The wide range of chemical behaviors observed among different elements is a direct consequence of the distinct internal structures of their atoms. Understanding the arrangement of particles within an atom is key to explaining their properties and interactions.

The idea that matter is composed of indivisible building blocks, atoms, dates back to ancient Indian and Greek philosophers (around 400 B.C.). They believed that repeatedly dividing matter would eventually lead to particles that could not be cut further, coining the term 'atom' from the Greek word 'a-tomio', meaning 'uncut-able'. However, these were purely philosophical ideas lacking experimental evidence.

Scientific investigation into the nature of atoms was rekindled in the 19th century. John Dalton's atomic theory (1808) provided a scientific foundation for these ideas, describing atoms as the fundamental, indivisible particles of matter. Dalton's theory successfully explained key laws of chemical combination, such as the Law of Conservation of Mass and the Law of Definite Proportions. However, it could not account for observations like the electrical charging of materials when rubbed, suggesting that atoms were not, in fact, the ultimate, indivisible particles.

Experiments conducted towards the end of the 19th and beginning of the 20th centuries revealed that atoms are composed of even smaller, sub-atomic particles. This unit explores the discovery and properties of these particles and the models developed to describe atomic structure.

---

Discovery Of Sub-Atomic Particles

Experiments involving electrical discharge through gases provided crucial insights into the composition of atoms.

Recall that particles with the same electrical charge repel each other, while particles with opposite charges attract each other. This fundamental principle helps understand the behavior of charged particles in electric and magnetic fields.

---

Discovery Of Electron

In the 1830s, Michael Faraday's experiments on electrolysis showed that passing electricity through electrolyte solutions caused chemical reactions at the electrodes, involving the deposition or liberation of matter. His findings suggested that electricity had a particulate nature.

Later, in the mid-1850s, scientists like Faraday studied electrical discharge in partially evacuated tubes, known as cathode ray discharge tubes.

A typical cathode ray tube is a glass tube with two metal electrodes sealed inside. Electrical discharge occurs at very low gas pressures and high voltages. When a sufficiently high voltage is applied, a stream of invisible particles flows from the negative electrode (cathode) to the positive electrode (anode). These were named cathode rays.

To visualise these rays, a hole was made in the anode, and the glass tube behind it was coated with a phosphorescent material like zinc sulfide. When the cathode rays passed through the hole and struck the coating, a bright spot appeared.

Observations about cathode rays:

1. They originate from the cathode and travel towards the anode.
2. They are invisible, but their presence is detected by materials that fluoresce or phosphoresce when struck by them (like a TV screen).
3. In the absence of external electric or magnetic fields, they travel in straight lines.
4. When electric or magnetic fields are applied, they behave like streams of negatively charged particles. The deflection direction is consistent with a negative charge.
5. The properties of cathode rays are independent of the material of the electrodes or the type of gas in the tube.

This led to the conclusion that these negatively charged particles, named electrons, are fundamental constituents of all atoms.

---

Charge To Mass Ratio Of Electron

In 1897, British physicist J.J. Thomson quantified the properties of electrons by measuring the ratio of their electrical charge ($e$) to their mass ($m_e$). He used a cathode ray tube with perpendicular electric and magnetic fields applied to the electron beam's path.

By applying either field, Thomson observed the deflection of the electrons. Applying both fields and adjusting their strengths, he could balance the deflecting forces so the electrons passed through undeflected (hitting point B). The amount of deflection was found to depend on:

• The magnitude of the charge ($e$) on the particle: Higher charge leads to greater deflection by the fields.
• The mass ($m_e$) of the particle: Lighter particles are deflected more easily.
• The strength of the electric or magnetic field: Stronger fields cause greater deflection.

Using precise measurements of field strengths and deflections, Thomson determined the charge-to-mass ratio ($e/m_e$) for the electron:

$\frac{e}{m_e} = 1.758820 \times 10^{11} \text{ C kg}^{-1}$

Since electrons are negatively charged, the charge is denoted as $-e$.

---

Charge On The Electron

While Thomson found the ratio $e/m_e$, he did not determine the individual values of $e$ or $m_e$. American physicist R.A. Millikan devised the famous oil drop experiment (1906-1914) to determine the charge of a single electron.

In this experiment, tiny oil droplets were introduced into a chamber containing two charged plates. Some droplets acquired electrical charge by colliding with ions produced by X-rays. By observing the motion of these charged droplets under the influence of gravity and an adjustable electric field, Millikan could determine the charge on each droplet. He found that the charge on any droplet ($q$) was always an integral multiple ($n$) of a fundamental unit of charge ($e$), i.e., $q = n \times e$.

Millikan's experiments yielded a value for the electronic charge: $-1.6 \times 10^{-19}$ C. The currently accepted value is $-1.602176 \times 10^{-19}$ C.

Knowing the charge ($e$) and Thomson's charge-to-mass ratio ($e/m_e$), the mass of the electron ($m_e$) could be calculated:

$m_e = \frac{e}{(e/m_e)} = \frac{1.602176 \times 10^{-19} \text{ C}}{1.758820 \times 10^{11} \text{ C kg}^{-1}} \approx 9.1094 \times 10^{-31} \text{ kg}$.

---

Discovery Of Protons And Neutrons

Following the discovery of electrons, experiments with modified cathode ray tubes led to the discovery of positively charged particles. When a perforated cathode was used, streams of positively charged particles were observed moving in the direction opposite to the cathode rays, passing through the holes in the cathode. These were called canal rays or anode rays.

Characteristics of these positively charged particles:

1. Unlike cathode rays, their mass depends on the type of gas in the discharge tube. These particles are essentially the positively charged ions formed from the gas molecules.
2. Their charge-to-mass ratio also varies depending on the gas.
3. Some particles carry a charge that is a multiple of the fundamental unit of charge, $e$.
4. They are deflected by electric and magnetic fields in the opposite direction compared to electrons, confirming their positive charge.

The smallest and lightest positive particle was observed when hydrogen gas was used. This particle was named the proton and was characterised in 1919. A proton carries a positive charge equal in magnitude but opposite in sign to the electron's charge ($+1.602176 \times 10^{-19}$ C).

Later, it became apparent that the total mass of an atom could not be explained by only protons and electrons. This suggested the presence of electrically neutral particles in the nucleus. In 1932, James Chadwick discovered these particles by bombarding a thin sheet of beryllium with alpha ($\alpha$) particles. Neutral particles with mass slightly greater than protons were emitted. These were named neutrons.

Key properties of the three fundamental sub-atomic particles are summarised below:

Name	Symbol	Absolute charge/C	Relative charge	Mass/kg	Mass/u	Approx. Mass/u
Electron	e	$-1.602176 \times 10^{-19}$	$-1$	$9.109382 \times 10^{-31}$	$0.00054$	$0$
Proton	p	$+1.602176 \times 10^{-19}$	$+1$	$1.6726216 \times 10^{-27}$	$1.00727$	$1$
Neutron	n	$0$	$0$	$1.674927 \times 10^{-27}$	$1.00867$	$1$

Besides these discoveries, other forms of radiation were identified. X-rays, discovered by Wilhelm Röntgen in 1895, are high-energy electromagnetic radiation produced when electrons hit a metal target. Radioactivity, discovered by Henri Becquerel and further studied by Marie and Pierre Curie and Rutherford, is the spontaneous emission of radiation by certain elements. This radiation consists of alpha ($\alpha$) particles (helium nuclei, $+2$ charge), beta ($\beta$) particles (electrons, $-1$ charge), and gamma ($\gamma$) rays (high-energy electromagnetic radiation, no charge).

Atomic Models

The experimental evidence for sub-atomic particles contradicted Dalton's model of an indivisible atom. Scientists faced the challenge of developing models that could explain the arrangement of electrons, protons, and neutrons within the atom and account for atomic properties.

Key questions atomic models needed to address:

• How is the atom stable?
• How do physical and chemical properties relate to atomic structure?
• How do atoms combine to form molecules?
• What is the origin of electromagnetic radiation absorbed or emitted by atoms (atomic spectra)?

Several models were proposed, but two early ones, by J.J. Thomson and Ernest Rutherford, were particularly significant.

---

Thomson Model Of Atom

Following his work on electrons, J.J. Thomson proposed the first atomic model in 1898, often called the "plum pudding" model.

According to this model:

• The atom is a sphere with a radius of approximately $10^{-10}$ m.
• The positive charge is spread uniformly throughout this sphere.
• The electrons are embedded within this positive sphere, like plums in a pudding or seeds in a watermelon, arranged to achieve the most stable electrostatic configuration.
• The total positive charge equals the total negative charge, making the atom electrically neutral.

An important assumption was that the mass of the atom was evenly distributed throughout the sphere. While this model explained the atom's overall neutrality, it could not be reconciled with the results of later experiments, particularly Rutherford's scattering experiment.

---

Rutherford’s Nuclear Model Of Atom

Ernest Rutherford, a former student of Thomson, conducted a landmark experiment in 1909 with his students Hans Geiger and Ernest Marsden, known as the alpha ($\alpha$)-particle scattering experiment.

A beam of high-energy $\alpha$-particles (which are positively charged helium nuclei, $\text{He}^{2+}$) was directed at a very thin gold foil (around 100 nm thick). A circular screen coated with zinc sulfide (a fluorescent material) was placed around the foil to detect the scattered $\alpha$-particles, as each strike produced a flash of light.

Based on Thomson's model, which proposed a uniform distribution of positive charge and mass, Rutherford expected the $\alpha$-particles to pass through the foil with minimal deflection. However, the results were surprising:

• Most $\alpha$-particles passed straight through the foil without any deflection.
• A small fraction of the $\alpha$-particles were deflected by noticeable, small angles.
• A very few $\alpha$-particles (about 1 in 20,000) were deflected back by large angles, some even exceeding 90$^\circ$, as if they had hit something dense and bounced off.

Based on these observations, Rutherford drew revolutionary conclusions about atomic structure:

1. Since most $\alpha$-particles passed through undeflected, most of the space within an atom must be empty.
2. The small angle deflections indicated a significant repulsive force, suggesting that the atom's positive charge is not spread out (as in Thomson's model) but is concentrated in a very small, dense region at the center.
3. The rare large-angle deflections and backward scattering meant that $\alpha$-particles sometimes hit this dense positive core directly. Calculations showed this central region, which he named the nucleus, occupies an extremely small volume compared to the atom's total volume. The radius of the atom is about $10^{-10}$ m, while the nucleus's radius is about $10^{-15}$ m. To appreciate the scale, if an atom were the size of a large sports stadium, the nucleus would be like a small pebble in the center.

Based on these conclusions, Rutherford proposed the Nuclear Model of the Atom:

• Most of the atom's mass and all its positive charge are concentrated in a tiny, dense central region called the nucleus.
• The electrons, which are negatively charged and have negligible mass, revolve rapidly around the nucleus in well-defined circular paths called orbits.
• The electrostatic force of attraction between the positively charged nucleus and the negatively charged electrons holds the atom together, similar to how planets orbit the sun due to gravitational force (hence, often called the planetary model).

---

Atomic Number And Mass Number

Rutherford's model established the nucleus as the location of the positive charge, carried by protons. For an atom to be electrically neutral, the number of protons in the nucleus must equal the number of electrons orbiting it.

The number of protons in the nucleus of an atom is defined as its atomic number ($Z$). This number uniquely identifies an element. In a neutral atom, the number of electrons equals the atomic number.

$Z = \text{Number of protons in the nucleus}$

$Z = \text{Number of electrons in a neutral atom}$

For example, Hydrogen has 1 proton ($Z=1$), Sodium has 11 protons ($Z=11$). A neutral Hydrogen atom has 1 electron, and a neutral Sodium atom has 11 electrons.

The mass of the nucleus comes from both protons and neutrons. Protons and neutrons collectively are called nucleons. The total number of nucleons (protons + neutrons) is the mass number ($A$) of the atom.

$A = \text{Number of protons (Z)} + \text{Number of neutrons (n)}$

An atom of an element is typically represented using the notation $^A_Z$X, where X is the element symbol, A is the mass number, and Z is the atomic number.

---

Isobars And Isotopes

Atoms can have variations in their composition, leading to different types:

• Isobars are atoms of different elements that have the same mass number ($A$) but different atomic numbers ($Z$). This means they have different numbers of protons and thus different chemical properties, but the total number of nucleons is the same. Example: Carbon-14 ($^{14}_6$C) and Nitrogen-14 ($^{14}_7$N).
• Isotopes are atoms of the same element (same atomic number, $Z$) but different mass numbers ($A$). This difference in mass number arises from having a different number of neutrons in the nucleus. Since isotopes of an element have the same number of protons and electrons, they exhibit very similar chemical properties.

Examples of isotopes:

• Hydrogen has three common isotopes:
  • Protium ($^1_1$H): 1 proton, 0 neutrons ($A=1$, $Z=1$). Most abundant ($\approx 99.985\%$).
  • Deuterium ($^2_1$D): 1 proton, 1 neutron ($A=2$, $Z=1$). (approx. 0.015%)
  • Tritium ($^3_1$T): 1 proton, 2 neutrons ($A=3$, $Z=1$). Radioactive, found in trace amounts.
• Carbon has common isotopes $^{12}_6$C, $^{13}_6$C, $^{14}_6$C (6 protons, 6, 7, and 8 neutrons respectively). Carbon-12 is the standard for atomic mass.
• Chlorine has isotopes $^{35}_{17}$Cl and $^{37}_{17}$Cl (17 protons, 18 and 20 neutrons respectively).

Since chemical behaviour is primarily determined by the number of electrons (equal to the number of protons, Z), isotopes of an element show almost identical chemical reactivity. Differences are mainly in physical properties influenced by mass (e.g., density, melting/boiling points, rates of reaction, especially for lighter elements like Hydrogen).

---

Answer:

---

Answer:

---

Drawbacks Of Rutherford Model

Despite being a significant advancement, Rutherford's nuclear model faced challenges based on classical physics:

• Instability of the atom: According to classical electromagnetic theory (Maxwell's equations), an accelerating charged particle (like an electron orbiting the nucleus) should continuously emit electromagnetic radiation. This emission would cause the electron to lose energy, slow down, and spiral inwards, eventually collapsing into the nucleus. Calculations showed this process should take a fraction of a second ($\approx 10^{-8}$ s), yet atoms are stable.
• Cannot explain the distribution and energies of electrons: Rutherford's model did not specify how electrons are arranged around the nucleus or what their specific energy levels might be. It only described them as orbiting the nucleus like planets.

These limitations highlighted the need for a new atomic model that could explain both the stability of atoms and their characteristic spectra.

Developments Leading To The Bohr’s Model Of Atom

The limitations of Rutherford's model paved the way for new theoretical developments. Niels Bohr's model of the hydrogen atom was significantly influenced by insights gained from the study of how radiation interacts with matter. Two major concepts were particularly important:

1. The dual nature of electromagnetic radiation (behaving as both waves and particles).
2. Experimental data on atomic spectra.

---

Wave Nature Of Electromagnetic Radiation

By the mid-19th century, physicists were actively studying thermal radiation emitted by heated objects. James Clerk Maxwell unified the theories of electricity and magnetism in the 1870s, proposing that accelerating electric charges produce and transmit alternating electric and magnetic fields. These disturbances propagate as electromagnetic waves or electromagnetic radiation.

Light, the most familiar form of radiation, was shown by Maxwell to be an electromagnetic wave, consisting of oscillating electric and magnetic fields.

Properties of electromagnetic waves:

• The oscillating electric and magnetic fields are mutually perpendicular to each other and also perpendicular to the direction the wave travels.
• They do not require a medium to propagate and can travel through a vacuum (unlike sound waves).
• There are many types of electromagnetic radiation, differing in their wavelengths ($\lambda$) and frequencies ($\nu$). This range of wavelengths/frequencies constitutes the electromagnetic spectrum.

Different regions of the spectrum have different names, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. The visible light spectrum is a small portion that our eyes can detect.

Electromagnetic radiation is characterised by:

• Frequency ($\nu$): The number of waves passing a point per second. SI unit is Hertz (Hz or s$^{-1}$).
• Wavelength ($\lambda$): The distance between two consecutive crests or troughs. SI unit is meter (m), though often smaller units like nanometer (nm) or angstrom (Å) are used.
• Velocity (c): In a vacuum, all electromagnetic radiation travels at the speed of light, $c = 3.0 \times 10^8$ m s$^{-1}$ (approx.).

These properties are related by the equation:

$c = \nu \lambda$

Another quantity used, especially in spectroscopy, is the wavenumber ($\bar{\nu}$), defined as the number of wavelengths per unit length. It's the reciprocal of wavelength:

$\bar{\nu} = \frac{1}{\lambda}$

SI unit for wavenumber is m$^{-1}$, but cm$^{-1}$ is also commonly used.

---

Answer:

---

Answer:

---

Answer:

---

Particle Nature Of Electromagnetic Radiation: Planck’s Quantum Theory

While wave nature explains phenomena like diffraction and interference, some experimental observations were not explained by classical wave theory. These included:

• Black-body radiation: The pattern of radiation emitted by ideal heated objects.
• Photoelectric effect: The ejection of electrons from a metal surface by light.
• The variation of heat capacity of solids with temperature.
• The line spectra of atoms (emission of light at specific wavelengths).

These phenomena suggested that energy is not continuously variable but can only be taken up or released in discrete amounts.

Max Planck provided a key explanation for black-body radiation in 1900. A hot object emits radiation over a range of wavelengths, and the intensity distribution changes with temperature. An ideal emitter/absorber is called a black body.

Classical physics failed to predict this intensity distribution accurately. Planck proposed that energy is emitted or absorbed by atoms in discrete bundles, not continuously. He called the smallest such bundle a quantum. The energy ($E$) of a quantum of radiation is directly proportional to its frequency ($\nu$).

$E = h\nu$

where $h$ is Planck's constant, with a value of $6.626 \times 10^{-34}$ J s.

This idea of energy quantisation is like climbing a staircase – you can only stand on the steps, not in between. Energy can only take specific values, $0, h\nu, 2h\nu, 3h\nu, \dots, nh\nu$.

Photoelectric Effect

In 1887, Heinrich Hertz observed that shining light on certain metal surfaces (like potassium, rubidium, caesium) caused electrons to be ejected. This is the photoelectric effect.

Key observations from the photoelectric effect:

1. Electron ejection is instantaneous when light strikes the surface; there's no time delay.
2. The number of ejected electrons is proportional to the intensity (brightness) of the light.
3. For each metal, there is a minimum frequency, called the threshold frequency ($\nu_0$), below which no electrons are ejected, regardless of the light intensity.
4. If the light frequency ($\nu$) is greater than the threshold frequency ($\nu_0$), the ejected electrons have kinetic energy. This kinetic energy increases linearly with the frequency of the light used, but is independent of the intensity of the light.

Classical physics predicted that electron ejection and kinetic energy should depend on light intensity, contradicting observations (especially the threshold frequency). Albert Einstein explained the photoelectric effect in 1905 using Planck's quantum theory.

Einstein proposed that light consists of a stream of particles called photons, each with energy $E = h\nu$. When a photon hits an electron in the metal, it transfers its energy to the electron. If the photon's energy ($h\nu$) is greater than the minimum energy required to remove an electron from the metal surface (called the work function, $W_0 = h\nu_0$), the electron is ejected. The excess energy becomes the kinetic energy of the ejected electron.

According to the law of conservation of energy:

Energy of incoming photon = Work function + Kinetic energy of ejected electron

$h\nu = W_0 + \frac{1}{2}m_e v^2$

or

$h\nu = h\nu_0 + \frac{1}{2}m_e v^2$

A more intense light beam has more photons, so it ejects more electrons, but the energy of each photon (and thus the maximum kinetic energy of each electron) depends only on the frequency, not the intensity.

Metal	Li	Na	K	Mg	Cu	Ag
$W_0$ / eV	2.42	2.3	2.25	3.7	4.8	4.3

Dual Behaviour of Electromagnetic Radiation

The successful explanation of phenomena like black-body radiation and the photoelectric effect using the particle nature of light, alongside phenomena like interference and diffraction explained by wave nature, led scientists to accept that light (and all electromagnetic radiation) exhibits dual behaviour – it can behave as both a wave and a stream of particles, depending on the experiment.

---

Answer:

---

Answer:

---

Answer:

---

Answer:

---

Evidence For The Quantized Electronic Energy Levels: Atomic Spectra

When white light passes through a prism, it splits into its constituent colors (spectrum) because the different wavelengths bend at different angles. This is a continuous spectrum because all wavelengths in the visible range are present and merge into each other.

However, when atoms in the gas phase are heated or subjected to electrical discharge, they emit light only at specific, discrete wavelengths. This produces a line emission spectrum, seen as bright lines on a dark background.

Conversely, if white light is passed through a gas of the same atoms, the atoms absorb light at those specific wavelengths they would normally emit. The resulting spectrum is a line absorption spectrum, showing dark lines at the same positions where the emission spectrum had bright lines (it's like a photographic negative).

The study of these spectra is called spectroscopy. Atomic line spectra are unique to each element, serving as a 'fingerprint' for identification. The discovery of elements like rubidium and caesium was made using spectroscopy.

Line Spectrum of Hydrogen

The hydrogen atom has the simplest line spectrum. When hydrogen gas is excited, it emits light at discrete frequencies. The hydrogen spectrum consists of several series of lines in different regions of the electromagnetic spectrum.

Johann Balmer discovered a formula in 1885 that described the wavelengths of the visible lines (Balmer series):

$\bar{\nu} = R_H (\frac{1}{2^2} - \frac{1}{n^2})$, where $n = 3, 4, 5, \dots$

Here, $\bar{\nu}$ is the wavenumber and $R_H$ is a constant (Rydberg constant for Hydrogen).

Johannes Rydberg later generalised this, finding that all series in the hydrogen spectrum fit the formula:

$\bar{\nu} = R_H (\frac{1}{n_1^2} - \frac{1}{n_2^2})$

where $n_1 = 1, 2, 3, \dots$ and $n_2 = n_1 + 1, n_1 + 2, \dots$.

The value of the Rydberg constant $R_H$ is approximately $109,677$ cm$^{-1}$.

Different values of $n_1$ correspond to different series:

Series	$n_1$	$n_2$	Spectral Region
Lyman	1	2, 3, 4, ...	Ultraviolet
Balmer	2	3, 4, 5, ...	Visible
Paschen	3	4, 5, 6, ...	Infrared
Bracket	4	5, 6, 7, ...	Infrared
Pfund	5	6, 7, 8, ...	Infrared
Humphreys	6	7, 8, 9, ...	Infrared

The existence of discrete lines in atomic spectra is strong evidence that electrons within atoms can only occupy specific energy levels, implying that their energy is quantised.

Bohr’s Model For Hydrogen Atom

In 1913, Niels Bohr combined Rutherford's nuclear model with Planck's quantum theory to propose a model for the hydrogen atom that could explain its stability and line spectrum quantitatively.

Bohr's model is based on the following postulates:

1. Quantised Orbits: Electrons revolve around the nucleus in specific circular paths called orbits, stationary states, or allowed energy states. These orbits have fixed radii and fixed energies. They are arranged concentrically around the nucleus.
2. Stationary States: Electrons in these specific orbits do not radiate energy, so their energy remains constant. Energy is only absorbed or emitted when an electron moves from one stationary state to another.
3. Energy Transitions: When an electron moves from a lower energy orbit ($E_1$) to a higher energy orbit ($E_2$), energy is absorbed ($\Delta E = E_2 - E_1 > 0$). When it moves from a higher energy orbit to a lower one, energy is emitted ($\Delta E = E_1 - E_2 < 0$). The frequency ($\nu$) of the emitted or absorbed radiation is given by Bohr's frequency rule:

   $\nu = \frac{|\Delta E|}{h} = \frac{|E_2 - E_1|}{h}$

   This implies that energy transitions are not continuous but occur only between discrete energy levels.
4. Quantisation of Angular Momentum: The angular momentum of an electron in a stationary orbit is quantised. It must be an integral multiple of $\frac{h}{2\pi}$:

   $m_e v r = n \frac{h}{2\pi}$, where $n = 1, 2, 3, \dots$

   $m_e$ is the electron mass, $v$ is its velocity, $r$ is the orbit radius, and $n$ is the principal quantum number (an integer). This condition selects only certain allowed orbits out of many classically possible ones.

According to Bohr's theory for the hydrogen atom (derivation complex, based on balancing electrostatic attraction and centripetal force, plus the angular momentum quantisation):

• The stationary states are numbered by the principal quantum number ($n$), where $n = 1, 2, 3, \dots$. $n=1$ is the lowest energy state (ground state).
• The radii of the stationary states are given by $r_n = n^2 a_0$, where $a_0$ is the Bohr radius ($52.9$ pm for $n=1$). The radius increases as $n$ increases.
• The energy of the stationary state is given by $E_n = -\frac{R_H}{n^2}$, where $R_H = 2.18 \times 10^{-18}$ J. The energy levels are negative, indicating the electron is bound to the nucleus. The lowest energy ($n=1$) is the most negative and thus most stable. As $n \rightarrow \infty$, $E_n \rightarrow 0$, which corresponds to a free electron (ionised atom).

Bohr's theory also applies to hydrogen-like species (ions with only one electron, like $\text{He}^+, \text{Li}^{2+}$). For these, the energy and radius depend on the atomic number ($Z$):

$E_n = -R_H \frac{Z^2}{n^2}$ J/atom

$r_n = a_0 \frac{n^2}{Z}$

Increasing $Z$ makes the energy more negative (tighter binding) and the radius smaller.

---

Explanation Of Line Spectrum Of Hydrogen

Bohr's model explained the observed line spectrum of hydrogen. Transitions between energy levels lead to the absorption or emission of photons with specific frequencies.

The energy difference between an initial state ($n_i$) and a final state ($n_f$) is:

$\Delta E = E_{n_f} - E_{n_i} = (-\frac{R_H}{n_f^2}) - (-\frac{R_H}{n_i^2}) = R_H (\frac{1}{n_i^2} - \frac{1}{n_f^2})$

= $2.18 \times 10^{-18} \text{ J } (\frac{1}{n_i^2} - \frac{1}{n_f^2})$

For emission ($n_i > n_f$), $\Delta E$ is negative, indicating energy is released as a photon. For absorption ($n_f > n_i$), $\Delta E$ is positive, meaning energy is absorbed.

The frequency ($\nu$) of the photon is $\nu = \frac{|\Delta E|}{h}$.

The wavenumber ($\bar{\nu}$) is $\bar{\nu} = \frac{1}{\lambda} = \frac{\nu}{c} = \frac{|\Delta E|}{hc} = \frac{R_H}{hc} (\frac{1}{n_i^2} - \frac{1}{n_f^2})$ (for $n_i > n_f$ in emission) or $\frac{R_H}{hc} (\frac{1}{n_f^2} - \frac{1}{n_i^2})$ (for $n_f > n_i$ in absorption).

Substituting the values of $R_H$, $h$, and $c$, the factor $\frac{R_H}{hc}$ matches the Rydberg constant for hydrogen ($\approx 1.09677 \times 10^7 \text{ m}^{-1}$ or $109677 \text{ cm}^{-1}$). This quantitatively explains the Rydberg formula and the existence of spectral series (Lyman, Balmer, etc.), where each line corresponds to a specific transition between two discrete energy levels.

The intensity of spectral lines relates to the number of photons emitted or absorbed during those specific transitions.

---

Answer:

---

Answer:

---

Limitations Of Bohr’s Model

While Bohr's model successfully explained the hydrogen atom and hydrogen-like ions, it had limitations:

• It could not explain the spectra of multi-electron atoms (atoms with more than one electron).
• It failed to account for the finer details (like the splitting into closely spaced lines, or doublets) observed in the hydrogen spectrum using high-resolution spectroscopes.
• It could not explain the Zeeman effect (splitting of spectral lines in a magnetic field) or the Stark effect (splitting in an electric field).
• It did not explain how atoms form chemical bonds to create molecules.

These limitations showed that Bohr's model, despite its success for hydrogen, was not a complete description of atomic structure and behavior, highlighting the need for a more advanced theory.

Towards Quantum Mechanical Model Of The Atom

The development of a more comprehensive atomic model was influenced by two significant ideas that emerged after Bohr's model:

1. The confirmation of the dual behaviour of matter (that particles also exhibit wave-like properties).
2. The formulation of the Heisenberg uncertainty principle.

---

Dual Behaviour Of Matter

Inspired by the wave-particle duality of light, French physicist Louis de Broglie proposed in 1924 that all matter, including particles like electrons, should also exhibit dual nature – possessing both particle-like and wave-like properties.

He suggested a relationship between the wavelength ($\lambda$) associated with a moving particle and its momentum ($p$). This is the de Broglie relation:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

where $h$ is Planck's constant, $m$ is the mass of the particle, and $v$ is its velocity.

De Broglie's hypothesis was experimentally confirmed a few years later when electron beams were shown to undergo diffraction, a phenomenon characteristic of waves. This wave nature of electrons is used in devices like the electron microscope, which achieves much higher magnifications than light microscopes.

According to de Broglie, every object in motion has a wavelength. However, for macroscopic objects (like a baseball or a car), the mass ($m$) is very large, making the de Broglie wavelength ($\lambda = h/mv$) extremely small – too small to be detected or have any significant effect on their behavior. For microscopic particles like electrons, protons, or neutrons, the mass is very small, resulting in a detectable wavelength that influences their behavior significantly.

---

Answer:

---

Answer:

---

Answer:

---

Heisenberg’s Uncertainty Principle

The wave-particle duality of matter has a profound consequence, described by Werner Heisenberg's Uncertainty Principle (1927):

It is impossible to determine simultaneously and precisely both the exact position and the exact momentum (or velocity) of a sub-atomic particle like an electron.

Mathematically, this is expressed as:

$\Delta x \cdot \Delta p_x \ge \frac{h}{4\pi}$

where:

• $\Delta x$ is the uncertainty in the position of the particle along the x-axis.
• $\Delta p_x$ is the uncertainty in the momentum along the x-axis ($\Delta p_x = m \Delta v_x$).
• $h$ is Planck's constant.

This principle means that the more accurately we know the position of an electron ($\Delta x$ is small), the less accurately we know its momentum or velocity ($\Delta p_x$ or $\Delta v_x$ is large), and vice versa. Our measurements are inherently limited; we get a "fuzzy" picture of the electron's state.

To illustrate, imagining trying to find an electron's position using light: The light used must have a wavelength smaller than the electron's "size" to locate it precisely. Photons of such short wavelength have high energy and momentum ($p=h/\lambda$). When a photon collides with the electron to determine its position, it transfers some of its momentum, inevitably changing the electron's velocity. So, while we might locate the electron, we lose certainty about its velocity after the observation.

Significance of Uncertainty Principle

The uncertainty principle is not noticeable for macroscopic objects. For a large object, even if there's a small uncertainty in momentum, because the mass is large, the uncertainty in velocity ($\Delta v_x = \Delta p_x / m$) becomes extremely small, and the uncertainty in position ($\Delta x$) remains effectively negligible for everyday purposes. Classical mechanics and the concept of a precise trajectory work perfectly for these objects.

However, for microscopic particles like electrons, the mass is tiny. A small uncertainty in momentum translates to a large uncertainty in velocity. Consequently, it's impossible to define a precise path or trajectory for an electron in an atom. The idea of electrons moving in fixed, well-defined orbits (like in Bohr's model) is incompatible with the uncertainty principle and the wave nature of matter.

Instead of trajectories, quantum mechanics describes the probability of finding an electron in a certain region of space around the nucleus.

---

Answer:

---

Answer:

---

The limitations of the Bohr model (ignoring wave nature and contradicting uncertainty principle) highlighted the need for a new theory that could account for the dual behaviour of matter and be consistent with the uncertainty principle. This led to the development of quantum mechanics.

Quantum Mechanical Model Of Atom

Classical mechanics, based on Newton's laws, successfully describes the motion of macroscopic objects. However, it fails when applied to microscopic particles because it doesn't account for their wave-like properties or the uncertainty principle. Quantum mechanics is the scientific framework that describes the motion and behavior of these microscopic objects, incorporating their dual nature.

Quantum mechanics was developed independently by Werner Heisenberg and Erwin Schrödinger in 1926. Schrödinger's approach, based on wave motion, is particularly relevant to understanding atomic structure. He developed the fundamental equation of quantum mechanics, the Schrödinger equation, which incorporates the wave-particle duality proposed by de Broglie.

For a system whose energy doesn't change over time (like an atom in a stable state), the time-independent Schrödinger equation is written as:

$\hat{H}\psi = E\psi$

where:

• $\hat{H}$ is the Hamiltonian operator, representing the total energy (kinetic + potential) of the system's particles.
• $\psi$ (psi) is the wave function, a mathematical function that describes the state of the system.
• $E$ is the total energy of the system.

Solving the Schrödinger equation for an atom provides the possible energy levels that electrons can occupy and the corresponding wave functions ($\psi$) for each energy level.

Hydrogen Atom and the Schrödinger Equation

Solving the Schrödinger equation for the hydrogen atom yields specific, quantised energy states and corresponding wave functions. These solutions are characterised by a set of three numbers called quantum numbers (principal $n$, azimuthal $l$, and magnetic $m_l$). The quantisation of energy levels arises naturally from the mathematics of the wave equation.

The wave function ($\psi$) itself doesn't have a direct physical meaning. However, the square of the wave function, $|\psi|^2$, at a particular point in space gives the probability density of finding the electron at that point. $|\psi|^2$ is always a positive value.

For one-electron systems like hydrogen or He$^+$, these wave functions are called atomic orbitals. Each atomic orbital corresponds to a specific energy level and describes a region in space where the probability of finding the electron is high.

For multi-electron atoms, the Schrödinger equation is more complex and cannot be solved exactly. Approximate methods are used, which show that the orbitals in multi-electron atoms are similar to hydrogen orbitals but are affected by the increased nuclear charge and electron-electron repulsions. Notably, in multi-electron atoms, the energy of an orbital depends not just on the principal quantum number ($n$) but also on the azimuthal quantum number ($l$).

Important Features of the Quantum Mechanical Model of Atom

The quantum mechanical model provides a more accurate description of atomic structure:

1. Electron energy in atoms is quantised (can only have specific discrete values).
2. The existence of these quantised energy levels is a direct result of the electron's wave-like properties, as described by the Schrödinger equation.
3. It's impossible to determine the exact position and velocity of an electron simultaneously (Heisenberg Uncertainty Principle). Therefore, the concept of a definite electron path or trajectory is replaced by the concept of probability.
4. An atomic orbital is the wave function ($\psi$) for an electron in an atom. It represents a region of space where the electron is likely to be found, not a fixed path. An atom can have many atomic orbitals, each corresponding to a different possible state for an electron. Each orbital can hold a maximum of two electrons with opposite spins. In multi-electron atoms, electrons occupy these orbitals starting from the lowest energy ones.
5. $|\psi|^2$ represents the probability density of finding the electron at a point. A boundary surface enclosing a region where the probability of finding the electron is high (e.g., 90%) is often used to depict the shape of an orbital.

---

Orbitals And Quantum Numbers

Atomic orbitals can be distinguished by their size, shape, and orientation in space. These characteristics are defined by a set of three quantum numbers ($n, l, m_l$) obtained from the solution of the Schrödinger equation for the hydrogen atom.

• Principal Quantum Number ($n$):
  • Represents the main energy level or shell the electron occupies.
  • Can be any positive integer: $n = 1, 2, 3, \dots$.
  • Determines the size and primarily the energy of the orbital. Higher $n$ means a larger orbital, further from the nucleus, and higher energy.
  • Shells are also designated by letters: K ($n=1$), L ($n=2$), M ($n=3$), N ($n=4$), etc.
  • The total number of orbitals within a shell $n$ is $n^2$.
• Azimuthal Quantum Number ($l$):
  • Also called the orbital angular momentum quantum number or subsidiary quantum number.
  • Defines the shape of the orbital (and thus the subshell).
  • For a given $n$, $l$ can have integer values from 0 to $n-1$. So, there are $n$ possible $l$ values for a given $n$.
  • Different $l$ values correspond to different subshells:
    • $l=0$: s subshell (spherical shape)
    • $l=1$: p subshell (dumbbell shape)
    • $l=2$: d subshell (more complex shapes)
    • $l=3$: f subshell (even more complex shapes)
    • And so on (g, h, etc. for $l=4, 5, \dots$)
• Magnetic Orbital Quantum Number ($m_l$):
  • Describes the spatial orientation of the orbital in a magnetic field.
  • For a given $l$, $m_l$ can have integer values from $-l$ through 0 to $+l$.
  • The number of possible $m_l$ values for a given $l$ is $2l+1$. This is the number of orbitals within a subshell:
    • $l=0$ (s subshell): $m_l = 0$ (1 s orbital)
    • $l=1$ (p subshell): $m_l = -1, 0, +1$ (3 p orbitals)
    • $l=2$ (d subshell): $m_l = -2, -1, 0, +1, +2$ (5 d orbitals)
    • $l=3$ (f subshell): $m_l = -3, -2, -1, 0, +1, +2, +3$ (7 f orbitals)
• Electron Spin Quantum Number ($m_s$):
  • An intrinsic property of the electron, like mass and charge.
  • Represents the electron's inherent angular momentum, often visualised as spinning on its axis.
  • Can only have two possible values: $+\frac{1}{2}$ or $-\frac{1}{2}$, representing the two possible spin orientations (often denoted by arrows $\uparrow$ and $\downarrow$).

These four quantum numbers ($n, l, m_l, m_s$) completely describe the state of an electron in an atom. The Pauli Exclusion Principle states that no two electrons in the same atom can have the exact same set of all four quantum numbers. This limits the capacity of orbitals and shells.

Quantum Number	Symbol	Allowed Values	Describes
Principal	$n$	1, 2, 3, ...	Shell, Size, Main energy level
Azimuthal (Orbital angular momentum)	$l$	0, 1, ..., $n-1$	Subshell, Shape
Magnetic	$m_l$	$-l, ..., 0, ..., +l$	Orientation
Spin	$m_s$	$+1/2, -1/2$	Electron spin orientation

Maximum number of electrons:

• In an orbital: 2 (with opposite spins).
• In a subshell $l$: $2 \times (2l+1)$.
• In a shell $n$: $2 \times n^2$.

For example, a shell with $n=2$ has $l=0$ and $l=1$. $l=0$ has $2(0)+1=1$ orbital (2s), holding 2 electrons. $l=1$ has $2(1)+1=3$ orbitals (2p), holding $3 \times 2 = 6$ electrons. Total electrons in $n=2$ shell = $2+6=8$, which is $2 \times 2^2 = 8$.

Orbit vs. Orbital: It's crucial to distinguish between Bohr's concept of a circular orbit (a definite path) and the quantum mechanical concept of an orbital (a region of probability). Due to the uncertainty principle, a definite trajectory for an electron cannot exist. An atomic orbital is a mathematical description ($\psi$) related to the probability of finding an electron in a certain volume of space around the nucleus.

---

Answer:

---

Answer:

---

Shapes Of Atomic Orbitals

While the wave function $\psi$ itself has no physical meaning, the probability density $|\psi|^2$ is important. Plots of $|\psi|^2$ show the regions where an electron is most likely to be found.

For 1s and 2s orbitals, the probability density varies with the distance from the nucleus. For 1s, it's highest at the nucleus and decreases outwards. For 2s, it decreases to zero, increases to a small peak, and then decreases again. The region where the probability density is zero is called a node or nodal surface. For ns orbitals, there are $(n-1)$ radial nodes.

To visualize orbital shapes, boundary surface diagrams are used. These surfaces enclose a region where the probability of finding the electron is high (e.g., 90%).

s Orbitals (l=0):

• s orbitals are spherically symmetric. The probability of finding the electron depends only on the distance from the nucleus, not the direction.
• The boundary surface diagram of an s orbital is a sphere centered on the nucleus.
• The size of s orbitals increases with the principal quantum number $n$ (e.g., 4s > 3s > 2s > 1s).

p Orbitals (l=1):

• p orbitals are not spherically symmetric. They have a dumbbell shape consisting of two lobes on either side of a plane passing through the nucleus.
• The plane where the two lobes meet and the probability density is zero is a nodal plane.
• For $l=1$, there are $2l+1 = 3$ p orbitals, oriented along the x, y, and z axes, designated as $p_x$, $p_y$, and $p_z$. They are identical in shape, size, and energy but differ in orientation.
• The size and energy of p orbitals increase with increasing $n$ (e.g., 4p > 3p > 2p).
• Number of radial nodes for np orbitals = $n-2$. Number of angular nodes = $l = 1$. Total nodes = $(n-1)$.

d Orbitals (l=2):

• d orbitals exist starting from $n=3$ (since $l \le n-1$).
• For $l=2$, there are $2l+1 = 5$ d orbitals.
• Four of the d orbitals ($d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}$) have similar shapes, typically depicted as four lobes. The fifth orbital ($d_{z^2}$) has a distinct shape (two lobes along the z-axis and a torus around the center).
• In an isolated atom, the five d orbitals within the same subshell are equivalent in energy (degenerate).
• The size and energy of d orbitals increase with increasing $n$.
• Number of radial nodes for nd orbitals = $n-3$. Number of angular nodes = $l = 2$. Total nodes = $(n-1)$.

In general, the total number of nodes in an orbital is $(n-1)$, which is the sum of radial nodes $(n-l-1)$ and angular nodes ($l$).

---

Energies Of Orbitals

The energy of an electron in an orbital depends on the attraction between the electron and the nucleus and the repulsion between the electron and other electrons.

• Hydrogen Atom (One-electron species): The energy of an orbital depends ONLY on the principal quantum number ($n$). For a given $n$, all orbitals ($s, p, d, f, \dots$) have the same energy; they are degenerate. Energy increases as $1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < \dots$.

• Multi-electron Atoms: The energy of an orbital depends on BOTH $n$ and $l$. This is due to electron-electron repulsions and the shielding effect. Inner electrons shield the outer electrons from the full positive charge of the nucleus. The net positive charge experienced by an outer electron is the effective nuclear charge ($Z_{eff}$), which is less than the actual nuclear charge ($Z$).

Different subshells within the same shell ($n$) experience different amounts of shielding. Electrons in s orbitals penetrate closer to the nucleus than p, d, or f electrons in the same shell. This means s electrons experience a higher $Z_{eff}$ and are more strongly attracted, having lower energy than p electrons, which have lower energy than d electrons, and so on.

Within a given shell, the order of increasing energy is typically:

$s < p < d < f$

For higher principal quantum numbers, energy levels of different shells can overlap or stagger (e.g., the 4s orbital has lower energy than the 3d orbitals in many atoms).

A useful rule to estimate the relative energies of orbitals in multi-electron atoms is the $(n+l)$ rule:

The orbital with the lower value of $(n+l)$ has lower energy. If two orbitals have the same $(n+l)$ value, the one with the lower principal quantum number ($n$) has lower energy.

Orbital	$n$	$l$	$n+l$	Order of Increasing Energy
1s	1	0	1	1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d ...
2s	2	0	2
2p	2	1	3
3s	3	0	3
3p	3	1	4
4s	4	0	4
3d	3	2	5
4p	4	1	5
5s	5	0	5
4d	4	2	6
5p	5	1	6

(Note: When $n+l$ is the same, e.g., 2p ($n=2, l=1, n+l=3$) and 3s ($n=3, l=0, n+l=3$), the orbital with lower $n$ (2p) has lower energy than the one with higher $n$ (3s)).

Energy of orbitals within the same subshell decreases with increasing atomic number ($Z$), as the nuclear attraction increases (though $Z_{eff}$ is affected by shielding).

---

Filling Of Orbitals In Atom

The distribution of electrons among the orbitals of an atom is called its electronic configuration. This filling process follows three fundamental rules:

1. Aufbau Principle: Electrons fill orbitals in the order of increasing energy. Lower energy orbitals are filled before higher energy ones. The approximate order of increasing energy (and thus filling order) is: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, \dots$ This order can be remembered using a diagram.

2. Pauli Exclusion Principle: No two electrons in the same atom can have the identical set of all four quantum numbers ($n, l, m_l, m_s$). This implies that an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins (one with $m_s = +1/2$, the other with $m_s = -1/2$).
3. Hund’s Rule of Maximum Multiplicity: For orbitals within the same subshell (which are degenerate, meaning they have the same energy, e.g., the three 2p orbitals), electron pairing does not occur until each orbital in the subshell is singly occupied. When orbitals are singly occupied, the electrons in them have parallel spins (all $+\frac{1}{2}$ or all $-\frac{1}{2}$) as far as possible. This arrangement minimises electron-electron repulsion and results in greater stability.

Applying these rules allows us to determine the electronic configuration of any atom.

---

Electronic Configuration Of Atoms

Electronic configuration can be written in two ways:

1. $nl^x$ notation: The principal quantum number ($n$) and subshell letter ($l$) are followed by a superscript ($x$) indicating the number of electrons in that subshell (e.g., $1s^2$).
2. Orbital diagram: Each orbital is represented by a box or a line. Electrons are represented by arrows ($\uparrow$ for spin $+1/2$, $\downarrow$ for spin $-1/2$). Paired electrons in an orbital are shown with opposite arrows ($\uparrow\downarrow$). Electrons in different orbitals of the same subshell are shown with parallel arrows ($\uparrow \uparrow \uparrow$) according to Hund's rule before pairing occurs ($\uparrow\downarrow \uparrow \uparrow$).

Examples:

• Hydrogen (Z=1): 1 electron. Lowest energy orbital is 1s. Configuration: $1s^1$. Orbital diagram: $\boxed{\uparrow}_{1s}$
• Helium (Z=2): 2 electrons. Both can go in 1s (Pauli principle allows opposite spins). Configuration: $1s^2$. Orbital diagram: $\boxed{\uparrow\downarrow}_{1s}$
• Lithium (Z=3): 3 electrons. $1s$ is full. Next lowest is 2s. Configuration: $1s^2 2s^1$. Orbital diagram: $\boxed{\uparrow\downarrow}_{1s} \boxed{\uparrow}_{2s}$
• Carbon (Z=6): 6 electrons. $1s^2 2s^2$. Remaining 2 electrons go in 2p. 2p has 3 orbitals. According to Hund's rule, they go into separate orbitals with parallel spins. Configuration: $1s^2 2s^2 2p^2$. Orbital diagram: $\boxed{\uparrow\downarrow}_{1s} \boxed{\uparrow\downarrow}_{2s} \boxed{\uparrow}_{2p_x} \boxed{\uparrow}_{2p_y} \boxed{\phantom{\uparrow}}_{2p_z}$

Electrons in the outermost shell (highest $n$) are called valence electrons, as they are involved in chemical bonding. Electrons in inner, filled shells are called core electrons. Electronic configurations can be abbreviated by using the noble gas configuration for the core electrons (e.g., [He] represents $1s^2$, [Ne] represents $1s^2 2s^2 2p^6$).

Filling continues following the energy order: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow \dots$. The 4s orbital is filled before 3d because it has lower energy according to the $(n+l)$ rule ($4+0=4$ for 4s, $3+2=5$ for 3d). This explains why Potassium ([Ar]$4s^1$) and Calcium ([Ar]$4s^2$) fill the 4s orbital.

Then, the 3d orbitals begin filling for the transition metals (Sc to Zn). Exceptions occur where small energy differences and the stability of half-filled or fully-filled subshells lead to slightly different configurations (e.g., Chromium: [Ar]$3d^5 4s^1$ instead of [Ar]$3d^4 4s^2$; Copper: [Ar]$3d^{10} 4s^1$ instead of [Ar]$3d^9 4s^2$).

Filling proceeds through the p orbitals (Ga to Kr), then the 5s, 4d, 5p, 6s, 4f (Lanthanides), 5d, 6p, 7s, 5f (Actinides), 6d, and 7p.

Understanding electronic configuration is fundamental in chemistry because it explains the chemical behavior and properties of elements, such as why some elements are reactive while others are inert, and how atoms form bonds.

See Table 2.6 for electronic configurations of elements (as provided in the original text).

---

Stability Of Completely Filled And Half Filled Subshells

Electronic configurations with completely filled or exactly half-filled subshells ($p^3, p^6, d^5, d^{10}, f^7, f^{14}$) tend to have extra stability compared to configurations with one electron more or less than half-filled/fully-filled. This extra stability is primarily due to two factors:

1. Symmetrical Distribution of Electrons: Configurations like $s^2, p^3, p^6, d^5, d^{10}, f^7, f^{14}$ have electrons symmetrically distributed within the degenerate orbitals of the subshell. This symmetry contributes to stability. Electrons in the same subshell have similar energies and spatial distributions. Their shielding of each other is relatively effective, but compared to unevenly filled subshells, the electron-electron repulsion is somewhat minimized, leading to slightly stronger attraction to the nucleus on average.
2. Exchange Energy: This stabilizing effect arises when electrons with the same spin can exchange positions within a set of degenerate orbitals. The energy released during these exchanges is called exchange energy. The more possible exchanges, the greater the exchange energy and the higher the stability. The number of possible exchanges is maximised when the subshell is exactly half-filled or completely filled.

For a $d^5$ configuration (5 electrons in 5 degenerate d orbitals, each with parallel spin), each electron can exchange positions with any of the other 4 electrons with the same spin. The number of such exchanges for electrons with the same spin is given by $\frac{n_e (n_e - 1)}{2}$, where $n_e$ is the number of electrons with parallel spins in the degenerate set. For $d^5$, if all 5 have parallel spin, exchanges = $\frac{5(5-1)}{2} = 10$. For a $d^{10}$ configuration, there are 5 spin-up and 5 spin-down electrons. Exchanges for spin-up = $\frac{5(5-1)}{2} = 10$. Exchanges for spin-down = $\frac{5(5-1)}{2} = 10$. Total exchanges = 20. For a $d^9$ configuration (say, 5 up, 4 down), exchanges = $\frac{5(4)}{2} + \frac{4(3)}{2} = 10 + 6 = 16$, less than $d^{10}$.

Hund's rule itself is a consequence of this exchange energy; placing electrons in separate degenerate orbitals with parallel spins maximises the number of same-spin electrons, thus maximising stabilizing exchange energy.

In summary, the extra stability of half-filled and completely filled subshells is attributed to their symmetrical electron distribution, relatively lower electron-electron repulsion, and maximum exchange energy.

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

Answer: