital angular momentum quantum number. It is denoted by the letter l and refers to the subshell to which the electron belongs. This quantum number describes the motion of the electron and tells us about the shape of the orbitals of a subshell. The values of l depend on the value of n (the principal quantum number) and may have all possible values from 0 to (n-1), i.e., l = 0, 1, 2, 3, etc. Thus, for a given value of n, the total number of l values is equal to n. For example, when n = 4, l = 0, 1, 2, 3 (a total of 4 values of l). For each value of l, separate notation is used to represent a particular subshell, as shown below: Azimuthal Quantum Number (l) 0 1 2 3 4 Notation for the Subshell s p d f g These notations of subshells have been taken from the characteristics of the spectral lines in atomic spectra. Thus, s stands for sharp, p for principal, d for diffuse, and f for fundamental. The subshells belonging to various shells are given below: n l notation for the subshell 1 0 0 2 1 0 3 1 2 0 4 1 2 18 | P a g e  n l notation for the subshell 3 The main points to be noted for the azimuthal quantum number are: (i) This gives an idea of the subshell to which the electron belongs. (ii) The total number of subshells in a given shell is equal to the numerical value of n (main shell). (iii) This quantum number corresponds to the orbital angular momentum of the electron. (iv) This gives an idea of the shape of the orbitals of the subshell. (v) The maximum number of electrons that can be accommodated in a given subshell is equal to 2(2l + 1). Thus, s, p, d, and f subshells with l = 0, 1, 2, and 3 can have a maximum of 2, 6, 10, and 14 electrons, respectively, i.e., s2, p6, d10 and f14. (iii) Magnetic Quantum Number (m) This quantum number determines the direction of angular momentum of the electrons, thereby describing the orientation of orbitals of a subshell in space. The value of m depends on the value of l, showing that each subshell consists of one or more regions in space with a maximum probability of finding the electron (i.e., orbitals). The number of such orbitals (or regions) is equal to the number of ways the electrons can orient themselves in space. This number is equal to 2l+1, and values of m are represented as (+) l to (–) l through 0. Thus, each value of m represents a particular orbital within a subshell, and the total number of m values gives the total number of orbitals in that subshell. For example, for the s-subshell, m = 0 corresponding to l = 0 means that m has only one value, indicating that the s-subshell has only one orbital or one possible orientation of electrons, which is spherically symmetrical around the nucleus. When l = 1 (i.e., p-subshell), m has three values, viz. +1, 0, -1, implying that the p-subshell has three orbitals or orientations that are perpendicular to each other and point towards the x, y, and z-axes. These are designated as px, py and pz. For l = 2 (i.e., d-subshell), m = +2, +1, 0, -1, -2 meaning that this subshell has five orbitals or orientations: dxy, dyz, dzx, dx2-y2, and dz2. On the same grounds, it can be shown for the f-subshell (l=3l = 3l=3) that it has seven orbitals or orientations corresponding to seven values of m, viz. +3, +2, +1, 0, -1, -2, and -3. For p, d, and f subshells (l = 1, 2, 3), various m values may be summarized as follows: Subshell Value of l Values of m Total m values p-subshell 1 +1, 0, -1 3 d-subshell 2 +2, +1, 0, -1, -2 5 f-subshell 3 +3, +2, +1, 0, -1, -2, -3 7 The main point to be noted for the magnetic quantum number is that it determines the total number of orbitals present in any subshell, corresponding to the preferred orientations of electrons in space. 19 | P a g e (iv) Spin Quantum Number (s) This quantum number arises from the direction of the spinning of an electron about its own axis. It is denoted by the letter s, which can have only two values represented as (+) ↑ and (–) ↓, representing clockwise spin (α-spin) or anticlockwise spin (β-spin). These values are also represented as upward (↑) and downward (↓) arrows. Being a charged particle, a spinning electron generates a so-called spin magnetic moment that can be oriented either upward 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. 20 | P a g e  UNIT 4 Effective nuclear charge and screening constant This term is related with shielding or screening effect. The electrons residing in the innermost shell experience the attraction of full charge of the nucleus (actual charge) but this is not true for the electrons contained in the outer shells. Actually, the electrons in the inner shells called intervening electrons, act as a shield or screen between the nucleus and outer shell electrons and thus reduce the force of attraction between them. This is called screening or shielding effect. This effect of inner electrons causes a decrease in the actual charge of the nucleus (atomic number, Z) acting on the outer electrons by a quantity σ (sigma) known as screening or shielding constant. The decreased nuclear charge (Z- σ) is called effective nuclear charge denoted by Zeff. It is to be noted that only inner electrons cause the shielding of nucleus from outer electrons and outer electrons do not produce any shielding effect on any of the inner electrons in question. The shielding constant is greater than zero and less than Zactual and is a measure of the degree to which the intervening electrons shield the outer shell electrons from the nuclear pull. The effective nuclear charge (Zeff.) is defined as “the difference between actual nuclear charge (Zactual) and the screening constant (σ) produced by the intervening electrons”. Zeff = Z−σ The shielding effect caused by inner electrons varies with the type of subshells to which these electrons belong, e.g. s > p > d > f. This shows that s-electrons cause maximum shielding effect followed by p, d and f-electrons which produce minimum shielding effect due to their arrangement around nucleus. There are certain factors which influence the magnitude of σ and Zeff. Number of intervening electrons: Greater is the number of intervening electrons, more will be the magnitude of σ and less is the value of Zeff. Down in a group, number of intervening electrons in elements increases and hence value of σ also increases. Consequently, Zeff value goes on decreasing. (ii) Size of atom: As the size of atoms increases, value of Zeff decreases, e.g. down in a group. Along a period, atomic size decreases and hence Zeff goes on increasing. Aufbau principle Aufbau is a German word which means building up or construction. The building up of orbitals implies the filling of orbitals with electrons. This principle gives us the sequence in which various orbitals are filled with electrons. 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 21 | P a g e energy”. This means the electrons are filled in the subshell of the lowest energy first followed by the higher energy subshells. There are certain rules which constitute the Aufbau principle: (i) In general, the subshells with lower n values are filled first followed by those with higher n-values (called lower n rule). (ii) For any given principal quantum number n, the order of filling up of subshells is s, p, d and f. (iii) (n +l) Rule; sometime lower (n + l) rule is violated. In such cases (n+l) rule is applicable according to which the subshells are filled in order of increasing (n+l) values, e.g., 4s- subshell [(n+l) = 4+0 equal to 4] is filled before 3d subshell [(n+l) =3+ 2 equal to 5) due to lower (n+l) values. Keeping in mind the above discussion, various subshells can be arranged in the order of increasing energy as follows: Energy sequence of subshells for electron filling This relative order of energy of various subshells of an atom may also be given as follows: 1s