Atomic Structure

1. Basic Ideas of Atomic Structure

An atom contains a small, dense nucleus surrounded by electrons. The nucleus contains positively charged protons and electrically neutral neutrons.

Fundamental particles

Atomic number

Atomic number is represented by Z. It is equal to the number of protons in the nucleus and identifies the element.

Mass number

Mass number is represented by A. It is the total number of protons and neutrons in the nucleus.

Example

Consider nitrogen written as:

Therefore:

• Atomic number = 7
• Protons = 7
• Electrons = 7 in a neutral atom
• Mass number = 14
• Neutrons = 14 - 7 = 7

Isotopes

Isotopes are atoms of the same element having the same atomic number but different mass numbers because they contain different numbers of neutrons.

2. Bohr Atomic Model

Rutherford's nuclear model established the existence of a small positive nucleus. However, the classical model could not explain why an electron moving around the nucleus would not continuously lose energy.

Niels Bohr proposed that electrons can exist only in certain permitted energy states.

Postulates of Bohr's model

1. Electrons can occupy certain permitted circular stationary orbits around the nucleus.
2. An electron in an allowed stationary orbit does not continuously radiate energy.
3. The angular momentum of an electron is quantized.
4. Radiation is emitted or absorbed when an electron moves from one allowed energy state to another.

Quantization of angular momentum

Energy of hydrogen-like species

The negative sign indicates that the electron has lower energy than the separated electron and nucleus at the ionization limit.

Radius of Bohr orbit

Energy transition

When an electron changes from one allowed energy level to another, the energy difference is related to the frequency of radiation.

Main limitation of Bohr's model

It also treats electrons as moving in definite circular orbits, which is inconsistent with the later wave-mechanical description.

3. Matter Waves and de Broglie Hypothesis

Louis de Broglie proposed that moving matter particles have an associated wavelength. This idea extended the wave-particle concept from radiation to matter.

de Broglie equation

Electron accelerated through a potential difference

If an electron is accelerated through a potential difference V, its kinetic energy is related to the accelerating potential.

Interpretation

Wavelength decreases when momentum increases. Electrons have very small mass, so their de Broglie wavelength can become comparable with atomic dimensions.

This reproduces Bohr's quantization condition and shows how the wave description provides a physical basis for the allowed-orbit condition.

4. Heisenberg Uncertainty Principle

Werner Heisenberg showed that position and momentum of a microscopic particle cannot both be known with unlimited precision at the same time.

Physical meaning

If the uncertainty in position is reduced, the minimum possible uncertainty in momentum increases. This is a fundamental property of quantum systems, not simply a defect of laboratory instruments.

5. Wave-Mechanical Model and Schrödinger Equation

The wave-mechanical model does not describe an electron as following a fixed circular path. Instead, the electron is described by a wave function.

Erwin Schrödinger developed the wave equation used to obtain the allowed quantum states and energies of a system.

Time-independent Schrödinger equation

Meaning of important symbols

What do the solutions provide?

Solving the Schrödinger equation gives allowed wave functions and their corresponding energy values. The mathematical solutions are associated with quantum numbers, atomic orbitals and the probability distribution of the electron.

6. Wave Function and Probability Density

The wave function, represented by ψ, is a mathematical function that describes the quantum state of an electron.

Probability of finding an electron

A larger value of |ψ|² at a particular point means a higher probability density there. It does not mean that the electron is permanently located at that point.

Conditions for an acceptable wave function

A physically acceptable wave function should be finite, single-valued and continuous in the appropriate region.

7. Radial and Angular Wave Functions

For a hydrogen-like atom, the wave function can be separated into a radial part and an angular part.

Radial wave function

The radial part depends on the distance r from the nucleus. It describes how the wave function changes with distance.

Angular wave function

The angular part depends on direction and is responsible for the directional characteristics and shapes of orbitals.

8. Radial Distribution Function

Probability density and radial probability are related but are not the same quantity.

Probability density

It describes probability per unit volume at a particular point or region of space.

Radial distribution

For a hydrogen-like orbital, if the wave function is written as ψ = R(r)Y(θ,φ), the radial probability distribution is proportional to the probability of finding the electron in a spherical shell at distance r.

The factor 4πr² represents the volume of a spherical shell of radius r and unit thickness.

Why can the radial distribution have a maximum?

Close to the nucleus the spherical shell has very small volume. As r increases, the shell volume increases. The radial distribution therefore depends on both the radial wave function and the available spherical volume.

Radial
probability
    |
    |             /\
    |            /  \
    |           /    \
    |          /      \
    |_________/        \________
    |
    +-----------------------------> r
               distance

Radial nodes

A radial node is a spherical region at a particular distance from the nucleus where the radial wave function becomes zero.

9. Atomic Orbitals and Their Shapes

An atomic orbital is a one-electron quantum state described by a wave function. The square of its magnitude, |ψ|², gives the probability density associated with that state.

s orbital

For an s orbital, l = 0. It is spherically symmetric around the nucleus.

             . . . . . .
          .               .
        .                   .
       .         ●           .
        .                   .
          .               .
             . . . . . .

              1s orbital

p orbitals

For p orbitals, l = 1. There are three p orbitals because mₗ can have three values: -1, 0 and +1.

They are commonly represented as pₓ, pᵧ and p𝓏 and have two lobes separated by an angular nodal plane.

              p orbital

                  ●
                  |
                  |
               ___|___
             /         \
            /           \
           /             \
          ●               ●
           \             /
            \           /
             \_________/

        Two lobes separated by
        an angular nodal plane

d orbitals

For d orbitals, l = 2 and five orbitals are possible.

Four d orbitals are commonly represented with four lobes, while d𝓏² has two lobes along the z-axis together with a toroidal region around the nucleus.

Number of orbitals in a subshell

10. Nodes and Nodal Surfaces

A node is a region in which the wave function is zero and therefore the probability density is zero.

Radial nodes

Radial nodes occur at particular distances from the nucleus and are spherical surfaces.

Angular nodes

Angular nodes are surfaces determined by the angular part of the wave function. Depending on the orbital, they can be planes or other angular nodal surfaces.

Examples

11. Quantum Numbers

Four quantum numbers are used to specify the quantum state of an electron in an atom.

Principal quantum number (n)

The principal quantum number identifies the main shell and is associated with the size and main energy level.

Azimuthal quantum number (l)

Magnetic quantum number (mₗ)

Spin quantum number (mₛ)

The spin quantum number has two possible values.

Quantum-number validity test

Consider:

This is valid because:

• For n = 3, l can be 0, 1 or 2.
• For l = 2, mₗ can be -2, -1, 0, +1 or +2.
• mₛ can be +1/2 or -1/2.

Now consider:

This is invalid because for n = 2, l can only be 0 or 1.

12. Energy Levels and Electronic Arrangement

Electrons occupy allowed quantum states. For a hydrogen atom, the energy depends on the principal quantum number n. In multi-electron atoms, electron-electron repulsion and shielding affect orbital energies.

Hydrogen-like energy levels

Energy
  ↑

   0 eV       ─────────────── n = ∞
               ionization limit

  -0.85 eV    ─────────────── n = 4

  -1.51 eV    ─────────────── n = 3

  -3.40 eV    ─────────────── n = 2

 -13.60 eV    ─────────────── n = 1

               lowest energy

Ground state

The lowest-energy state of an atom is called its ground state.

Excited state

When an electron absorbs an appropriate amount of energy and moves to a higher allowed state, the atom is in an excited state.

Emission of radiation

When an electron returns to a lower energy state, the energy difference can be released as electromagnetic radiation.

Hydrogen spectrum

Hydrogen produces discrete spectral lines because only specific energy transitions are allowed.

13. Worked Numerical Problems

For numerical questions, use the sequence: Given → Formula → Substitution → Calculation → Unit → Meaning of answer.

Numerical 1: de Broglie wavelength

Calculate the de Broglie wavelength of an electron moving with a velocity of 2.00 × 10⁶ m/s.

Numerical 2: Energy of hydrogen electron

Calculate the energy of an electron in the second energy level of hydrogen.

Numerical 3: Radial nodes in 4p

Determine the number of radial nodes in a 4p orbital.

Numerical 4: Nodes in 4d

Determine the total, angular and radial nodes in a 4d orbital.

Numerical 5: Number of orbitals in a d subshell

Numerical 6: Minimum uncertainty in velocity

An electron has an uncertainty in position of 1.0 × 10⁻¹⁰ m. Estimate the minimum uncertainty in its velocity.

Numerical 7: Energy of hydrogen at n = 3

14. Important Comparisons

Orbit and orbital

Probability density and radial probability

Bohr model and wave-mechanical model

15. Chemistry and Laboratory Connections

Atomic structure becomes useful beyond this chapter because electron energy and electronic arrangement help explain several later areas of chemistry.

Spectroscopy

Atoms and molecules can interact with electromagnetic radiation. Absorption or emission at particular energies can provide information about the species being studied.

Flame and emission observations

Characteristic colours produced by some metal ions are associated with electronic transitions between allowed energy states.

Periodic properties

Electron arrangement helps explain trends such as atomic size, ionization energy and electronegativity.

Chemical bonding

The concept of atomic orbitals is important for understanding orbital overlap, molecular orbitals, hybridization and bond formation in later chemistry topics.

Analytical chemistry

Many analytical methods depend on the interaction of matter with electromagnetic radiation. The detailed instruments differ, but the underlying ideas of energy levels and transitions connect back to atomic structure.

16. Common Mistakes Students Make

1. Orbit and orbital are different. Do not use the two terms interchangeably.
2. ψ is not probability density. Probability density is |ψ|².
3. Radial probability is not simply |ψ|². The spherical volume factor must be considered.
4. Do not remove the negative sign from hydrogen energy. The negative value indicates a bound state.
5. For n = 3, l cannot be 3. Allowed l values are 0, 1 and 2.
6. A p subshell contains three orbitals, not three electrons. Each orbital can hold two electrons.
7. Do not confuse total nodes with radial nodes. Total nodes = n - 1, while radial nodes = n - l - 1.
8. Do not treat the uncertainty principle as an instrument problem. It is a fundamental quantum-mechanical relation.
9. Check all four quantum numbers. One invalid value makes the complete set invalid.
10. Do not skip units in numerical problems. Units often help identify calculation mistakes.
11. Do not write only equations in a long-answer question. Explain what the equations mean.
12. Do not treat an excited electron as permanently remaining in the excited state. Excited states can return to lower-energy states with energy release.

17. Practice Yourself

Try these questions without looking at the answers above. For numerical problems, show all working steps.

Concept questions

1. Define atomic number and mass number.
2. State the main postulates of Bohr's atomic model.
3. Why was the Bohr model unable to explain all atomic spectra?
4. State de Broglie's matter-wave hypothesis.
5. State Heisenberg's uncertainty principle.
6. What is the physical significance of |ψ|²?
7. What is an atomic orbital?
8. Define radial node and angular node.
9. State the four quantum numbers.
10. Explain why the hydrogen spectrum consists of discrete lines.

Numerical and application questions

1. Calculate the energy of hydrogen in n = 4.
2. Calculate the de Broglie wavelength of an electron when its velocity is given.
3. Determine the number of radial nodes in a 5p orbital.
4. Determine total, radial and angular nodes in a 5d orbital.
5. How many orbitals and electrons are present in an f subshell?
6. Check whether n = 3, l = 2, mₗ = +3, mₛ = -1/2 is valid.
7. Check whether n = 4, l = 1, mₗ = 0, mₛ = +1/2 is valid.

18. TU Examination Preparation

Atomic Structure contains definitions, explanations, equations, diagrams and numerical problems. The length of the answer should match the marks available.

2-mark preparation

• Define de Broglie wavelength.
• State Heisenberg's uncertainty principle.
• What is an atomic orbital?
• What is a radial node?
• State the possible values of l for n = 3.
• How many orbitals are present in a d subshell?
• What is the significance of |ψ|²?
• Distinguish between orbit and orbital.

5-mark preparation

• Explain the postulates of Bohr's atomic model.
• Explain de Broglie's matter-wave hypothesis.
• Explain the physical significance of ψ and |ψ|².
• Explain the four quantum numbers.
• Explain radial and angular wave functions.
• Explain radial distribution function.
• Describe s and p orbitals.

10-mark preparation

• Explain Bohr's atomic model, derive the energy expression for hydrogen and discuss its limitations.
• Explain de Broglie's matter-wave hypothesis and show its connection with Bohr's quantization condition.
• Explain Heisenberg's uncertainty principle and discuss its significance in the quantum-mechanical description of electrons.
• Explain the wave-mechanical model and discuss the significance of the Schrödinger equation.
• Explain the four quantum numbers and their role in describing electrons.
• Explain atomic orbitals and discuss the shapes and nodal characteristics of s, p and d orbitals.

Answer structure by marks

19. Quick Revision

Four quantum numbers at a glance

Node shortcut

Before Moving to the Next Chapter

• Can you explain why the Bohr model was proposed?
• Can you state the main limitations of the Bohr model?
• Can you derive the de Broglie wavelength equation?
• Can you explain the uncertainty principle in your own words?
• Can you distinguish ψ from |ψ|²?
• Can you distinguish probability density from radial probability?
• Can you explain the difference between an orbit and an orbital?
• Can you identify s, p, d and f subshells from l?
• Can you calculate total, angular and radial nodes?
• Can you determine whether a set of quantum numbers is valid?
• Can you calculate the energy of a hydrogen electron?
• Can you solve a de Broglie numerical with correct units?
• Can you write a properly structured 10-mark answer?