QUANTUM MECHANICAL MODEL OF THE ATOM:

 

1. QUANTUM MECHANICAL MODEL OF THE ATOM:

 Quantum Mechanical Model of the Atom:

1. Limitations of Classical Mechanics:
• Classical mechanics, based on Newton's laws, works well for macroscopic objects with particle-like behavior.
• It fails when applied to microscopic objects like electrons, atoms, and molecules due to their dual nature and the uncertainty principle.
2. Introduction of Quantum Mechanics:
• Quantum mechanics is a theoretical science that accounts for the dual behavior of matter in microscopic objects.
• It deals with objects that exhibit both wave-like and particle-like properties.
3. Quantum Mechanics vs. Classical Mechanics:
• Quantum mechanics specifies the laws of motion for microscopic objects.
• When applied to macroscopic objects where wave-like properties are negligible, quantum mechanics yields the same results as classical mechanics.
4. Development of Quantum Mechanics:
• Quantum mechanics was independently developed in 1926 by Werner Heisenberg and Erwin Schrödinger.
• This discussion focuses on the wave-based approach to quantum mechanics.
5. Schrödinger's Equation:
• Erwin Schrödinger formulated the fundamental equation of quantum mechanics, winning him the Nobel Prize in Physics in 1933.
• The Schrödinger equation incorporates the wave-particle duality of matter, as proposed by Louis de Broglie.
6. Complexity of Schrödinger's Equation:
• Schrödinger's equation is a complex mathematical equation.
• Solving it for different systems requires a strong foundation in higher mathematics.
7. Hamiltonian Operator:
• In the Schrödinger equation, there's a mathematical operator called the Hamiltonian operator denoted by H.
• Schrödinger provided a method for constructing this operator from the total energy expression of the system.
8. Total Energy Consideration:
• The total energy of a system, like an atom or a molecule, accounts for various factors.
• It includes the kinetic energies of subatomic particles (electrons, nuclei), attractive potentials between electrons and nuclei, and repulsive potentials among individual electrons and nuclei.
9. Solving Schrödinger's Equation:
• By solving Schrödinger's equation, physicists can determine the allowed energy levels (E) and wave functions (ψ) for a given system.
• These solutions provide insights into the behavior of electrons in atoms and molecules.

The Quantum Mechanical Model of the Atom was developed to explain the behavior of microscopic particles, taking into account their wave-particle duality and the uncertainty principle. Schrödinger's equation is a central element of this model, providing a mathematical framework to describe the energy levels and wave functions of particles in various systems, such as atoms and molecules.

 

2. HYDROGEN ATOM AND THE SCHRODINGER EQUATION

Hydrogen Atom and the Schrödinger Equation:

1. Solving the Schrödinger Equation for Hydrogen Atom:
• When the Schrödinger equation is solved for the hydrogen atom, it yields information about the possible energy levels that an electron can occupy and the corresponding wave functions (ψ) associated with each energy level.
2. Quantization of Energy Levels:
• The solutions to the Schrödinger equation for the hydrogen atom naturally result in quantized energy states.
• These quantized energy levels are characterized by a set of three quantum numbers: principal quantum number (n), azimuthal quantum number (l), and magnetic quantum number (ml).
3. Information Contained in Wave Functions:
• When an electron is in any energy state, the corresponding wave function (ψ) contains all the information about that electron's behavior within the atom.
• These wave functions are mathematical functions that depend on the coordinates of the electron in the atom and do not have physical meaning by themselves.
4. Atomic Orbitals:
• Wave functions for hydrogen or hydrogen-like species with a single electron are called atomic orbitals.
• These wave functions pertain to one-electron systems and describe the electron's probability distribution within the atom.
• The probability of finding an electron at a specific point within an atom is proportional to the square of the absolute value of ψ (|ψ|²) at that point.
5. Prediction of Hydrogen Atom Spectrum:
• Quantum mechanical results for the hydrogen atom successfully predict all aspects of the hydrogen atom's spectrum.
• This includes phenomena that couldn't be explained by the Bohr model, such as fine spectral lines.
6. Challenges in Multi-Electron Atoms:
• When applying the Schrödinger equation to multi-electron atoms, a challenge arises: it cannot be solved exactly for such systems due to the interactions between multiple electrons.
• Approximate methods are used to overcome this difficulty.
7. Similarity to Hydrogen Orbitals:
• Calculations with modern computers show that the orbitals in multi-electron atoms, while not identical, do not differ radically from hydrogen orbitals.
• The primary difference is due to the increased nuclear charge in multi-electron atoms, which causes all orbitals to be somewhat contracted.
8. Energy Dependence on Quantum Numbers:
• In contrast to hydrogen or hydrogen-like species, where orbital energies depend only on the principal quantum number (n), the energies of orbitals in multi-electron atoms depend on both the principal quantum number (n) and the azimuthal quantum number (l).

The Schrödinger equation provides a powerful framework for understanding the behavior of electrons in the hydrogen atom, leading to quantized energy levels and corresponding wave functions called atomic orbitals. These principles are essential for explaining the hydrogen atom's spectrum and are extended, with some modifications, to understand multi-electron atoms.

3. IMPORTANT FEATURES OF THE QUANTUM MECHANICAL MODEL OF ATOM:

Important Features of the Quantum Mechanical Model of Atom 

1. Quantization of Electron Energy:
• Electrons in atoms have quantized energy levels, meaning they can only occupy specific, discrete energy values.
• This quantization is evident when electrons are bound to the atomic nucleus.
2. Wave-Like Properties and Energy Levels:
• The existence of quantized electronic energy levels is a direct consequence of the wave-like properties of electrons.
• These energy levels are solutions of the Schrödinger wave equation.
3. Heisenberg Uncertainty Principle:
• The exact position and exact velocity of an electron in an atom cannot be simultaneously determined with absolute precision, as stated by the Heisenberg uncertainty principle.
• Consequently, the precise path of an electron within an atom cannot be accurately known.
• Instead, we discuss the probability of finding an electron at various points within an atom.
4. Atomic Orbitals:
• An atomic orbital is described by the wave function ψ for an electron in an atom.
• Each unique wave function corresponds to a specific orbital.
• Multiple atomic orbitals are possible within an atom, forming the basis of its electronic structure.
• In each orbital, an electron has a definite energy.
• An orbital can contain a maximum of two electrons, each with opposite spins.
• In multi-electron atoms, electrons fill various orbitals following the order of increasing energy.
• Each electron in a multi-electron atom is associated with an orbital wave function that characterizes the orbital it occupies.
• All information about an electron in an atom is stored within its orbital wave function ψ.
5. Probability Density:
• The probability of finding an electron at a particular point within an atom is proportional to the square of the orbital wave function, denoted as |ψ|².
• |ψ|² is known as the probability density and is always a positive value.
• By examining the values of |ψ|² at different points within an atom, it becomes possible to predict the regions around the nucleus where electrons are most likely to be found.

The Quantum Mechanical Model of the Atom introduces quantized energy levels for electrons in atoms, which arise from their wave-like properties and are solutions of the Schrödinger equation. It also emphasizes the fundamental concept of wave functions, known as atomic orbitals, which describe the behavior of electrons within atoms. The Heisenberg uncertainty principle underscores the limitations in simultaneously determining the precise position and velocity of an electron, leading to the concept of probability density in quantum mechanics.

4. ORBITALS AND QUANTUM NUMBERS:

Orbitals and Quantum Numbers:

1. Orbitals:

• Orbitals are regions in an atom where electrons are likely to be found.
• They can be distinguished by their size, shape, and orientation.
• Smaller orbitals have a higher probability of finding an electron near the nucleus.

2. Quantum Numbers:

• Quantum numbers are used to precisely describe the properties of electrons in an atom.
• They consist of four quantum numbers: n, l, m_l, and m_s.

3. Principal Quantum Number (n):

• Denoted as "n," it is a positive integer (n = 1, 2, 3, ...) representing the main energy level or shell of an electron.
• Determines the size and, to a large extent, the energy of the orbital.
• Each value of "n" corresponds to a shell, with "n = 1" corresponding to the first shell (K), "n = 2" to the second shell (L), and so on.
• The number of orbitals within a shell is "n²."

4. Azimuthal Quantum Number (l):

• Denoted as "l," it determines the shape or type of orbital within a given shell.
• For a given "n," "l" can have values ranging from 0 to (n-1).
• Different values of "l" correspond to different types of orbitals, such as "s," "p," "d," and "f."
• The number of sub-shells in a principal shell is equal to the value of "n."

5. Magnetic Quantum Number (m_l):

• Denoted as "m_l," it specifies the spatial orientation of an orbital within a sub-shell.
• For a given "l," there are (2l+1) possible values of "ml."
• These values determine how many ways the orbitals can be oriented within a sub-shell.

6. Electron Spin Quantum Number (m_s):

• Denoted as "m_s," it describes the intrinsic spin of an electron.
• It has two possible values: +½ (spin up) and -½ (spin down).
• Electrons in an orbital must have opposite spin values.

7. Relation between Quantum Numbers and Orbitals:

• Each orbital is defined by a unique set of quantum numbers (n, l, ml).
• For example, "n = 2, l = 1, ml = 0" represents a "2p" orbital in the second shell.
• The number of orbitals associated with a sub-shell is determined by the value of "l," and it is (2l+1).

8. Electron Spin and Line Spectra:

• To explain certain line spectra patterns observed in multi-electron atoms, a fourth quantum number, the electron spin quantum number (ms), was introduced.
• Electrons have intrinsic spin angular momentum with two possible orientations: +½ and -½.
• Orbitals can hold a maximum of two electrons, and these electrons must have opposite spins.

Quantum numbers (n, l, ml, and ms) provide a comprehensive description of electrons within an atom, including their energy levels, orbital shapes, orientations, and intrinsic spins. These quantum numbers help explain the electronic structure of atoms and their behavior in atomic orbitals.

5. SHAPES OF ATOMIC ORBITALS:

Shapes of Atomic Orbitals:

1. Nature of Orbital Wave Function (ψ):

• The orbital wave function (ψ) for an electron in an atom is a mathematical function of electron coordinates and has no physical meaning by itself.

2. Probability Density (ψ²):

• Max Born's interpretation states that the square of the wave function (ψ²) at a point represents the probability density of finding the electron at that point within an orbital.
• Probability density varies with distance (r) from the nucleus.

3. Nodal Surfaces (Nodes):

• Nodal surfaces are regions within an orbital where the probability density is zero.
• For example, 1s orbitals have a maximum probability density at the nucleus and decrease rapidly away from it, whereas 2s orbitals have a more complex probability density pattern.
• The region where the probability density reduces to zero is called nodal surfaces.

4. Visualization of Orbitals:

• Charge cloud diagrams visualize probability density. Dots represent electron probability density.
• Boundary surface diagrams are drawn for constant probability density values, enclosing regions where the probability of finding an electron is very high (e.g., 90%).
• Boundary surface diagrams of constant probability density for different orbitals give a fairly good representation of the shapes of the orbitals.

5. Spherical s-Orbitals:

• s-Orbitals are spherically symmetric, meaning the probability of finding an electron at a given distance is equal in all directions.
• The size of s-orbitals increases with increasing principal quantum number (n): 4s > 3s > 2s > 1s.
• 
  
  
  
  

6. p-Orbitals:

• p-Orbitals have two lobes on either side of a plane that passes through the nucleus.
• Each p orbital has two sections (lobes) with a nodal plane in between, where the probability density is zero.
• The three 2p orbitals (p_x, p_y, p_z) have the same energy but differ in orientation along the x, y, and z axes.
• 
  
  
  
  
  
  
  
  
  

7. Number of Nodes:

• The number of nodes in an orbital is determined by both radial nodes and angular nodes.
• Radial nodes are where the probability density function is zero.
• Angular nodes are nodal planes that pass through the origin (nucleus).
• The total number of nodes is given by (n-1), where "n" is the principal quantum number.
• For example, 3p orbitals have one angular node and one radial node, totaling two nodes.

8. d-Orbitals:

• d-Orbitals have shapes that include five lobes, and their orientation is more complex than s and p orbitals.
• There are five d orbitals (d_xy, d_yz, d_xz, d_x²-y², d_z²), and they have the same energy but different orientations.

9. Node Types:

• Besides radial nodes, some orbitals have nodal planes (nodal surfaces), which are called angular nodes.
• The number of angular nodes is given by the azimuthal quantum number (l).

10. Energy and Size Variation: - The energy and size of orbitals change with the principal quantum number (n). Higher "n" values lead to larger and higher-energy orbitals.

The shapes of atomic orbitals are defined by their wave functions and probability density patterns. These shapes are represented using boundary surface diagrams, and the number and types of nodes within an orbital depend on its quantum numbers. Different orbitals (s, p, d) have distinct shapes and orientations, and their sizes and energies vary with the principal quantum number (n).

6.  BOUNDARY SURFACE DIAGRAMS:

1. Boundary Surface Diagrams:

• Boundary surface diagrams are representations of orbitals that depict surfaces in space where the probability density |ψ|² is constant.
• These diagrams help visualize the shape of orbitals and provide a way to understand the three-dimensional distribution of electron probability within an orbital.

2. Selection of Boundary Surfaces:

• Multiple boundary surfaces are possible for each orbital because |ψ|² can take on various constant values.
• However, only the boundary surface that encloses a region where the probability of finding the electron is very high, typically around 90%, is considered a good representation of the orbital's shape.

3. Why Not 100% Probability?:

• The reason boundary surfaces do not enclose regions with a 100% probability of finding the electron is due to the nature of the quantum mechanical description of electrons in atoms.
• |ψ|² always has some value, no matter how small, at any finite distance from the nucleus.
• In other words, there is always a non-zero probability of finding the electron at some distance from the nucleus.
• As a result, it is not possible to draw a boundary surface diagram that bounds a region with a 100% probability.

4. Spherical Shape of s-Orbitals:

• The boundary surface diagram for a 1s orbital is a sphere centered on the nucleus, and in two dimensions, it appears as a circle.
• This spherical shape is representative of all s-orbitals, which are spherically symmetric.
• Spherically symmetric means that the probability of finding the electron at a given distance is the same in all directions from the nucleus.

5. Relationship to Principal Quantum Number (n):

• The size of s-orbitals increases with an increase in the principal quantum number (n).
• For example, 4s > 3s > 2s > 1s, indicating that the electron is located further away from the nucleus as n increases.
• Higher n values correspond to larger and more extended s-orbitals.

In summary, boundary surface diagrams are used to visualize the shapes of atomic orbitals. These diagrams represent surfaces in space where the probability density |ψ|² is constant. While it's not possible to enclose regions with 100% probability, the selected boundary surface typically encompasses a region where the probability of finding the electron is around 90%. S-orbitals, such as 1s and 2s, are spherical and spherically symmetric, with their sizes increasing as the principal quantum number (n) increases.

 

 

7. STRUCTURE OF p-ORBITALS:

1. Structure of 2p Orbitals:

• 2p orbitals belong to the principal quantum number (n = 2) and have an azimuthal quantum number (l = 1), which indicates their shape and orientation.
• Unlike s-orbitals, 2p orbitals are not spherical but have a more complex shape.
• Each 2p orbital consists of two lobes (distinct regions) that are separated by a nodal plane.

2. Lobes and Nodal Plane:

• The lobes of 2p orbitals are found on either side of a plane that passes through the nucleus, known as the nodal plane.
• The probability density function (|ψ|²) of 2p orbitals is zero precisely on the nodal plane where the two lobes touch each other.

3. Identical Size, Shape, and Energy:

• All three 2p orbitals (2px, 2py, and 2pz) have identical size, shape, and energy.
• Their only difference lies in the orientation of their lobes along the x, y, or z-axis, respectively.

4. Perpendicular Axes:

• The lobes of the 2p orbitals can be thought of as lying along the x, y, or z-axis, which are mutually perpendicular to each other.
• This means that the lobes of 2p_x, 2p_y, and 2p_z orbitals are oriented along the x-axis, y-axis, and z-axis, respectively.

5. Order of Energy and Size:

• Similar to s orbitals, the energy and size of p orbitals increase with an increase in the principal quantum number (n).
• The order of energy and size of various p orbitals is 4p > 3p > 2p.
• In other words, 4p orbitals are larger and higher in energy compared to 3p, and 3p is larger and higher in energy compared to 2p.

6. Radial Nodes:

• The number of radial nodes in p orbitals is given by (n - 1).
• For example, there is one radial node in the 3p orbital, two in the 4p orbital, and so on.
• Radial nodes are regions where the probability density function (|ψ|²) is zero as the distance from the nucleus increases.

2p orbitals are distinct from s-orbitals in that they have two lobes separated by a nodal plane. These lobes are oriented along the x, y, or z-axis, depending on whether it's 2p_x, 2p_y, and 2p_z, respectively. Despite their different orientations, all three 2p orbitals have the same size, shape, and energy. The number of radial nodes in p orbitals depends on the principal quantum number (n), with one radial node in 3p, two in 4p, and so on.

8. STRUCTURE OF d-ORBITALS:

1. Quantum Numbers for d-Orbitals:

• D-orbitals are associated with an azimuthal quantum number (l = 2), which means they have a more complex shape compared to s and p orbitals.
• The principal quantum number (n) must be at least 3 for d-orbitals to exist because l cannot be greater than (n-1).

2. Number of ml Values and Types:

• For l = 2, there are five possible values of the magnetic quantum number (ml), which are -2, -1, 0, +1, and +2.
• These five ml values correspond to five distinct d orbitals.

3. Designation of d Orbitals:

• Each of the five d orbitals is designated by a specific label that indicates its orientation. These labels are:
• d_xy
• d_yz
• d_xz
• d_x²-y²
• d_z²

4. Similarities and Differences:

• The first four d orbitals (d_xy, d_yz, d_xz, d_x²-y²) have similar shapes and energy levels. They are often collectively referred to as "degenerate."
• The fifth d orbital, dz², has a different shape compared to the other four. It is oriented along the axis perpendicular to the plane of the other four d orbitals.
• Despite their different shapes, all five 3d orbitals (for n = 3) have equivalent energy levels.

5. Energy and Size in Higher Principal Shells:

• D orbitals for higher principal quantum numbers (4d, 5d, etc.) have shapes similar to the 3d orbital but differ in energy and size. The energy increases with higher n values.

6. Radial Nodes:

• Similar to other orbitals, d orbitals have radial nodes where the probability density function (|ψ|²) is zero.
• The number of radial nodes is determined by the principal quantum number (n) and is given by (n - 1). For example, 3d orbitals have two radial nodes.

7. Nodal Planes and Angular Nodes:

• In addition to radial nodes, d orbitals also have nodal planes.
• The probability density functions for np and nd orbitals are zero at the plane(s) that pass through the nucleus (origin). These planes are known as nodal planes.
• In the case of d orbitals, there are two angular nodes, which are regions where the probability density function is zero. The number of angular nodes is equal to the value of the azimuthal quantum number (l). Therefore, d orbitals have two angular nodes.

8. Total Number of Nodes:

• The total number of nodes in an orbital is the sum of angular nodes (l) and radial nodes (n - l - 1).
• For d orbitals, the total number of nodes is given by the sum of two angular nodes (l = 2) and the radial nodes (n - 3 - 1).

d orbitals are characterized by their complex shapes and orientations. There are five distinct d orbitals, with the first four being degenerate in energy and shape, while the fifth (d_z²) is different. The number of nodes in d orbitals depends on both the principal quantum number (n) and the azimuthal quantum number (l), leading to a combination of radial and angular nodes.

 

9. ENERGIES OF AN ELECTRON IN A HYDROGEN ATOM :

1. Energy Dependence on Principal Quantum Number (n):

• The energy of an electron in a hydrogen atom is primarily determined by the principal quantum number (n).
• As the value of n increases, the energy of the orbital also increases.

2. Order of Increasing Energies:

• The order of increasing energies for orbitals in a hydrogen atom is as follows:
• 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f

3. Degenerate Orbitals:

• Orbitals that have the same energy level are called "degenerate" orbitals.
• In the case of the hydrogen atom, the 2s and 2p orbitals have the same energy.
• This means that electrons in the 2s and 2p orbitals of a hydrogen atom have the same energy despite their different shapes.
• The term "degenerate" implies that these orbitals are equivalent in energy.

4. Ground State and Excited States:

• The 1s orbital in a hydrogen atom represents the lowest energy level and is called the "ground state."
• An electron residing in the 1s orbital is in its most stable condition and is most strongly attracted to the nucleus.
• When an electron occupies any other orbital with higher energy (e.g., 2s, 2p, 3s, etc.), it is said to be in an "excited state."
• Excited states indicate that the electron has absorbed energy and moved to a higher orbital. These states are less stable than the ground state.

5. Energy Diagram:

• The relative energies of the hydrogen atom orbitals are often depicted in an energy diagram, as shown in the figure below

The energy of an electron in a hydrogen atom depends primarily on its principal quantum number (n). The orbitals with the same energy level are called degenerate orbitals. The 1s orbital is the ground state with the lowest energy, while any electron in a higher orbital (e.g., 2s, 2p, 3s, etc.) is considered to be in an excited state, indicating a less stable condition. The energy order of orbitals in a hydrogen atom is a fundamental concept in quantum mechanics and helps explain the electronic structure of atoms.

 

 

 

10. ENERGIES OF AN ELECTRON IN A MULTI- ELECTRON ATOM :

1. Energy Dependence on Principal Quantum Number (n) and Azimuthal Quantum Number (l):

• Unlike in the hydrogen atom, the energy of an electron in a multi-electron atom depends not only on its principal quantum number (shell, n) but also on its azimuthal quantum number (subshell, l).
• Within a given principal quantum number (shell), such as the same n value, different subshells (s, p, d, f, etc.) have different energies.
• The order of increasing energies within a given shell is s < p < d < f. For example, 2s has lower energy than 2p, and 3d has lower energy than 4s.

2. Attractive and Repulsive Interactions:

• In multi-electron atoms, there are both attractive and repulsive interactions among electrons.
• The attractive interactions exist between electrons and the nucleus, while repulsive interactions occur between electrons themselves due to their like charges.
• The stability of an electron in a multi-electron atom results from the fact that the total attractive interactions are greater than the repulsive interactions.

 

 

3. Shielding Effect:

• Electrons in inner shells shield or partially screen outer-shell electrons from the full positive charge of the nucleus.
• This shielding effect means that the net positive charge experienced by the outer electrons is less than the actual nuclear charge, and it's called the effective nuclear charge (Z_eff e).
• (Z_eff e )is the nuclear charge experienced by an electron after accounting for shielding by inner electrons.

4. Energy and Shielding:

• Both attractive and repulsive interactions depend on the shape and shell of the orbital where the electron is located.
• Electrons in s orbitals shield outer electrons from the nucleus more effectively than electrons in p orbitals.
• Electrons in p orbitals shield outer electrons better than electrons in d orbitals, even if all these orbitals are in the same shell.
• For a given shell (principal quantum number), (Z_eff e) experienced by electrons decreases with an increase in the azimuthal quantum number (l).

5. Reason for Different Energies:

• The main reason for different energy levels of subshells is the mutual repulsion among the electrons in multi-electron atoms.
• In multi-electron atoms, there are both attractive interactions between electrons and the nucleus and repulsive interactions between electrons.
• Electrons in the outer shell experience more repulsion from the inner shell electrons, and this repulsion affects their energy.
• The attractive interactions depend on the effective nuclear charge (Z_eff e), which is the net positive charge experienced by outer electrons after considering the shielding effect of inner shell electrons.
• As the azimuthal quantum number (l) increases, electrons spend less time close to the nucleus, leading to weaker attractive interactions and higher orbital energies.

6. Energy Splitting Within the Same Shell:

• Electrons in different subshells of the same shell (or principal quantum number) have different energies in multi-electron atoms.
• The extent of shielding from the nucleus varies for electrons in different orbitals, leading to energy splitting within the same shell.
• Mathematically, the energy of an orbital depends on the values of n and l, with a simple rule that lower values of (n + l) correspond to lower energy orbitals.
• If two orbitals have the same (n + l) value, the one with a lower n value has lower energy.

7. (n + l) Rule:

• The (n + l) rule is a simple guideline to predict the relative energies of orbitals in multi-electron atoms. Lower (n + l) values indicate lower energy orbitals.
• If two orbitals have the same (n + l) value, the orbital with the lower n value will have lower energy.

8. Variation of Energies with Atomic Number (Z_eff):

• The energies of orbitals within the same subshell decrease with an increase in the effective nuclear charge (Z_eff e).
• Z_eff e is influenced by the presence of inner shell electrons, which partially shield outer electrons from the full positive charge of the nucleus.
• Therefore, as you move across the periodic table (from left to right), the energy of the same type of orbital (e.g., 2s) decreases due to the increasing atomic number (Z).
• For example, energy of 2s orbital of hydrogen atom is greater than that of 2s orbital of lithium and that of lithium is greater than that of sodium and so on, that is, E_2s (H) > E_2s (Li) > E_2s (Na) > E_2s (K).

 

In multi-electron atoms, the energy of an electron depends on both its principal quantum number (n) and azimuthal quantum number (l). Different subshells within the same shell have different energies due to repulsive interactions among electrons and the shielding effect of inner shell electrons. The (n + l) rule helps predict relative orbital energies, and the effective nuclear charge (Z_eff e) affects the energy levels of electrons in the same subshell. These principles explain the energy ordering of orbitals in multi-electron atoms and their variation across the periodic table.

1. Shielding Effect of Spherical s Orbitals: Electrons in spherical-shaped s orbitals are distributed more symmetrically around the nucleus. Due to this spherical symmetry, s orbitals tend to shield the outer electrons more effectively than other types of orbitals, such as p or d orbitals. Electrons in s orbitals create a more uniform distribution of negative charge, which partially screens the positive charge of the nucleus from outer electrons. 2. Variation in Shielding by Different Orbitals: Electrons in p orbitals, although in the same shell as s electrons, do not shield the outer electrons as effectively. This is because p orbitals have a more complex, dumbbell-shaped distribution, and the electron density is concentrated in two regions rather than being spherical. Electrons in d orbitals, which are even more complex in shape, shield the outer electrons less effectively than s or p electrons.