Structure of Atom — NEET notes

Discovery of Subatomic Particles

Atoms were once thought to be indivisible — until Thomson found electrons (1897), Rutherford found the nucleus (1911), and Chadwick found neutrons (1932). The cathode ray and alpha-scattering experiments are NEET staples. Know the charge-to-mass ratios and who discovered what.

e m_e = 1.758 10^ 11 C kg^ -1 (Thomson's value)

Cathode Rays & Discovery of Electron

Thomson passed electric discharge through a gas at low pressure in a discharge tube and discovered cathode rays — streams of negatively charged particles later called electrons.

When a high voltage (~10,000 V) is applied across a gas at very low pressure (~0.01 mm Hg) in a discharge tube, a stream of particles shoots from the cathode (negative electrode) toward the anode (positive electrode). These are cathode rays.

• Cathode rays travel in straight lines (cast sharp shadows)
• They are deflected by electric and magnetic fields → they carry charge
• Deflection direction → they are negatively charged
• They can rotate a paddle wheel → they have mass (they're particles, not radiation)
• Properties are independent of cathode material and gas → electrons are universal

e m_e = 1.758820 10^ 11 C kg^ -1

Thomson measured the charge-to-mass ratio ( e/m_e ) by balancing electric and magnetic fields. He couldn't separately find charge or mass — only their ratio. The actual charge was measured later by Millikan (oil drop experiment).

Anode Rays & Discovery of Proton

Modified cathode ray experiments revealed positively charged particles streaming toward the cathode. The lightest of these (from hydrogen) was named the proton.

When cathode rays (electrons) knock electrons off gas atoms, the resulting positive ions accelerate toward the cathode. Using a perforated (holed) cathode, these ions pass through and form anode rays (or canal rays), discovered by Goldstein (1886).

• Anode ray properties depend on the gas — unlike cathode rays (which are universal)
• Lightest anode ray particle = from hydrogen gas → called proton (discovered by Rutherford, named by him in 1920)
• Proton: charge = +1.6022 10^ -19 C, mass = 1.6726 10^ -27 kg
• Proton mass ≈ 1836 times electron mass

Discovery of Neutron

James Chadwick bombarded beryllium with alpha particles and detected uncharged particles with mass nearly equal to protons — neutrons.

Chadwick (1932) bombarded a thin beryllium sheet with alpha particles ( , i.e., He nuclei). A highly penetrating radiation emerged that wasn't deflected by electric or magnetic fields → it was neutral. Its mass was close to that of a proton → these were neutrons.

^ 9 _ 4 Be + ^ 4 _ 2 He arrow ^ 12 _ 6 C + ^ 1 _ 0 n

• Neutrons are uncharged → not deflected by electric or magnetic fields
• Neutron mass ≈ proton mass (slightly heavier: 1.6749 vs 1.6726 10^ -27 kg)
• Neutrons are present in all atoms except ^ 1 _ 1 H (protium — the only atom with zero neutrons)
• Atomic mass ≈ protons + neutrons (electrons are ~1/1836 the mass, negligible)

Quick recall

Arrange in order of discovery: proton, electron, neutron.

Electron (Thomson, 1897) → Proton (Rutherford, 1920) → Neutron (Chadwick, 1932). Remember: E-P-N = alphabetical order of first letters!

Atomic Models: Thomson & Rutherford

Thomson proposed the plum pudding model (electrons embedded in positive sphere), but Rutherford's alpha-scattering experiment blew it apart — most of the atom is empty space with a tiny, dense, positive nucleus. Know the observations, conclusions, and limitations of both models.

b = Ze^2 ( /2) 4 _0 1 2 m_ v^2

Thomson's Plum Pudding Model

Thomson proposed that the atom is a uniform sphere of positive charge with electrons embedded in it like plums in a pudding. It explained electrical neutrality but failed to explain scattering experiments.

After discovering the electron, Thomson proposed (1904): the atom is a sphere of uniform positive charge in which electrons are embedded like plums in a Christmas pudding (or seeds in a watermelon).

• Atom is electrically neutral: total positive charge = total negative charge of electrons
• Mass of atom is spread uniformly throughout the sphere
• Also called the watermelon model or raisin bread model
• Explained neutrality of atoms — but that's about it

Limitations: Thomson's model could NOT explain (1) Rutherford's alpha-scattering results, (2) the stability of atoms, or (3) the origin of atomic spectra (line spectra of elements). It was abandoned after Rutherford's gold foil experiment.

Rutherford's Alpha Scattering Experiment

Rutherford, Geiger, and Marsden bombarded a thin gold foil with alpha particles. The shocking observations led to the nuclear model of the atom.

Setup (1911): A beam of alpha particles ( ^4_2 He ^ 2+ , mass = 4 u, charge = +2e) was fired at a very thin gold foil (~0.004 mm). A movable ZnS screen detected scattered particles by tiny flashes (scintillations).

• Most alpha particles passed straight through the foil → atom is mostly empty space
• A small fraction were deflected by small angles → they experienced some repulsive force
• Very few (~1 in 20,000) bounced back at angles > 90° → they hit something very dense and positive
• The larger the angle of deflection, the closer the alpha particle approached the nucleus

Rutherford's Nuclear Model

From the scattering data, Rutherford concluded that all positive charge and nearly all mass is concentrated in a tiny nucleus, with electrons orbiting around it.

Rutherford's conclusions from the alpha-scattering experiment:

• All positive charge and nearly all mass is concentrated in a very tiny region called the nucleus
• Nucleus radius ~ 10^ -15 m vs atom radius ~ 10^ -10 m → nucleus is 10^5 times smaller
• Electrons revolve around the nucleus in circular orbits (like planets around the Sun)
• Atom is mostly empty space — the reason most alpha particles passed through
• The attractive electrostatic force between nucleus (+) and electrons (−) provides the centripetal force for orbiting

Atom radius 10^ -10 m ; Nucleus radius 10^ -15 m

Limitations of Rutherford's model:

• Stability problem: An accelerating charge (orbiting electron) should emit electromagnetic radiation continuously → lose energy → spiral into the nucleus. Atoms should collapse — but they don't!
• No explanation of line spectra: Continuous energy loss should produce a continuous spectrum, not the discrete line spectrum actually observed for elements
• No electron arrangement: The model says nothing about how electrons are arranged or distributed in orbits

Bohr's Model of the Atom

Bohr fixed Rutherford's stability problem by proposing quantized orbits — electrons can only exist in specific energy levels where they DON'T radiate. Energy is absorbed/emitted only when jumping between levels. The formulas for E_n, r_n, and v_n are NEET gold — memorize them cold.

E_n = - 13.6 Z^2 n^2 eV ; r_n = 0.529 n^2 Z Å

Bohr's Postulates

Bohr proposed three revolutionary postulates: quantized orbits, stationary states (no radiation), and frequency condition for transitions. These explained hydrogen's stability and line spectrum.

Niels Bohr (1913) proposed a model for the hydrogen atom (and hydrogen-like ions: He ^+ , Li ^ 2+ , etc.) based on three postulates that broke from classical physics:

• Postulate 1 (Quantized orbits): Electrons revolve around the nucleus in certain fixed circular orbits called stationary states or energy levels
• Postulate 2 (No radiation in orbits): While in a stationary state, the electron does NOT radiate energy — it has fixed energy. This solves Rutherford's stability problem
• Postulate 3 (Angular momentum quantization): The angular momentum of the electron is quantized: mvr = n h 2 , where n = 1, 2, 3,
• Frequency condition: When an electron jumps between levels, it absorbs or emits a photon: E = h = E_ higher - E_ lower

mvr = nh 2 (n = 1, 2, 3, )

E = h = E_ n_2 - E_ n_1 (n_2 > n_1)

Energy & Radius Formulas

For hydrogen-like atoms, the energy, radius, and velocity of the nth orbit are all expressible in terms of n and Z. These are among the most tested formulas in NEET chemistry.

E_n = - 13.6 Z^2 n^2 eV = - 2.18 10^ -18 Z^2 n^2 J

r_n = 0.529 n^2 Z Å = 52.9 n^2 Z pm

v_n = 2.18 10^6 Z n m/s

• E_n is always negative → electron is bound to nucleus. E = 0 at n = (ionized)
• Energy increases (becomes less negative) as n increases → higher orbits have more energy
• Radius increases as n^2 → orbits get farther apart quickly (4× at n=2 , 9× at n=3 )
• Velocity decreases as 1/n → electrons in outer orbits move slower
• For H atom ( Z = 1 ): E_1 = -13.6 eV, r_1 = 0.529 Å (Bohr radius, a_0 )

Hydrogen Spectrum & Spectral Series

When an electron transitions between energy levels, it emits/absorbs a photon of specific wavelength. The transitions are grouped into named series (Lyman, Balmer, Paschen, etc.).

1 = R_H Z^2 ( 1 n_1^2 - 1 n_2^2 ) (n_2 > n_1)

Where R_H = 1.097 10^7 m ^ -1 is the Rydberg constant. Each spectral series corresponds to transitions ending at a specific lower level n_1 .

• Balmer series is the only one in the visible region — this is tested repeatedly
• Total number of spectral lines from level n : n(n-1) 2
• First line of a series = transition from n_1 + 1 to n_1 (smallest energy gap → longest )
• Series limit = transition from n_2 = to n_1 (largest energy gap → shortest )
• Line spectrum (not continuous) → energy levels are quantized (Bohr's key proof)

Quick recall

An electron in H atom falls from n=5 to n=2. Which series? How many total lines possible from n=5?

Falls to n_1 = 2 → Balmer series (visible). Total lines from n = 5 : 5 4 2 = 10 lines.

Limitations of Bohr's Model

Bohr's model works perfectly for hydrogen and hydrogen-like ions but fails for multi-electron atoms. It also can't explain fine structure, Zeeman effect, or chemical bonding.

• Only works for single-electron species (H, He ^+ , Li ^ 2+ ) — fails for He, Li, etc.
• Cannot explain fine structure of spectral lines (closely spaced doublets seen at high resolution)
• Cannot explain Zeeman effect (splitting of lines in magnetic field) or Stark effect (splitting in electric field)
• Assumes circular orbits — actual orbitals have complex 3D shapes (s, p, d, f)
• Violates Heisenberg uncertainty principle: assumes electron has definite orbit (fixed r and v simultaneously)
• Cannot explain chemical bonding or molecular structure

Dual Nature & de Broglie Wavelength

Light behaves as both wave and particle. de Broglie extended this duality to matter: every moving particle has a wavelength λ = h/mv. Heisenberg's uncertainty principle says you can NEVER know both position and momentum exactly at the same time. These ideas demolished classical orbits and paved the way for quantum mechanics.

= h mv = h p ; x p h 4

Wave-Particle Duality of Light

Light shows wave behavior (interference, diffraction) AND particle behavior (photoelectric effect). Neither wave theory alone nor particle theory alone explains all observations.

Wave nature: Interference, diffraction, polarization → explained by Maxwell's equations. Particle nature: Photoelectric effect, Compton effect → explained by Einstein's photon concept ( E = h ). Light is BOTH — this is wave-particle duality.

E = h = hc ; p = h = h c

• h = 6.626 10^ -34 J·s (Planck's constant)
• Photon has zero rest mass but has momentum p = h/
• Higher frequency → higher energy photon → more particle-like behavior
• Lower frequency → longer wavelength → more wave-like behavior

de Broglie Hypothesis

Louis de Broglie (1924) proposed that if light (a wave) can behave like a particle, then particles (like electrons) should also have wave character. The wavelength is λ = h/mv.

de Broglie's bold idea (1924): Every moving particle has an associated matter wave with wavelength inversely proportional to its momentum.

= h mv = h p = h 2mKE

The last form uses KE = 1 2 mv^2 p = 2m KE . For an electron accelerated through potential V : KE = eV .

_e = h 2m_e eV = 1.226 V nm (for electron)

• Heavier particles → smaller → wave nature becomes undetectable
• A cricket ball (0.15 kg) at 30 m/s: 10^ -34 m — immeasurably small!
• An electron at v = 10^6 m/s: 0.73 nm — comparable to atomic spacings → detectable!
• Davisson-Germer experiment (1927) confirmed electron diffraction, proving de Broglie was right
• de Broglie wavelength connects Bohr's angular momentum quantization: mvr = nh/2 means the orbit fits exactly n wavelengths

Heisenberg's Uncertainty Principle

You CANNOT simultaneously know the exact position and exact momentum of a particle. The more precisely you know one, the less precisely you can know the other. This is not a measurement limitation — it's a law of nature.

x p h 4 or equivalently x m v h 4

x = uncertainty in position, p = m v = uncertainty in momentum. The product can NEVER be smaller than h/4 . The more precisely you pin down position ( x 0 ), the more uncertain the momentum becomes ( p ), and vice versa.

• This is NOT about poor instruments — it's a fundamental limit of nature
• Significant only for microscopic particles (electrons, photons) — negligible for macroscopic objects
• Destroys the concept of 'orbit': if we can't know exact x and v simultaneously, we can't trace a definite path
• Replaces orbits with orbitals — regions of probability where an electron is likely to be found
• Energy-time form: E t h 4 (less commonly tested in NEET)

Quick recall

If the uncertainty in position of an electron is 0.1 nm, what is the minimum uncertainty in velocity?

v h 4 m_e x = 6.626 10^ -34 4 3.14 9.109 10^ -31 10^ -10 5.79 10^ 5 m/s. That's enormous — nearly the speed of the electron itself!

Quantum Numbers

Every electron in an atom has a unique 'address' given by four quantum numbers: n (shell), l (subshell shape), ml (orbital orientation), ms (spin). These determine the allowed energy levels, orbital shapes, and maximum electron capacity. Valid vs invalid combinations are tested HEAVILY in NEET.

n 1 , l = 0 to (n-1) , m_l = -l to +l , m_s = 1 2

Principal Quantum Number (n)

n determines the shell, the average distance from the nucleus, and the energy of the electron. Higher n = farther from nucleus = higher energy.

The principal quantum number n tells you which shell (energy level) the electron is in. It can take any positive integer value: n = 1, 2, 3, 4,

• n determines the size of the orbital and energy of the electron
• Higher n → larger orbital → electron farther from nucleus → higher energy (for H atom)
• Maximum electrons in a shell = 2n^2
• Number of subshells in a shell = n (shell n=3 has 3 subshells: s, p, d)
• Total orbitals in a shell = n^2 (shell n=3 has 9 orbitals)

Azimuthal Quantum Number (l)

l determines the subshell (s, p, d, f), the shape of the orbital, and contributes to the energy in multi-electron atoms. For a given n, l ranges from 0 to (n−1).

The azimuthal (angular momentum) quantum number l determines the subshell and shape of the orbital. For a given n : l = 0, 1, 2, , (n-1) .

• l also gives the orbital angular momentum: L = l(l+1) h 2
• For l = 0 (s orbital): angular momentum = zero (the electron has no angular motion — surprising!)
• Number of orbitals in a subshell = 2l + 1
• Maximum electrons in a subshell = 2(2l + 1) : s = 2, p = 6, d = 10, f = 14
• The letters s, p, d, f come from spectral line descriptions: sharp, principal, diffuse, fundamental

Magnetic Quantum Number ( m_l )

ml determines the orientation of the orbital in space. For a given l, ml ranges from −l to +l, giving (2l+1) possible orientations (orbitals).

The magnetic quantum number m_l specifies the orientation of the orbital in space. For a given l : m_l = -l, -(l-1), , 0, , (l-1), +l .

• Total orbitals for a given l = 2l + 1
• In the absence of a magnetic field, all orbitals with the same n and l are degenerate (same energy)
• In a magnetic field, different m_l values split into different energies (Zeeman effect)
• The three p orbitals ( p_x, p_y, p_z ) point along the three axes — mutually perpendicular

Spin Quantum Number ( m_s )

Each electron spins on its own axis, creating a small magnetic field. Spin can only be +½ or −½. Two electrons in the same orbital MUST have opposite spins (Pauli exclusion).

The spin quantum number m_s describes the intrinsic spin of the electron on its own axis. It has only two possible values:

m_s = + 1 2 ( , spin up) or m_s = - 1 2 ( , spin down)

• Each orbital can hold a maximum of 2 electrons with opposite spins ( )
• This directly gives: max electrons = 2 (number of orbitals)
• Spin is an intrinsic quantum property — the electron isn't literally spinning like a top
• Two electrons with opposite spins in the same orbital are called an electron pair
• Pauli Exclusion Principle: No two electrons in an atom can have all four quantum numbers identical → max 2 electrons per orbital

Quick recall

Is the set of quantum numbers (2, 2, 0, +½) valid?

NO! For n = 2 , the maximum value of l is n - 1 = 1 . So l = 2 is not allowed. Invalid set.

Shapes of Orbitals

Orbitals are NOT orbits — they are 3D regions where there's high probability (say 90%) of finding the electron. s orbitals are spherical, p are dumbbell-shaped, d are cloverleaf. Nodes (regions of zero probability) follow clear formulas: radial nodes = n−l−1, angular nodes = l, total nodes = n−1.

Radial nodes = n - l - 1 ; Angular nodes = l ; Total nodes = n - 1

Orbital vs Orbit

An orbital is a mathematical function that describes the probability of finding an electron in a given region of space. It's fundamentally different from Bohr's 'orbit' — there's no definite path.

The probability of finding an electron at a point is given by | |^2 (square of the wave function). This is Max Born's interpretation. The orbital shape is the surface where this probability density is constant.

s Orbitals

s orbitals (l = 0) are spherical. They have no angular nodes, only radial nodes. The 1s orbital has no nodes at all, 2s has 1 radial node, 3s has 2 radial nodes.

• Shape: spherical (looks the same from every direction)
• Non-directional — no preferred axis
• Only ONE s orbital per shell ( m_l = 0 only)
• Size increases with n : 1s < 2s < 3s (electron is farther from nucleus)
• 1s has 0 nodes, 2s has 1 radial node (a sphere where = 0 ), 3s has 2 radial nodes

At the node, = 0 and | |^2 = 0 — zero probability of finding the electron there. The electron exists on both sides of the node but never AT the node. (Quantum tunneling, not classical motion!)

p Orbitals

p orbitals (l = 1) are dumbbell-shaped with one angular (nodal) plane. There are three mutually perpendicular p orbitals in each shell (starting from n = 2).

• Shape: dumbbell (two lobes on opposite sides of the nucleus)
• Three p orbitals: p_x , p_y , p_z — pointing along x, y, z axes respectively
• All three are degenerate (same energy) in the absence of external fields
• Each p orbital has 1 angular node (a nodal plane through the nucleus). p_x has the yz-plane as its node
• p orbitals start from n = 2 (no 1p orbital exists since l < n )
• The two lobes have opposite signs of (+ and −) but same | |^2 (probability)

d Orbitals

d orbitals (l = 2) have five orientations. Four are cloverleaf-shaped with lobes between or along axes, and one (dz²) looks like a doughnut with two lobes. They start from n = 3.

• Five d orbitals: d_ xy , d_ yz , d_ xz , d_ x^2-y^2 , d_ z^2
• d_ xy , d_ yz , d_ xz : lobes lie between the axes (in the respective planes)
• d_ x^2-y^2 : lobes lie along the x and y axes
• d_ z^2 : unique shape — doughnut (torus) in the xy-plane with two lobes along z-axis
• All five are degenerate in a free atom; they split in crystal fields (basis of CFT in coordination chemistry)
• Each d orbital has 2 angular nodes; d orbitals start from n = 3

Nodes

Nodes are regions where the probability of finding the electron is exactly zero. Two types: radial (spherical shells) and angular (planes/cones). The formulas are simple and heavily tested.

Radial nodes = n - l - 1 Angular nodes = l Total nodes = n - 1

• Radial nodes = spherical shells of zero probability (like layers in an onion)
• Angular nodes = planes or cones passing through the nucleus
• s orbitals have only radial nodes (no angular nodes since l = 0 )
• Total nodes always = n - 1 regardless of orbital type
• At a node: = 0 and therefore | |^2 = 0 (zero probability)

Quick recall

How many radial and angular nodes does a 4d orbital have?

n = 4, l = 2 . Radial = 4 - 2 - 1 = 1 . Angular = 2 . Total = 3 (= n - 1 ). Check: 1 + 2 = 3 .

Electronic Configuration

How electrons fill up orbitals follows three rules: Aufbau (lowest energy first), Pauli exclusion (max 2 per orbital with opposite spins), and Hund's rule (maximize unpaired electrons in degenerate orbitals). The (n+l) rule gives the filling order. Cr and Cu are FAMOUS exceptions — memorize them.

Filling order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d

Aufbau Principle & (n+l) Rule

Electrons fill orbitals in order of increasing energy. When two subshells have the same (n+l) value, the one with lower n fills first.

Aufbau (German for 'building up'): Electrons are added one at a time to the orbital with the lowest available energy. The energy order is determined by the (n+l) rule:

• Lower (n+l) → lower energy → fills first
• If two subshells have the same (n+l), the one with lower n fills first
• Example: 3d (n+l = 5) vs 4p (n+l = 5) → 3d fills first (lower n )
• This is why 4s fills before 3d: 4s has n+l = 4 , 3d has n+l = 5

Pauli Exclusion Principle

No two electrons in an atom can have the same set of all four quantum numbers. This limits each orbital to a maximum of 2 electrons with opposite spins.

If (n, l, m_l) are the same for two electrons m_s MUST be different

Since m_s can only be + 1 2 or - 1 2 , each orbital (defined by n, l, m_l ) can hold at most 2 electrons — one spin-up ( ) and one spin-down ( ).

• Max electrons per orbital = 2 (with opposite spins)
• Max electrons per subshell = 2(2l+1) : s = 2, p = 6, d = 10, f = 14
• Max electrons per shell = 2n^2 : K = 2, L = 8, M = 18, N = 32
• Proposed by Wolfgang Pauli (1925) — Nobel Prize 1945
• Violation example: writing 1s^3 — impossible! 1s has only 1 orbital → max 2 electrons

Hund's Rule of Maximum Multiplicity

Electrons fill degenerate orbitals (same energy) singly with parallel spins first, before any pairing occurs. This minimizes electron-electron repulsion.

Hund's rule: In a set of degenerate orbitals (same subshell), electrons occupy each orbital singly with parallel spins first. Pairing begins only after all orbitals are half-filled.

• Filling 3 electrons in 2p: (one in each orbital, all spin-up) — CORRECT
• NOT: _ (pairing before all orbitals are singly filled) — WRONG
• Parallel spins have lower energy than paired spins due to exchange energy
• Maximum unpaired electrons → maximum spin multiplicity = 2S + 1 where S = total spin

Quick recall

Draw the orbital filling diagram for nitrogen (Z = 7).

1s^2 2s^2 2p^3 : 1s → , 2s → , 2p → (3 unpaired electrons, all with same spin). This is why N is very stable — half-filled p subshell!

Writing Electronic Configurations

Combine Aufbau + Pauli + Hund's to write configurations for any element. Use noble gas core shorthand for efficiency. Know the key exceptions: Cr and Cu.

Steps to write electronic configuration:

• Step 1: Find the atomic number (Z = total electrons for neutral atom)
• Step 2: Fill orbitals following Aufbau order: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → ...
• Step 3: Apply Pauli (max 2 per orbital) and Hund's (spread before pair in degenerate orbitals)
• Step 4: Use noble gas shorthand: [He] = 1s^2 , [Ne] = 1s^2 2s^2 2p^6 , [Ar] = 1s^2 2s^2 2p^6 3s^2 3p^6

Stability of Half-Filled & Fully-Filled Orbitals

Half-filled (d⁵, p³) and fully-filled (d¹⁰, p⁶) subshells have EXTRA stability due to symmetrical electron distribution and exchange energy. This explains the famous exceptions: Cr is [Ar] 3d⁵4s¹ (not 3d⁴4s²) and Cu is [Ar] 3d¹⁰4s¹ (not 3d⁹4s²). NEET tests this concept almost every year.

Exchange energy n(n-1) 2 (n = number of parallel spin electrons)

Exchange Energy

Electrons with the same spin in degenerate orbitals can 'exchange' positions, releasing exchange energy. More parallel-spin electrons = more exchange pairs = more stability.

When two or more electrons with the same spin occupy degenerate orbitals, they can exchange their positions. Each such exchange lowers the total energy by a small amount called exchange energy.

Number of exchange pairs = n 2 = n(n-1) 2

Where n = number of electrons with parallel spin. More exchange pairs → more stabilization.

Symmetrical Distribution

Half-filled and fully-filled subshells have a symmetrical distribution of electrons — each orbital has the same number of electrons. This symmetry provides additional stability beyond exchange energy.

• Half-filled: Every orbital in the subshell has exactly 1 electron → perfectly symmetrical
• Fully-filled: Every orbital has exactly 2 electrons → also perfectly symmetrical
• Symmetrical charge distribution → lower electron-electron repulsion → more stable
• Unsymmetrical distributions (like d^4 or d^9 ) have uneven charge density → less stable
• This is an additional factor on top of exchange energy — both work together

Chromium & Copper

Cr (Z=24) adopts [Ar] 3d⁵4s¹ instead of 3d⁴4s², and Cu (Z=29) adopts [Ar] 3d¹⁰4s¹ instead of 3d⁹4s². Both sacrifice a 4s electron to achieve the extra stability of half-filled or fully-filled 3d.

In both cases, one electron shifts from 4s to 3d to achieve the extra stability of half-filled ( d^5 ) or fully-filled ( d^ 10 ) configuration. The energy gained from exchange energy and symmetry more than compensates for the 'violation' of Aufbau.

• Cr: 3d^4 4s^2 3d^5 4s^1 — gains half-filled d stability (10 exchange pairs vs 6)
• Cu: 3d^9 4s^2 3d^ 10 4s^1 — gains fully-filled d stability (20 exchange pairs vs 16)
• Other notable exceptions: Mo ( 4d^5 5s^1 ), Ag ( 4d^ 10 5s^1 ) — same logic, different period
• Pd ( Z = 46 ) is extreme: [Kr] 4d^ 10 5s^0 — moves BOTH 5s electrons into 4d

Quick recall

What is the electronic configuration of Cu²⁺ (Z = 29)?

Cu = [Ar] 3d^ 10 4s^1 . Remove 2 electrons: first from 4s ( 4s^0 ), then from 3d. Cu ^ 2+ = [Ar] 3d^9 . It has 1 unpaired electron. (This is why Cu ^ 2+ salts are colored!)

Consequences of Extra Stability

Half-filled and fully-filled stability explains many trends: why N has higher ionization energy than O, why Mn²⁺ is more stable than expected, and why Cu⁺ prefers d¹⁰ configuration.

• N > O in IE₁ (first ionization energy): N has half-filled 2p^3 → extra stable → harder to remove electron. O has 2p^4 → one paired electron → easier to remove
• Be > B in IE₁: Be has fully-filled 2s^2 , B starts filling 2p → removing from 2p is easier
• Mn²⁺ (d ^5 ) is very stable → Mn ^ 3+ is a strong oxidizer (wants to go back to d ^5 )
• Cu⁺ (d ^ 10 ) is stable in solid state but disproportionates in aqueous solution: 2Cu^+ Cu^ 2+ + Cu
• Zn (d ^ 10 s ^2 ) is NOT a typical transition metal — fully-filled d means no variable oxidation states, no color