Subatomic Particles and Quantum Theory
Subatomic Particles of the Atom
In 1886, German physicist E. Goldstein discovered canal rays using a discharge tube fitted with a perforated cathode. When high-speed cathode rays (electrons) strike gas molecules inside the tube, they knock out electrons from those molecules, producing positive ions. These positive ions move toward the cathode and some pass through the fine holes, creating a glow on the wall opposite the anode. Since these rays travel through canals in the cathode, they are called canal rays, and because they carry a positive charge, they are also called positive rays. The reaction producing these ions is:
$$M + e^- \rightarrow M^+ + 2e^-$$
Goldstein's Experiment: A discharge tube with a cathode containing fine perforations was used. When high voltage was applied, both cathode rays (moving away from cathode) and positive rays (moving toward cathode through the perforations) were produced simultaneously.
Origin of Positive Rays: The positive ions are created when fast-moving electrons from the cathode collide with and ionize neutral gas molecules in the tube.
Identification of the Proton: When hydrogen gas is used, the positive particle obtained has the highest charge-to-mass ratio of any gas, confirming it as the lightest positive particle. Rutherford named this particle the proton.
Mass Comparison: The mass of a proton is 1836 times the mass of an electron, making it the lightest among all positively charged particles obtained from different gases.
Properties of Positive Rays
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Deflected by electric and magnetic fields (positive charge)
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Travel in straight lines opposite to cathode rays
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Produce flashes on a ZnS plate
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e/m value depends on the gas used — heavier gas gives smaller e/m
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The e/m value is always smaller than that of electrons
An atom is composed of three fundamental subatomic particles: the proton, the neutron, and the electron. Protons and neutrons are concentrated in the tiny central nucleus, while electrons occupy regions of space around it. Each particle has distinct charge and mass properties that define atomic structure and chemical behaviour.
Proton: Carries a relative charge of +1 and a mass of approximately 1.0073 amu. It is found inside the nucleus and determines the atomic number of an element.
Neutron: Carries no charge (electrically neutral) and has a mass of approximately 1.0087 amu, slightly heavier than a proton. It also resides in the nucleus and contributes to atomic mass.
Electron: Carries a relative charge of -1 and has a mass of approximately $5.486 \times 10^{-4}$ amu — about 1836 times lighter than a proton. Electrons occupy the space outside the nucleus.
Fundamental Particle Properties
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Proton: charge = +1.6022 × 10⁻¹⁹ C, mass = 1.6726 × 10⁻²⁷ kg (1.0073 amu)
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Neutron: charge = 0, mass = 1.6750 × 10⁻²⁷ kg (1.0087 amu)
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Electron: charge = -1.6022 × 10⁻¹⁹ C, mass = 9.1095 × 10⁻³¹ kg (5.486 × 10⁻⁴ amu)
Key Mass Relationships
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Mass of proton ≈ 1836 × mass of electron
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Mass of neutron ≈ mass of proton (slightly greater)
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Mass of electron ≈ 1/1836 of proton mass
Planck's Quantum Theory and the Photon
In 1900, Max Planck proposed the quantum theory to explain the emission and absorption of radiation. According to this revolutionary theory, energy is not emitted or absorbed continuously. Instead, it travels in discrete packets called quantuma (plural: quanta). In the case of light, each quantum of energy is called a photon. The energy associated with each photon is directly proportional to its frequency.
$$E = h\nu$$
Relates the energy of a photon to its frequency — higher frequency radiation carries more energy per photon
$E$=Energy of one photon(joules (J))
$h$=Planck's constant(6.626 × 10⁻³⁴ J·s)
$\nu$=Frequency of the radiation(hertz (Hz, s⁻¹))
Energy is Quantized: A body can emit or absorb energy only in whole-number multiples of the quantum. Fractional quanta cannot exist.
Photon Concept: The photon is the smallest discrete unit of electromagnetic energy. It is massless and always travels at the speed of light.
Planck's Constant: The constant $h = 6.626 \times 10^{-34}$ J·s is the ratio of energy to frequency for any photon. It is a fundamental constant of nature.
Frequency Dependence: The energy of a photon increases with increasing frequency. Blue light (higher frequency) carries more energy per photon than red light (lower frequency).
The energy of a photon can be expressed in three equivalent forms depending on which property of the radiation is known. The frequency $\nu$, wavelength $\lambda$, and wave number $\bar{\nu}$ are all interrelated through the speed of light. The wavelength is the distance between two adjacent crests (or troughs) of a wave, and the wave number is the number of waves per unit length — the reciprocal of wavelength.
$$E = \frac{hc}{\lambda}$$
Relates photon energy to wavelength — longer wavelength means lower energy per photon
$E$=Energy of one photon(joules (J))
$h$=Planck's constant(6.626 × 10⁻³⁴ J·s)
$c$=Speed of light(3.0 × 10⁸ m/s)
$\lambda$=Wavelength of the radiation(metres (m))
$\lambda$ is large (radio waves)
→Photon energy is very small
$\lambda$ is small (gamma rays)
→Photon energy is very large
Frequency–Wavelength Link: The frequency and wavelength of any electromagnetic wave are related by $\nu = c / \lambda$. Doubling the frequency halves the wavelength.
Wave Number Form: The wave number $\bar{\nu} = 1/\lambda$ gives energy as $E = hc\bar{\nu}$. Greater wave number means greater photon energy.
Inverse Relationships: Greater wavelength → smaller frequency → lower energy. Greater wave number → smaller wavelength → higher energy.
Wavelength Units: Common units are Å ($10^{-10}$ m), nm ($10^{-9}$ m), and pm ($10^{-12}$ m). Always convert to metres for calculations with SI constants.
Energy Relationships Summary
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$E = h\nu$ (from frequency)
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$E = hc/\lambda$ (from wavelength)
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$E = hc\bar{\nu}$ (from wave number)
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$\bar{\nu} = 1/\lambda$ (wave number from wavelength)
Quantum Numbers and Energy Levels
An electron in an atom is completely described by four quantum numbers. Three of these arise from solving the Schrödinger wave equation for the hydrogen atom, and the fourth (spin) was added experimentally. These four numbers define the shell, subshell, orbital orientation, and spin direction of an electron.
Principal Quantum Number ($n$): Defines the shell or main energy level. Takes integer values $n = 1, 2, 3, ...$. The first shell ($n=1$) is nearest to the nucleus and has the lowest energy. Maximum electrons in a shell = $2n^2$.
Azimuthal Quantum Number ($l$): Defines the subshell and the shape of the orbital. Takes values $l = 0, 1, 2, ..., n-1$. Each value corresponds to a subshell type: $l = 0$ (s), $l = 1$ (p), $l = 2$ (d), $l = 3$ (f).
Magnetic Quantum Number ($m_l$): Defines the orientation of the orbital in space. Takes integer values from $-l$ to $+l$, including zero. For a p-subshell ($l=1$), $m_l = -1, 0, +1$ (three orbitals).
Spin Quantum Number ($m_s$): Describes the direction of electron spin. Can only be $+1/2$ (clockwise, ↑) or $-1/2$ (anticlockwise, ↓). Two electrons in the same orbital must have opposite spins.
Quantum Number Values for Common Subshells
1
1s: $n=1,\; l=0,\; m_l=0$ — 1 orbital
2
2s: $n=2,\; l=0,\; m_l=0$ — 1 orbital
3
2p: $n=2,\; l=1,\; m_l=-1,\; 0,\; +1$ — 3 orbitals
4
3d: $n=3,\; l=2,\; m_l=-2,\; -1,\; 0,\; +1,\; +2$ — 5 orbitals
5
4f: $n=4,\; l=3,\; m_l=-3,\; -2,\; -1,\; 0,\; +1,\; +2,\; +3$ — 7 orbitals
The energy of a subshell depends on both the principal quantum number $n$ and the azimuthal quantum number $l$. The $(n+l)$ rule states that subshells are filled in order of increasing $(n+l)$ value. If two subshells have the same $(n+l)$ value, the one with the smaller $n$ is filled first. This gives the energy ordering: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s.
$$\text{Maximum electrons in shell} = 2n^2$$
Gives the total number of electrons that can be accommodated in a shell with principal quantum number $n$
$n$=Principal quantum number (shell number)(dimensionless integer)
$2n^2$=Total electron capacity of the shell(electrons)
$n = 1$
→Maximum 2 electrons (K shell)
$n = 2$
→Maximum 8 electrons (L shell)
$n = 3$
→Maximum 18 electrons (M shell)
$n = 4$
→Maximum 32 electrons (N shell)
(n+l) Rule: Calculate $n + l$ for each subshell. Lower $(n+l)$ means lower energy and is filled first. For equal $(n+l)$, lower $n$ is preferred.
Energy Ordering: 4s ($n+l=4$) fills before 3d ($n+l=5$) despite 3d having a lower principal quantum number. This is why transition metals fill the 4s orbital before 3d.
Subshell Capacities: s-subshells hold 2 electrons (1 orbital), p-subshells hold 6 (3 orbitals), d-subshells hold 10 (5 orbitals), f-subshells hold 14 (7 orbitals).
Shell Names: $n=1$ (K), $n=2$ (L), $n=3$ (M), $n=4$ (N). The K shell is closest to the nucleus and has the lowest energy.
Subshell Energy Ordering by (n+l) Rule
1
1s ($1+0=1$)
2
2s ($2+0=2$)
3
2p ($2+1=3$)
4
3s ($3+0=3$)
5
3p ($3+1=4$), 4s ($4+0=4$) — same sum, 3s before 4s
6
3d ($3+2=5$), 4p ($4+1=5$)
7
4d ($4+2=6$), 5s ($5+0=5$), 5p ($5+1=6$)
Three rules govern how electrons are distributed among orbitals within subshells. Together with the $(n+l)$ energy ordering, these rules allow the complete electronic configuration of any element to be written.
Aufbau Principle: Electrons fill subshells in order of increasing energy. The lowest-energy subshell available is always filled before moving to the next higher one.
Pauli Exclusion Principle: No two electrons in the same orbital can have all four quantum numbers identical. Consequently, each orbital holds a maximum of two electrons, and these must have opposite spins (↑↓).
Hund's Rule: When electrons are placed in degenerate orbitals (orbitals of the same energy within a subshell), they occupy separate orbitals with parallel spins before any orbital is doubly occupied. For carbon, the 2p electrons enter two different p-orbitals with the same spin rather than pairing in one orbital.
Orbitals and Their Shapes
An orbital is a three-dimensional region around the nucleus where there is a high probability of finding an electron. The shape of an orbital is determined by the azimuthal quantum number $l$. The probability of finding the electron at a particular point is given by $|\psi|^2$, where $\psi$ is the wave function from Schrödinger's equation. Unlike the fixed circular paths (orbits) in Bohr's model, orbitals represent probability distributions.
s-Orbitals ($l = 0$): Spherically symmetrical around the nucleus. The probability of finding an electron depends only on the distance from the nucleus, not the direction. All s-orbitals have this same spherical shape, but higher $n$ values give larger spheres.
p-Orbitals ($l = 1$): Dumbbell-shaped with a node (zero electron density) at the nucleus. There are three p-orbitals oriented along the three coordinate axes: $p_x$ (along x-axis), $p_y$ (along y-axis), and $p_z$ (along z-axis). These three are degenerate orbitals — same energy, different orientation.
d-Orbitals ($l = 2$): Five orbitals with more complex shapes: $d_{xy}$ (lobes between x and y axes), $d_{yz}$ (between y and z axes), $d_{xz}$ (between x and z axes), $d_{x^2-y^2}$ (lobes along x and y axes), and $d_{z^2}$ (two lobes along z-axis with a doughnut-shaped region in the xy-plane).
f-Orbitals ($l = 3$): Seven orbitals with even more complex shapes. These are relevant for lanthanides and actinides but are less commonly needed at this level.
Orbital Shapes at a Glance
1
s-orbital: spherical, 1 per subshell, 2 electrons max
2
p-orbital: dumbbell, 3 per subshell ($p_x, p_y, p_z$), 6 electrons max
3
d-orbital: five distinct shapes, 5 per subshell, 10 electrons max
4
f-orbital: seven complex shapes, 7 per subshell, 14 electrons max
Every orbital contains regions where the probability of finding an electron is zero. These are called nodes. The number and type of nodes depend on the quantum numbers $n$ and $l$. Understanding nodes is essential for visualising how orbitals change with increasing energy. The total number of nodes increases with the principal quantum number, meaning higher-energy orbitals have more complex shapes.
$$\text{Number of orbitals in a subshell} = 2l + 1$$
Gives the number of distinct orbitals within a subshell based on its azimuthal quantum number
$l$=Azimuthal quantum number(dimensionless integer)
$2l + 1$=Number of orbitals in the subshell(orbitals)
$l = 0$ (s-subshell)
→1 orbital
$l = 1$ (p-subshell)
→3 orbitals
$l = 2$ (d-subshell)
→5 orbitals
$l = 3$ (f-subshell)
→7 orbitals
Radial Nodes: Spherical surfaces where electron probability drops to zero. The number of radial nodes equals $n - l - 1$. For example, the 2s orbital ($n=2, l=0$) has $2 - 0 - 1 = 1$ radial node.
Angular Nodes: Planes or cones where electron probability is zero. The number of angular nodes equals $l$. All s-orbitals have zero angular nodes (hence their spherical symmetry), p-orbitals have one, and d-orbitals have two.
Total Nodes: The total number of nodes in any orbital is $n - 1$. This is the sum of radial and angular nodes: $(n - l - 1) + l = n - 1$.
Electron Capacity: Each orbital holds at most 2 electrons (Pauli exclusion). The total electrons in a subshell is $2(2l + 1)$. The total electrons in a shell is $2n^2$.
Node Counting for Selected Orbitals
1
1s ($n=1, l=0$): 0 radial + 0 angular = 0 total nodes
2
2s ($n=2, l=0$): 1 radial + 0 angular = 1 total node
3
2p ($n=2, l=1$): 0 radial + 1 angular = 1 total node
4
3s ($n=3, l=0$): 2 radial + 0 angular = 2 total nodes
5
3p ($n=3, l=1$): 1 radial + 1 angular = 2 total nodes
6
3d ($n=3, l=2$): 0 radial + 2 angular = 2 total nodes