Structure of Atom

By the late nineteenth century, experiments with electric discharge through gases at very low pressure (in a discharge tube, also called a Crookes tube) revealed that atoms are not indivisible — they are built from smaller charged particles.

When a high voltage (of the order of 10,000 V) is applied across a discharge tube containing gas at very low pressure (below about 0.01 mm Hg), invisible rays are found to travel from the cathode to the anode; these are called cathode rays, and their passage can be observed because they cause certain materials, such as zinc sulphide, to fluoresce. Cathode rays: (i) start from the cathode and move towards the anode; (ii) travel in straight lines in the absence of an electric or magnetic field; (iii) are deflected by external electric and magnetic fields in the manner expected of negatively charged particles; and (iv) have characteristics that do not depend on the material of the electrodes or the nature of the gas present in the tube. These observations showed that cathode rays consist of negatively charged particles that are a universal constituent of all matter. J. J. Thomson named this particle the electron and, using the deflections produced by electric and magnetic fields, measured its charge-to-mass ratio, e/m.

R. A. Millikan's oil drop experiment independently measured the actual charge, e, carried by the electron. Combining Millikan's value of e with Thomson's e/m ratio gives the mass of the electron.

Using a discharge tube with a perforated cathode, Eugen Goldstein observed a new kind of rays travelling in a direction opposite to the cathode rays, called canal rays or anode rays, made up of positively charged particles. Unlike cathode rays, the charge-to-mass ratio of these particles depended on the gas taken in the tube. The smallest and lightest positively charged particle was obtained when the tube was filled with hydrogen gas; this particle was named the proton.

Because atoms are electrically neutral overall but the proton accounts for only part of the atomic mass in heavier atoms, the existence of a third, neutral particle was suspected. In 1932, James Chadwick bombarded a thin sheet of beryllium with fast alpha particles and obtained a new, highly penetrating radiation that was not deflected by electric or magnetic fields. This neutral particle, of mass almost equal to that of the proton, was named the neutron.

Worked example. Determine the number of protons, neutrons and electrons in the chloride ion, written as ³⁵₁₇Cl⁻ (mass number A = 35, atomic number Z = 17, ionic charge = −1).

• Number of protons = atomic number, Z = 17.
• Number of neutrons = A − Z = 35 − 17 = 18.
• Number of electrons in the neutral atom would equal the number of protons, i.e. 17; since the species carries a charge of −1, it has one extra electron, so the number of electrons in Cl⁻ = 17 + 1 = 18.

So the chloride ion ³⁵Cl⁻ contains 17 protons, 18 neutrons and 18 electrons.

J. J. Thomson proposed that an atom is a sphere of uniform positive charge with electrons embedded in it, much like seeds in a watermelon (often called the plum-pudding model). The positive and negative charges were assumed to be equal in magnitude, making the atom electrically neutral, and the mass was assumed to be evenly spread throughout the sphere. While this model could account for the overall neutrality of the atom, it could not explain later experimental observations on how particles scatter off atoms.

Ernest Rutherford tested the structure of the atom by directing a beam of fast-moving, positively charged alpha (α) particles from a radioactive source at a very thin sheet of gold foil, with a circular fluorescent zinc sulphide screen around the foil to detect the scattered particles. He observed that:

• most of the α-particles passed straight through the gold foil without any deflection;
• a small fraction of the α-particles were deflected through small angles;
• a very small fraction (about 1 in 20,000) bounced back, i.e. were deflected through angles close to 180°.

From these observations, Rutherford proposed the nuclear model of the atom: most of the space inside an atom is empty (which is why most α-particles pass straight through); almost the entire mass of the atom and all of its positive charge are concentrated in an extremely small region at the centre, called the nucleus (which repels and strongly deflects the rare α-particle that travels close to or almost head-on into it); and the electrons revolve around this nucleus in circular paths at high speed. The radius of the nucleus (of the order of 10⁻¹⁵ m) is about 10⁵ times smaller than the radius of the atom (of the order of 10⁻¹⁰ m).

Rutherford's model, however, had a serious limitation: it could not explain the stability of the atom. Classical electromagnetic theory requires that a charged particle undergoing acceleration (as an electron does while revolving in a circular orbit) must continuously emit radiation and lose energy. An electron losing energy in this way should spiral inward and fall into the nucleus almost instantly, which contradicts the observed stability of atoms. The model also said nothing about how electrons are arranged around the nucleus or what energies they possess, so it could not account for the line spectra of atoms.

Worked example. Using Rutherford's observations, explain why only a tiny fraction of α-particles were deflected through large angles while the vast majority passed straight through the foil.

• Since the atom is mostly empty space, an α-particle travelling anywhere except very close to a nucleus experiences almost no force and continues in a straight line — this explains why the large majority pass through undeflected.
• The nucleus occupies only an extremely small fraction of the atom's volume, so very few α-particles happen to travel close enough to it to be strongly repelled.
• An α-particle that does travel close to, or almost directly at, a nucleus experiences a very large electrostatic repulsion from the concentrated positive charge and large mass of the nucleus, and can be deflected through a large angle or even bounced straight back — this explains the rare large-angle deflections.

To overcome the failure of Rutherford's model to explain atomic stability and line spectra, Niels Bohr proposed a model for the hydrogen atom (and other one-electron/hydrogen-like species) based on three postulates:

1. An electron in an atom moves around the nucleus only in certain fixed circular paths of definite energy, called stationary states or orbits. As long as an electron remains in a given stationary state, its energy does not change, i.e. it does not radiate energy.
2. Only those orbits are permitted for which the angular momentum of the electron is quantized, i.e. an integral multiple of h/2π: mvr = nh/2π, where n = 1, 2, 3, … is called the principal quantum number.
3. Energy is emitted or absorbed by the atom only when an electron moves (jumps) from one stationary state to another, and the energy of the radiation absorbed or emitted equals the difference in energy between the two states: ΔE = E₂ − E₁ = hν.

Solving the model for a hydrogen-like species of atomic number Z gives the radius, velocity and energy of the electron in its nth orbit:

• radius, r_n = 0.529 × (n²/Z) Å;
• velocity, v_n = 2.18 × 10⁶ × (Z/n) m s⁻¹;
• energy, E_n = −2.18 × 10⁻¹⁸ × (Z²/n²) J = −13.6 (Z²/n²) eV.

The negative sign shows that the electron is bound to the nucleus; as n increases, the energy becomes less negative (higher), and at n = ∞ the electron is completely free of the nucleus (E = 0, corresponding to ionization). When an electron falls from a higher orbit n₂ to a lower orbit n₁, radiation of a definite wavelength is emitted, given by the Rydberg formula, 1/λ = R_H Z² (1/n₁² − 1/n₂²), where R_H = 1.097 × 10⁷ m⁻¹ is the Rydberg constant. This correctly accounted for the observed line spectrum of the hydrogen atom, including the Lyman (n₁ = 1), Balmer (n₁ = 2), Paschen (n₁ = 3), Brackett (n₁ = 4) and Pfund (n₁ = 5) series.

Despite this success, Bohr's model has several limitations: it fails to explain the line spectra of atoms or ions containing more than one electron; it cannot account for the splitting of spectral lines into finer lines in the presence of a magnetic field (Zeeman effect) or an electric field (Stark effect); it does not explain the ability of atoms to form molecules with covalent bonds; and it does not incorporate the wave nature of the electron or the uncertainty principle, since it assumes the electron follows a well-defined circular path.

Worked example. Calculate the energy needed to excite an electron in a hydrogen atom from n = 1 to n = 2, and the wavelength of radiation associated with this transition.

• E₁ = −2.18 × 10⁻¹⁸ × (1²/1²) = −2.18 × 10⁻¹⁸ J.
• E₂ = −2.18 × 10⁻¹⁸ × (1²/2²) = −5.45 × 10⁻¹⁹ J.
• ΔE = E₂ − E₁ = (−5.45 × 10⁻¹⁹) − (−2.18 × 10⁻¹⁸) = 1.635 × 10⁻¹⁸ J, which must be absorbed.
• Using ΔE = hc/λ: λ = hc/ΔE = (6.626 × 10⁻³⁴ × 3 × 10⁸) / (1.635 × 10⁻¹⁸) = 1.216 × 10⁻⁷ m = 121.6 nm.

This wavelength (121.6 nm, in the ultraviolet region) corresponds to the first line of the Lyman series.

Experiments on black-body radiation and the photoelectric effect showed that light (and electromagnetic radiation in general) behaves as a stream of energy packets called photons, each of energy E = hν, in addition to its well-established wave nature (seen in diffraction and interference). Light therefore shows dual behaviour — it behaves as a wave in some experiments and as a stream of particles in others.

In 1924, Louis de Broglie proposed that this dual behaviour is not special to light but is a general property of all moving matter — electrons, protons, or even larger particles. He proposed that a wave, called a matter wave or de Broglie wave, of wavelength λ = h/mv (h = Planck's constant, m = mass, v = velocity) is associated with every moving particle. Because Planck's constant h is extremely small, the de Broglie wavelength is only significant (measurable) for particles of very small mass, such as electrons; for macroscopic objects (like a ball or a car), the associated wavelength is far too small to be observed.

Werner Heisenberg's uncertainty principle (1927) states that it is impossible to determine, simultaneously and with perfect accuracy, both the exact position and the exact momentum (or velocity) of a microscopic moving particle such as an electron. Mathematically, if Δx is the uncertainty in position and Δp is the uncertainty in momentum, then Δx · Δp ≥ h/4π. The more precisely the position of the electron is known, the less precisely its momentum can be known, and vice versa. This principle rules out the idea of an electron following a well-defined path or orbit (as Bohr had assumed), and instead leads to the idea that we can only speak of the probability of finding an electron at a given point in space — the foundation of the quantum mechanical model of the atom.

Worked example. Calculate the de Broglie wavelength associated with an electron (mass = 9.11 × 10⁻³¹ kg) moving with a velocity of 2.05 × 10⁷ m s⁻¹.

• λ = h/(mv) = (6.626 × 10⁻³⁴) / (9.11 × 10⁻³¹ × 2.05 × 10⁷).
• mv = 9.11 × 10⁻³¹ × 2.05 × 10⁷ = 1.868 × 10⁻²³ kg m s⁻¹.
• λ = 6.626 × 10⁻³⁴ / 1.868 × 10⁻²³ = 3.548 × 10⁻¹¹ m ≈ 35.5 pm.

This wavelength is comparable to atomic dimensions, which is why wave-like behaviour of electrons (such as diffraction) is observable, unlike for everyday macroscopic objects.

Erwin Schrödinger developed a mathematical equation, based on the wave nature of the electron, whose solutions (called wave functions, ψ) describe the electron in an atom. The wave function ψ itself has no direct physical meaning, but ψ² gives the probability density of finding the electron at a point in space. An orbital is defined as the three-dimensional region of space around the nucleus within which the probability of finding an electron is maximum (typically taken as 90%) — this is quite different from Bohr's idea of a fixed, well-defined, two-dimensional circular orbit.

The size, shape, orientation and energy of orbitals, as well as the spin of the electron occupying them, are described by four quantum numbers:

• The principal quantum number, n (n = 1, 2, 3, …) determines the size and energy of the shell; the number of subshells in a shell equals n, the number of orbitals in a shell equals n², and the maximum number of electrons a shell can hold is 2n².
• The azimuthal (angular momentum) quantum number, l (l = 0 to n − 1, for a given n) determines the shape of the subshell/orbital: l = 0, 1, 2, 3 correspond to s, p, d, f subshells respectively. The number of orbitals in a subshell is 2l + 1, and the orbital angular momentum of the electron is √[l(l + 1)] · h/2π.
• The magnetic quantum number, ml (ml = −l to +l, including 0) describes the orientation of the orbital in space; there are (2l + 1) values of ml for a given l.
• The spin quantum number, ms (+1/2 or −1/2) describes the two possible spin orientations of an electron about its own axis.

The shapes of orbitals depend on l: s orbitals (l = 0) are spherically symmetric about the nucleus, with size increasing as n increases (1s < 2s < 3s); a 2s orbital, for example, has a spherical node in addition to being larger than 1s. p orbitals (l = 1) are dumbbell-shaped, with two lobes on either side of the nucleus separated by a nodal plane passing through it; the three p orbitals of a given shell (p_x, p_y, p_z) are identical in shape and energy but point along the three different axes. d orbitals (l = 2) have five orbitals (d_xy, d_yz, d_xz, d_x²−y², d_z²); four of these have a four-lobed, clover-leaf shape lying along or between the axes, while d_z² has two lobes along the z-axis together with a doughnut-shaped ring of electron density around the centre in the xy-plane. The total number of nodes (regions of zero probability) in an orbital equals n − 1, of which l are angular nodes and (n − l − 1) are radial nodes.

Worked example. Identify which of the following sets of quantum numbers (n, l, ml, ms) are not permissible for an electron, giving reasons.

1. n = 1, l = 1, ml = 0, ms = +1/2
2. n = 2, l = 1, ml = 0, ms = −1/2
3. n = 3, l = 0, ml = 0, ms = +1/2
4. n = 2, l = 2, ml = −1, ms = +1/2

For a given n, l can only range from 0 to n − 1. In set (1), n = 1 allows only l = 0, but l = 1 is given, so this set is not permissible. Set (2) is valid: for n = 2, l can be 0 or 1, and for l = 1, ml can be −1, 0 or +1, so ml = 0 is allowed. Set (3) is valid: for n = 3, l can be 0, 1 or 2, and l = 0 requires ml = 0. In set (4), n = 2 allows only l = 0 or 1, but l = 2 is given, so this set is also not permissible. So sets (1) and (4) are not permissible; sets (2) and (3) are permissible.

Once the shapes and relative energies of orbitals are known, the ground-state electronic configuration of an atom — how its electrons are distributed among the various orbitals — is built up using three rules.

The Aufbau principle states that, in the ground state of an atom, orbitals are filled in order of increasing energy. The order of filling is obtained from the (n + l) rule: an orbital with a lower value of (n + l) is filled before one with a higher value; if two orbitals have the same (n + l) value, the orbital with the lower value of n is filled first. This gives the familiar filling order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, ….

The Pauli exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers. Equivalently, an orbital can hold a maximum of two electrons, and these two electrons must have opposite (paired) spins.

Hund's rule of maximum multiplicity states that electron pairing in the orbitals of a subshell (such as the three p orbitals or the five d orbitals, which all have the same energy) does not begin until each orbital of that subshell contains one electron each, and all these singly occupied electrons have parallel spins.

Applying these rules generally gives the correct configuration, but for a few elements the configuration predicted by the simple Aufbau filling order is not the one actually observed, because completely filled (s², p⁶, d¹⁰, f¹⁴) and exactly half-filled (s¹, p³, d⁵, f⁷) sets of degenerate orbitals have extra stability. This extra stability arises from (i) the symmetrical distribution of electrons among the orbitals of a subshell, and (ii) a larger exchange energy released when electrons of parallel spin occupy such a symmetrical arrangement.

Worked example. Write the ground-state electronic configuration of chromium (Cr, Z = 24), and explain why it differs from the configuration predicted by simply filling orbitals in order of increasing energy.

• Simple Aufbau filling for Z = 24 would predict: [Ar] 3d⁴ 4s².
• The actual, experimentally observed configuration is: [Ar] 3d⁵ 4s¹.
• One electron shifts from the 4s orbital into the 3d orbitals so that the 3d subshell becomes exactly half-filled (3d⁵), with all five d orbitals singly occupied and parallel spins (following Hund's rule). This symmetrical, half-filled arrangement, together with the larger exchange energy it allows, makes 3d⁵4s¹ more stable than 3d⁴4s².

A similar shift occurs for copper (Cu, Z = 29): instead of the expected [Ar] 3d⁹ 4s², the observed configuration is [Ar] 3d¹⁰ 4s¹, because a completely filled 3d¹⁰ subshell is more stable.