Structure of atom-Notes

In the middle of the nineteenth century, scientists became increasingly interested in understanding the nature of electricity and matter. Around the 1850s, pioneers such as Michael Faraday carried out experiments on the passage of electricity through gases at low pressure. For this purpose, they used specially designed glass tubes called cathode ray discharge tubes. These tubes contained two metal plates, known as electrodes, sealed inside the glass. One electrode was connected to the negative terminal (cathode) and the other to the positive terminal (anode) of a high-voltage source.

It was observed that electrical discharge through gases occurs only when the gas inside the tube is maintained at very low pressure and a very high voltage is applied across the electrodes. The pressure inside the tube could be reduced by evacuating air from it. When the applied voltage became sufficiently high, a stream of particles was seen moving from the cathode (negative electrode) toward the anode (positive electrode). These streams were called cathode rays.

To confirm the direction and presence of these rays, scientists modified the experimental setup. A small hole was made in the anode, and the glass wall behind it was coated with a fluorescent material such as zinc sulphide. When cathode rays passed through the hole in the anode and struck the zinc sulphide coating, a bright glowing spot appeared. This glow provided clear evidence of the movement of particles from the cathode toward the anode.

The key observations from these experiments can be summarized as follows:

• Origin and direction:
  Cathode rays originate from the cathode and travel toward the anode.
• Invisibility and detection:
  Cathode rays themselves are invisible. However, their presence can be detected because they cause certain materials (fluorescent or phosphorescent substances) to glow when struck. Modern television picture tubes are based on the same principle; images are produced due to fluorescence on a screen coated with suitable materials.
• Straight-line motion:
  In the absence of electric and magnetic fields, cathode rays travel in straight lines.
• Nature of charge:
  When subjected to electric or magnetic fields, cathode rays behave as negatively charged particles. They are deflected toward the positive plate in an electric field. This behaviour proved that cathode rays consist of negatively charged particles, later named electrons.
• Independence from gas and electrodes:
  The properties of cathode rays remain the same regardless of the type of gas present in the tube or the material used for the electrodes.

From these consistent experimental results, scientists concluded that electrons are fundamental particles present in all atoms. This discovery marked a major milestone in the development of atomic structure theory and laid the foundation for modern atomic physics.

After the discovery of cathode rays, scientists were keen to determine the fundamental properties of the particles present in these rays. One of the most significant breakthroughs was the measurement of the charge to mass ratio \(\left(\frac{e}{m}\right)\) of the electron. This was accomplished through carefully designed experiments carried out by J. J. Thomson in the late nineteenth century.

Experimental Approach

Thomson used a discharge tube similar to the one employed in cathode ray experiments. When a high potential difference was applied across the electrodes under low pressure, a beam of cathode rays was produced. These rays were allowed to pass through both electric and magnetic fields arranged perpendicular to each other and also perpendicular to the path of the beam.

The central idea was simple yet powerful:

• An electric field exerts a force on a charged particle.
• A magnetic field also exerts a force when the particle is in motion.

By adjusting the strengths of these fields, Thomson observed how the cathode ray beam was deflected. When the electric and magnetic forces were made equal and opposite, the beam travelled in a straight line without any deflection. Under this balanced condition:

If \(E\) is the electric field strength and \(B\) is the magnetic field strength, and \(v\) is the velocity of the particles, then: \[ \begin{aligned} eE&=evB\\ v&=\dfrac{E}{B} \end{aligned} \]

Next, when only the magnetic field was applied, the electron beam followed a circular path due to magnetic force. By measuring the radius of curvature \(r\), Thomson used the relation: \[ \begin{align} evB&=\dfrac{mv^2}{r}\\ eB&=\dfrac{mv}{r}\\ \dfrac{e}{m}&=\dfrac{v}{Br}\tag{1} \end{align} \]

Typical measured values in Thomson’s setup:

Now calculate velocity:

This is an illustrative set of values. Using more precise experimental measurements, Thomson obtained:

The electron is a fundamental subatomic particle that carries a negative electric charge. The magnitude of this charge was determined experimentally by Robert A. Millikan through the famous oil drop experiment.

The charge on one electron is:

Where:
\(e\) = elementary charge
The negative sign (–) indicates that the electron carries negative charge
Unit: coulomb (C)

Discovery of the Proton

The discovery of positively charged particles is associated with the work of Eugen Goldstein in 1886. While studying electrical discharge in a modified cathode ray tube, Goldstein used a cathode perforated with small holes. When a high voltage was applied under low-pressure conditions, he observed a new type of radiation emerging from the holes in the cathode and moving in a direction opposite to that of cathode rays.

These rays were called canal rays or anode rays. Detailed observations revealed the following characteristics:

• They travel in straight lines.
• They consist of positively charged particles.
• Their mass is much greater than that of electrons.
• The value of charge-to-mass ratio \((e/m)\) varies depending upon the gas present in the discharge tube.

When hydrogen gas was used in the discharge tube, the particles obtained had the smallest mass and carried a positive charge equal in magnitude to the charge of an electron. These particles were later identified as hydrogen nuclei and named protons.

Thus, a proton is defined as a fundamental particle carrying one unit positive charge \(\left(+1.602\times{10}^{-19}\mathrm{\ C} \right)\) and having a mass of approximately \(1.673\times{10}^{-27}\mathrm{\ kg}\). The proton is present in the nucleus of every atom, and the number of protons in an atom determines its atomic number and chemical identity.

Discovery of the Neutron

In 1932, James Chadwick performed a crucial experiment. He bombarded a thin sheet of beryllium with alpha particles. During this process, a new type of radiation was emitted. This radiation was electrically neutral and had a penetrating power greater than that of protons. When it was allowed to strike paraffin wax, it knocked out high-speed protons.

From careful measurements and calculations, Chadwick concluded that the radiation consisted of neutral particles having a mass nearly equal to that of a proton. These particles were named neutrons.

A neutron has:

• No electric charge.
• A mass of approximately \(1.675\times{10}^{-27}\mathrm{\ kg}\), slightly greater than that of a proton.

Significance of the Discovery

The discovery of protons and neutrons established that the atom contains a dense central nucleus composed of these particles, collectively called nucleons. Electrons revolve around this nucleus in the extranuclear region.

The number of protons determines the atomic number \(\left(Z\right)\), while the total number of protons and neutrons gives the mass number \(\left(A\right)\):

where \(N\) represents the number of neutrons.
The concept of neutrons also explained the existence of isotopes—atoms of the same element having identical atomic numbers but different mass numbers.

Isotopes

Isotopes are atoms of the same element having the same atomic number but different mass numbers.

In other words, isotopes possess:

• Same number of protons
• Same number of electrons (in neutral atoms)
• Different number of neutrons
• Different mass numbers \((A)\)

The term isotope was introduced by Frederick Soddy, who showed that chemically identical atoms can differ in mass.

Hydrogen exists naturally in three isotopic forms:

• Protium: \(^1_1H\) (no neutron)
• Deuterium: \(_1^2H\) (one neutron)
• Tritium: \(_1^3H\) (two neutrons)

All three have atomic number 1, so they behave almost identically in chemical reactions, but their masses are different because of varying neutron content.

Key Characteristics of Isotopes

• Chemical properties are nearly the same
  Since chemical behaviour depends mainly on electronic configuration, isotopes of an element show similar chemical reactions.
• Physical properties are different
  Properties such as density, melting point, diffusion rate, and boiling point differ because these depend on mass.
• Same position in the periodic table
  All isotopes of an element occupy the same place because they have the same atomic number.

Average Atomic Mass

The atomic mass of an element shown in the periodic table is not the mass of a single isotope. It is a weighted average of the masses of all naturally occurring isotopes, based on their relative abundances.

This explains why atomic masses are often fractional rather than whole numbers.

Isobars

Isobars are atoms of different elements having the same mass number but different atomic numbers.

Thus, isobars have:

• Same mass number \(\left(A\right)\)
• Different atomic numbers \(\left(Z\right)\)
• Different numbers of protons, electrons, and neutrons
• Different chemical properties
• Example\[^{40}_{18}\text{Ar} \quad \text{and} \quad ^{40}_{20}\text{Ca}\]Both have mass number 40, but argon has 18 protons while calcium has 20. Because their atomic numbers differ, they are different elements with entirely different chemical behaviour.

Characteristics of Isobars

• They belong to different elements.
• They have different electronic configurations.
• Their physical and chemical properties are completely different.
• They occupy different positions in the periodic table.

According to classical physics, an electron moving in a circular path should continuously lose energy and spiral into the nucleus, causing the atom to collapse. This clearly contradicts experimental observations, since atoms are stable. Moreover, Rutherford’s model failed to account for the sharp spectral lines observed in hydrogen.

Bohr addressed these problems by introducing the idea that electrons can occupy only certain permitted energy states.

Fundamental Postulates of Bohr’s Model

Bohr proposed the following key assumptions for hydrogen-like atoms:

1. Fixed Circular Orbits (Stationary States)
   Electrons revolve around the nucleus only in certain allowed circular paths called stationary orbits or energy levels. While moving in these orbits, electrons do not radiate energy.
2. Quantisation of Angular Momentum
   Only those orbits are permitted for which the angular momentum of the electron is an integral multiple of \(\frac{h}{2\pi}\): \[mvr=\frac{nh}{2\pi}\] where
   \(m\) = mass of electron,
   \(v\) = velocity,
   \(r\) = radius of orbit,
   \(h\) = Planck’s constant,
   \(n\) =1,2,3,\ldots(principal quantum number).
3. Emission and Absorption of Energy
   Radiation is emitted or absorbed only when an electron jumps from one allowed orbit to another. The energy change is given by: \[\Delta E=h\nu\] An electron emits energy when it falls to a lower orbit and absorbs energy when it moves to a higher orbit.
4. Discrete Energy Levels
   Each permitted orbit has a definite energy. These energies increase as the distance from the nucleus increases.

Energy Levels in Hydrogen Atom

For the hydrogen atom, Bohr derived expressions for the radius and energy of electrons in different orbits. Radius of the \(n^{th}\) orbit:

Energy of the electron in the \(n^{th}\) orbit:

The negative sign indicates that the electron is bound to the nucleus. The lowest energy state n1is called the ground state, while higher values of nrepresent excited states.

Merits of Bohr’s Model

Bohr’s model provided several important insights:

• It explained the stability of atoms.
• It accounted quantitatively for the hydrogen emission spectrum.
• It introduced the concept of quantised energy levels.
• It gave mathematical expressions for atomic radii and energies.

These successes marked a significant advance over earlier models.

Limitations of Bohr’s Model

Despite its achievements, Bohr’s model has notable limitations:

• It applies accurately only to hydrogen and hydrogen-like species (such as He⁺ and Li²⁺).
• It cannot explain spectra of multi-electron atoms.
• It fails to account for fine spectral details and effects of magnetic or electric fields.
• It treats electrons as particles moving in definite orbits, which conflicts with later wave-based descriptions.

Energy can travel through space in the form of electromagnetic radiation. Light from the Sun, radio waves used in communication, X-rays in medical imaging—all belong to this same family. To understand atomic structure and spectra, it is essential to study the wave nature of this radiation.

Electromagnetic radiation behaves like a wave and can propagate even in vacuum, without the need for any material medium.

Nature of Electromagnetic Waves

An electromagnetic wave consists of two oscillating fields:

• an electric field
• a magnetic field

These fields vibrate at right angles to each other and also perpendicular to the direction in which the wave travels. The continuous mutual variation of these fields allows the wave to move forward through space.

This wave description of radiation was mathematically explained by James Clerk Maxwell, who showed that changing electric and magnetic fields give rise to electromagnetic waves travelling with the speed of light.

Important Wave Characteristics

To describe electromagnetic radiation, certain wave parameters are used.

1. Wavelength \(\left(\lambda\right)\)
   Wavelength is the distance between two successive crests or troughs of a wave. It is usually expressed in metres (m), nanometres (nm), or ångströms (Å).
2. Frequency \(\left(\nu\right)\)
   Frequency is the number of complete waves passing a point in one second. Its unit is hertz (Hz or s⁻¹).
3. Velocity \(\left(c\right)\)
   All electromagnetic waves travel in vacuum with the same speed, known as the speed of light: \[c=3.0\times{10}^8{\mathrm{\ m\ \operatorname{s}}}^{-1}\] Wavelength and frequency are related by: \[c=\lambda\nu\] This means that if wavelength increases, frequency decreases, and vice versa.
4. Amplitude
   Amplitude represents the height of the wave from its mean position. It is related to the intensity (brightness) of radiation. Higher amplitude means higher intensity.

Electromagnetic Spectrum

Electromagnetic radiation exists over a wide range of wavelengths and frequencies, called the electromagnetic spectrum. It includes:

• Radio waves
• Microwaves
• Infrared radiation
• Visible light
• Ultraviolet radiation
• X-rays
• Gamma rays

Among these, visible light occupies only a very small region, yet it is the part detectable by the human eye.

As we move from radio waves to gamma rays:

• wavelength decreases
• frequency increases
• energy increases

Energy Associated with Radiation

Although electromagnetic radiation shows wave behaviour, its energy is closely linked with frequency. Higher-frequency radiation carries more energy. This relationship is expressed as:

where
\(E=\) energy of radiation,
\(h=\) Planck’s constant,
\(\nu=\) frequency.

This connection between wave properties and energy becomes crucial while explaining atomic spectra and electronic transitions.

Significance in Atomic Structure

The wave nature of electromagnetic radiation explains:

• propagation of light through space
• formation of spectra
• interaction of radiation with matter

When atoms absorb or emit radiation, electrons move between energy levels. These processes involve specific wavelengths, producing characteristic line spectra for each element. Thus, understanding electromagnetic waves is fundamental to interpreting atomic behaviour.

In 1900, Planck suggested that radiant energy is not exchanged continuously but in small, discrete packets called quanta.

The main points of Planck’s quantum theory are:

• Energy is emitted or absorbed in discrete units
  Radiation is exchanged in the form of tiny bundles of energy known as quanta.
• Each quantum has a definite amount of energy
  The energy of one quantum is directly proportional to the frequency of radiation: \[E=h\nu\] where
  \(E=\) energy of one quantum,
  \(h=\) Planck’s constant \(\mathrm{6.626\times 10^{-34}\, J\ s}\),
  \(\nu=\) frequency of radiation.
• Higher frequency means higher energy
  Radiation of higher frequency carries more energetic quanta, while lower frequency radiation carries less energetic quanta.
• Emission or absorption occurs in whole numbers of quanta
  Energy exchange takes place as integral multiples of \(h\nu\), such as \(h\nu,\;2h\nu,\;3h\nu\),and so on.

Photons: Particles of Radiation

Later developments introduced the term photon for one quantum of electromagnetic radiation. A photon represents a particle-like unit of light energy.

Important characteristics of photons include:

• They have energy but no rest mass.
• They travel with the speed of light.
• Their energy depends only on frequency.
• Greater intensity of radiation corresponds to a larger number of photons, not higher energy per photon.

Significance in Atomic Structure

Planck’s quantum theory plays a vital role in understanding atomic phenomena:

• It explains why atoms absorb or emit radiation only at specific frequencies.
• It forms the basis for explaining line spectra.
• It provides the foundation for later atomic models and quantum mechanics.
• It helps interpret electronic transitions between energy levels.

Whenever an electron moves from a higher energy state to a lower one, energy is released in the form of photons. Conversely, absorption of photons allows electrons to jump to higher energy states.

The photoelectric effect refers to the emission of electrons from the surface of a metal when electromagnetic radiation of suitable frequency falls on it.

In simple terms, when light strikes certain metals (such as potassium, sodium, or cesium), electrons are ejected from the metal surface. These emitted electrons are called photoelectrons.

This effect was carefully analysed and successfully explained by Albert Einstein, who applied Planck’s quantum concept to radiation.

Important facts were established from photoelectric experiments

1. Threshold Frequency For every metal, there exists a minimum frequency of incident radiation below which no electrons are emitted, regardless of how intense the light is. This minimum frequency is called the threshold frequency \(\left(\nu_0\right)\). If \(\nu<\nu_0\) no photoelectric emission occurs.
2. 2. Instantaneous Emission When radiation of frequency greater than the threshold frequency strikes the metal, electrons are emitted immediately, without any measurable time lag. This observation contradicts classical theory, which predicted a gradual accumulation of energy.
3. 3. Effect of Intensity Increasing the intensity of light (at fixed frequency) increases the number of emitted electrons, but does not increase their kinetic energy.
4. 4. Effect of Frequency Increasing the frequency of incident radiation increases the kinetic energy of emitted electrons, but not their number.

Einstein’s Explanation of the Photoelectric Effect

Einstein proposed that light consists of discrete packets of energy called photons. Each photon has energy: \[E=h\nu\] where
\(h=\) Planck’s constant,
\(\nu=\) frequency of radiation.

According to Einstein:

• One photon transfers its entire energy to one electron.
• A part of this energy is used to overcome the attractive forces holding the electron in the metal. This minimum required energy is called the work function \(\left(\phi\right)\).
• The remaining energy appears as kinetic energy of the emitted electron. Thus, \[h\nu=\phi+\frac{1}{2}mv^2\] This equation is known as the photoelectric equation.

Meaning of Work Function

The work function is the minimum energy needed to remove an electron from the surface of a metal. It depends on the nature of the metal and is usually expressed in electron volts (eV).

Different metals have different work functions, which explains why some metals emit electrons easily while others do not.

Significance in Atomic Structure

The photoelectric effect provides direct proof that electromagnetic radiation behaves like particles during interaction with matter. Its importance includes:

• Confirmation of Planck’s quantum theory.
• Strong evidence for the particle nature of light.
• Explanation of how atoms exchange energy with radiation.
• Foundation for understanding electronic transitions.
• Basis of devices such as photoelectric cells, light sensors, and solar panels.

When hydrogen gas is excited in an electric discharge tube, it does not produce a continuous band of colours. Instead, it gives a set of sharp, discrete lines at specific wavelengths. This pattern is called the line spectrum of hydrogen, and it provides direct experimental evidence that electrons in atoms occupy quantized energy levels.

If the emitted light from excited hydrogen is passed through a prism or spectroscope, several bright coloured lines appear against a dark background. Each line corresponds to radiation of a definite wavelength. These lines arise because electrons fall from higher energy levels to lower ones, releasing energy in the form of photons: \[\Delta E=h\nu\] Since only certain energy levels exist, only certain frequencies (and hence wavelengths) are produced.

Balmer Series (Visible Region)

The visible lines of hydrogen were first expressed mathematically by Johann Jakob Balmer. This set of lines is called the Balmer series.

In this series, electrons fall from higher levels \(\left(n=3,4,5,\ldots\right)\) to the second energy level \(n_2\). Typical colours include red, blue-green, blue, and violet.

Balmer showed that the wavelengths follow the relation: \[\frac{1}{\lambda}=R\left(\dfrac{1}{2^2}-\dfrac{1}{n^2}\right)\quad n=3,4,5,…\] where \(R\) is the Rydberg constant.

Other Spectral Series of Hydrogen

Further studies revealed that hydrogen has additional line series outside the visible region:

Each series corresponds to electrons returning to a fixed lower energy level.

Bohr introduced a set of revolutionary postulates:

1. Fixed Energy Orbits (Stationary States)
   

   Electrons revolve around the nucleus only in certain permitted circular paths called stationary orbits. Each orbit has a definite energy. As long as an electron remains in one of these allowed orbits, it does not lose energy.
   
   Thus, unlike classical theory, Bohr assumed that only selected energy states are possible.

2. Quantisation of Angular Momentum
   Only those orbits are allowed in which the angular momentum of the electron is an integral multiple of \(\frac{h}{2\pi}\): \[\boxed{\bbox[indigo,5pt]{mvr=\frac{nh}{2\pi}}}\] where
   \(m=\) mass of electron,
   \(v=\) velocity,
   \(r=\) radius of orbit,
   \(h=\) Planck’s constant,
   \(n=1,2,3,\ldots\) (principal quantum number).
   This condition restricts electrons to discrete orbits.
3. Emission and Absorption of Radiation
   

   Radiation is emitted or absorbed only when an electron jumps from one allowed orbit to another:

   • When an electron falls from a higher energy level to a lower one, energy is emitted.
   • When an electron moves from a lower level to a higher one, energy is absorbed.
   
   

   The energy change is given by:

   \[\Delta E=h\nu\]

   This explains why atoms interact with radiation only at specific frequencies.

Electronic Energy Levels in Hydrogen

Using these postulates, Bohr derived expressions for the energy of electrons in hydrogen: \[E_n=-\frac{13.6}{n^2}\mathrm{\ eV}\] where \(n\) is the principal quantum number.

Important points:

• The negative sign indicates that the electron is bound to the nucleus.
• The lowest level n1is called the ground state.
• Higher levels \(\left(n=2,3,4,\ldots\right)\)are excited states.
• As nincreases, the energy levels come closer together.
• At \(n=\infty\), the electron becomes free from the atom (ionisation).

In classical physics, waves and particles were treated as entirely different entities. However, experimental evidence showed that electrons, which were long considered particles, also exhibit wave-like properties. This revolutionary concept was proposed by Louis de Broglie, who suggested that every moving particle is associated with a wave, called a matter wave.

According to de Broglie, the wavelength \(\left(\lambda\right)\) of a particle is given by: \[\boxed{\bbox[indigo,5pt]{\lambda=\frac{h}{mv}}}\] where
\(h=\) Planck’s constant,
\(m=\) mass of the particle,
\(v=\) velocity of the particle.

This relation shows that microscopic particles such as electrons have measurable wavelengths, while macroscopic objects have extremely small wavelengths and therefore do not show observable wave behaviour.

Electronic Energy Levels

Because electrons behave as waves, only certain wavelengths—and hence only certain energies—are possible within an atom. Each allowed energy corresponds to a specific electronic state.

Important features of these energy levels are:

• Electrons can occupy only definite energies.
• Intermediate energies are not allowed.
• Lower energy states are more stable.
• Energy levels become closer together as we move away from the nucleus.

The lowest energy level is called the ground state, while higher ones are known as excited states.

Atomic Spectra as Evidence of Quantization

When atoms absorb energy, electrons jump to higher energy levels. As they return to lower levels, energy is released in the form of radiation.

This radiation does not appear as a continuous band but as sharp spectral lines, known as atomic spectra. Each line corresponds to a specific electronic transition.

The energy of emitted or absorbed radiation is given by:

Since electrons can change energy only in fixed steps, atoms produce radiation only at certain frequencies. This explains why every element has its own characteristic spectrum.

Thus, atomic spectra provide direct experimental evidence that:

• electronic energies are quantized,
• transitions occur between definite energy levels,
• matter at the atomic scale obeys wave–particle duality.

Connection Between Dual Behaviour and Spectra

The dual behaviour of electrons explains why atomic spectra consist of discrete lines:

• The wave nature of electrons restricts them to specific standing-wave states.
• These states correspond to fixed energy levels.
• Transitions between these levels produce photons of definite energies.

If electrons behaved only as classical particles, atoms would emit continuous spectra. The observed line spectra confirm that electrons also possess wave character.

Heisenberg showed that it is impossible to determine simultaneously and exactly both the position and the momentum of a microscopic particle such as an electron.

Mathematically, this is written as: \[\boxed{\bbox[indigo, 5pt]{\Delta x\,\Delta p\geq\frac{h}{4\pi}}}\] where
\(\Delta x=\) uncertainty in position,
\(\Delta p=\) uncertainty in momentum,
\(h=\) Planck’s constant.

This relation does not arise from experimental imperfections; it is a fundamental property of nature. The more accurately we try to locate an electron, the more uncertain its momentum becomes, and vice versa.

Meaning at the Atomic Scale

Electrons are extremely light and move very rapidly. Even a small attempt to observe their position (for example, by shining light on them) disturbs their motion significantly. As a result:

• electrons cannot follow precise circular paths,
• their exact location at any instant cannot be fixed,
• only probabilities of finding electrons in certain regions can be discussed.

This explains why modern atomic theory avoids the idea of sharply defined orbits and instead uses the concept of allowed energy states.

Significance in Atomic Structure

Heisenberg’s uncertainty principle:

• rules out precise electron orbits,
• supports the concept of probability-based electron distribution,
• explains why electrons occupy discrete energy levels,
• provides a quantum foundation for atomic spectra,
• bridges experimental observations with theoretical models.

It establishes that atomic behaviour cannot be understood using classical ideas alone.

Schrödinger proposed that the behaviour of an electron in an atom can be described by a wave equation. Instead of tracking an electron along a definite path, this equation provides a wave function \(\left(\psi\right)\), which contains all information about the electron.

The square of this wave function, \(\psi^2\), represents the probability of finding the electron in a particular region of space. Thus, electrons are no longer pictured as moving in circular orbits; instead, they occupy three-dimensional regions around the nucleus called orbitals.

For the hydrogen atom (which has only one electron), the Schrödinger equation can be solved exactly. These solutions give precise values of allowed energies and corresponding wave functions.

Quantized Electronic Energy Levels

When the Schrödinger equation is applied to hydrogen, it leads to a remarkable result: the electron can possess only certain fixed energies. These energies are given by: \[\boxed{\bbox[indigo,5pt]{E_n=-\frac{13.6}{n^2}\mathrm{\ eV}}}\] where \(n=1,\,2,\,3,\,\ldots\) is the principal quantum number.

Important features of these energy levels are:

• Only specific energies are permitted (quantization).
• The lowest level \(n_1\) is the ground state.
• Higher values of ncorrespond to excited states.
• As nincreases, energy levels come closer together.
• At \(n=\infty\), the electron becomes free from the atom (ionisation).

Thus, the Schrödinger equation naturally explains why electronic energies in hydrogen are discrete rather than continuous.

Atomic Orbitals from the Wave Equation

The solutions of the Schrödinger equation also describe the shapes and orientations of electron clouds, called orbitals. Each orbital is defined by a set of quantum numbers and represents a region where the probability of finding the electron is high. Unlike classical orbits, orbitals:

• have no sharp boundaries,
• represent probability distributions,
• differ in shape and size depending on energy.

This probabilistic picture replaces earlier orbit-based models and provides a more realistic description of atomic structure.

An orbital does not represent a definite path. Rather, it describes a region of space where an electron is most likely to be present. Each orbital corresponds to a specific allowed energy and shape obtained from solutions of the Schrödinger wave equation.

Key features of orbitals are:

• They represent probability distributions, not trajectories.
• Each orbital has a definite energy.
• Orbitals differ in size, shape, and orientation.
• At most two electrons can occupy one orbital, and they must have opposite spins.

Thus, orbitals provide a realistic quantum picture of how electrons are arranged within atoms.

Quantum Numbers: Addresses of Electrons

To specify completely the state of an electron in an atom, four quantum numbers are required. Together, they define the energy, shape, orientation, and spin of an electron.

1. Principal Quantum Number \((n)\)
   The principal quantum number determines the main energy level and size of the orbital.
   • \(n=1,2,3,\ldots\)
   • Higher nmeans higher energy and larger orbital size.
   • Each value of ncorresponds to a shell.
   As \(n\) increases, energy levels come closer together, eventually leading to ionisation when \(n\rightarrow\infty\).
2. Azimuthal Quantum Number \(\left(l\right)\)
   This quantum number defines the shape of the orbital and the subshell.
   For a given n,
   \[l=0,1,2,\ldots(n-1)\]

   Each value of lcorresponds to a subshell:

   • \(l=0\rightarrow \mathrm{s}\)
   • \(l=1\rightarrow \mathrm{p}\)
   • \(l=2\rightarrow \mathrm{d}\)
   • \(l=3\rightarrow \mathrm{f}\)
   
   

   Different subshells have distinct shapes and slightly different energies (except in hydrogen, where energy depends only on \(n\)).

3. Magnetic Quantum Number \(\left(m_l\right)\)
   This quantum number describes the orientation of an orbital in space. For a given \(l\): \[m_l=-l\mathrm{\ to\ }+l\] It explains why subshells split into multiple orbitals (for example, three p orbitals and five d orbitals).
4. Spin Quantum Number \(\left(m_s\right)\)
   Electrons possess an intrinsic angular momentum called spin. \[m_s=+\frac{1}{2}\mathrm{\ or\ }-\frac{1}{2}\] This allows only two electrons per orbital, with opposite spins.

An orbital does not represent a circular track. Instead, it represents a probability distribution for an electron. Each orbital is associated with:

• a definite energy,
• a characteristic shape,
• a particular orientation in space.

The energy and geometry of orbitals are governed by quantum numbers. While the principal quantum number\((n)\) mainly controls energy and size, the azimuthal quantum number\((l)\) determines the shape of the orbital.

Shapes of Common Atomic Orbitals

s-Orbitals (Spherical)

For \(l=0\), the orbitals are called s-orbitals.

• Shape:
  perfectly spherical around the nucleus.
• Orientation:
  same in all directions.
• Examples:
  1s, 2s, 3s, etc.

As the value of nincreases, s-orbitals become larger and may show regions of zero probability called nodes. Despite this internal structure, their overall symmetry remains spherical.

p-Orbitals (Dumbbell-Shaped)

For \(l=1\), the orbitals are known as p-orbitals.

• \Shape: dumbbell-like, with two lobes on opposite sides of the nucleus.
• \Orientation: three possible directions—along \(x,\; y,\;\) and \(z\) axes—called \(p_x,\; p_y,\text{ and }p_z\).
• Each p-subshell therefore contains three orbitals.

A nodal plane passes through the nucleus, separating the two lobes. Compared to s-orbitals of the same shell, p-orbitals are more directional in nature.

d-Orbitals (Cloverleaf and Related Shapes)

For \(l=2\), the orbitals are termed d-orbitals.

• Number: five in each d-subshell.
• Shapes: four have cloverleaf structures, while one \((d_{z^2})\) has a dumbbell shape with a ring around the centre.

These orbitals are larger and more complex, playing an important role in bonding and the chemical behaviour of transition elements.

Relationship Between Orbital Shape and Energy

In hydrogen, all orbitals with the same nhave equal energy. In multi-electron atoms, however, orbital energy depends on both \(n\) and \(l\). Generally:

\[\mathrm{s}\lt\mathrm{p}\lt\mathrm{d}\lt \mathrm{f} \] within the same shell.

Thus, shape and energy are closely linked, influencing how electrons fill orbitals and how atoms interact.

The Pauli Exclusion Principle states:

No two electrons in an atom can have the same set of all four quantum numbers.

In simpler terms, electrons are unique. Even if two electrons occupy the same orbital, they must differ in at least one quantum property.

Meaning in Terms of Orbitals

Each electron in an atom is described by four quantum numbers:

• principal quantum number \(\left(n\right)–\) main energy level
• azimuthal quantum number \(\left(l\right)–\) shape of orbital
• magnetic quantum number \(\left(m_l\right)–\) orientation of orbital
• spin quantum number \(\left(m_s\right)–\) direction of electron spin

When two electrons are present in the same orbital, they automatically have the same values of \(n,\; l\) and \(m_l\). Therefore, to satisfy Pauli’s rule, they must differ in spin.

As a result:

• Each orbital can accommodate a maximum of two electrons.
• These two electrons must have opposite spins \(\left( \frac{1}{2}\right)\text{ and }\left(-\frac{1}{2}\right)\).

This explains why orbitals never contain more than two electrons.

While studying electronic configuration, it is observed that certain atoms do not always follow the expected order of filling of orbitals strictly. For example, some elements show slight variations in their configurations. These variations can be explained on the basis of the special stability associated with completely filled and half-filled subshells.

To understand this stability, we must recall that electrons in an atom occupy orbitals in such a way that the total energy of the atom becomes minimum. The arrangement of electrons within the same subshell also plays an important role in determining stability.

A subshell is said to be completely filled when all the orbitals of that subshell are occupied by the maximum possible number of electrons. For example, the s subshell can hold 2 electrons, the p subshell 6 electrons, the d subshell 10 electrons, and the f subshell 14 electrons. When these subshells are fully occupied, they exhibit extra stability.

Similarly, a subshell is said to be half-filled when each orbital of that subshell contains exactly one electron. For example, a p subshell with 3 electrons (p³) or a d subshell with 5 electrons (d⁵) represents a half-filled condition.

The extra stability of completely filled and half-filled subshells can be explained mainly due to two factors:

Symmetrical Distribution of Electrons

When a subshell is either completely filled or half-filled, the distribution of electrons among the orbitals is symmetrical. Symmetrical distribution leads to a more balanced arrangement of charge within the atom. This balanced arrangement lowers the energy of the atom, making it more stable.

For example:

• p³ configuration has one electron in each of the three p-orbitals.
• d⁵ configuration has one electron in each of the five d-orbitals.

In both cases, the electrons are evenly distributed, resulting in symmetry.

Exchange Energy

Another important factor responsible for stability is exchange energy. When electrons with parallel spins are present in different orbitals of the same subshell, they can exchange their positions. This exchange of electrons leads to a decrease in energy, known as exchange energy.

The greater the number of possible exchanges, the greater is the exchange energy, and hence the greater is the stability.

In a half-filled subshell, the number of possible exchanges between electrons with parallel spins is maximum. Similarly, in a completely filled subshell, although electrons are paired, there are still significant exchange interactions within each set of parallel spin electrons.

Therefore, half-filled and completely filled subshells possess lower energy and hence greater stability compared to other arrangements.