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Química Geral · Atoms, electrons and the periodic table

Números quânticos e orbitais atômicos

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O que você vai aprender

  • State the allowed values of n, l, mℓ and ms and recognise a forbidden set of quantum numbers.
  • Convert between quantum numbers and subshell notation, and count the orbitals and electrons in a shell or subshell.
  • Describe the shapes of s, p and d orbitals and count their radial and angular nodes.
  • Compare orbital energies in hydrogen and hydrogen-like ions with those in many-electron atoms.

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Four quantum numbers

Each electron in an atom is described by four quantum numbers. The principal quantum number n = 1, 2, 3, … sets the shell, the size of the orbital and, in hydrogen, its energy. The angular momentum quantum number l takes the integer values 0 to n − 1 and sets the subshell and its shape; l = 0, 1, 2, 3 are written s, p, d, f (l = 4 is g). The magnetic quantum number mℓ takes the integer values from −l to +l, one for each orbital in the subshell, and sets the orbital's orientation. The spin quantum number ms is +½ or −½. The first three numbers name an orbital: n = 3, l = 2 is the 3d subshell, and n = 3, l = 2, mℓ = −1 is one of its five orbitals. A set such as n = 2, l = 2 is forbidden because l must be smaller than n, so a 2d subshell does not exist; n = 2, l = 1, mℓ = −2 is forbidden because |mℓ| cannot exceed l. The Pauli exclusion principle says no two electrons in an atom share all four quantum numbers, so one orbital holds at most two electrons, one with ms = +½ and one with ms = −½.

Counting orbitals and electrons

Shell n contains n subshells (l = 0 to n − 1). A subshell with quantum number l contains 2l + 1 orbitals and holds up to 2(2l + 1) electrons: s 1 orbital and 2 electrons, p 3 and 6, d 5 and 10, f 7 and 14. Adding the subshells of shell n gives n² orbitals and 2n² electrons. For n = 3: 3s + 3p + 3d = 1 + 3 + 5 = 9 orbitals, which hold 18 electrons, and exactly half of those electrons, 9, can have ms = +½. To count orbitals that share one quantum number, list every allowed combination: with n = 3 and mℓ = 0 the subshell can be 3s, 3p or 3d, so there are 3 such orbitals and 6 electrons. In a filled subshell every orbital is full, so a neon atom (drawn), 1s² 2s² 2p⁶, has 4 electrons with l = 0 and 6 with l = 1.

Shapes of s, p and d orbitals

An orbital's boundary surface encloses most (typically 90%) of the electron density. Every s orbital is spherical. Each p orbital has two lobes on opposite sides of the nucleus along one axis, with opposite signs of the wavefunction, separated by a nodal plane through the nucleus: 2pₓ lies along the x axis and its nodal plane is the yz plane. Four of the five d orbitals have four lobes in a plane. The d_xy, d_xz and d_yz lobes point between the axes, and each has two nodal planes; for d_xy these are the xz and yz planes. The d_x²−y² lobes point along the x and y axes, and its nodal planes are the two vertical planes that bisect the angles between those axes. The fifth, d_z², has two lobes along the z axis and a ring (torus) around the nucleus in the xy plane with the opposite sign; its two angular nodes are cones, not planes. The orbitals of a set are distinguished by orientation, not capacity: every orbital holds at most two electrons.

Radial and angular nodes

A node is a surface where the wavefunction, and so the probability of finding the electron, is zero. An orbital has n − 1 nodes in total. Of these, l are angular nodes (planes or cones through the nucleus) and n − l − 1 are radial nodes (spheres around the nucleus). Examples: 1s has none; 2s has one radial node; 3p has one radial and one angular node; 4d has one radial and two angular nodes; 5f has one radial and three angular nodes. Every orbital with l ≥ 1 has an angular node passing through the nucleus, so only s orbitals have non-zero electron density at the nucleus. Radial nodes give the radial distribution an inner maximum: a 2s electron has a small probability peak close to the nucleus, inside the 1s region, while a 2p electron has none. Getting close to the nucleus in this way is called penetration, and it decreases in the order s > p > d > f for the same n.

Orbital energies: hydrogen versus many-electron atoms

In hydrogen and hydrogen-like ions (one electron, nuclear charge Z) the energy depends only on n: Eₙ = −13.6 eV × Z²/n². In hydrogen E₁ = −13.6 eV, E₂ = −3.40 eV and E₃ = −1.51 eV, so the levels crowd together as n rises, and all n² orbitals of one shell are degenerate: 3s, the three 3p and the five 3d orbitals have the same energy. In He⁺ (Z = 2) E₁ = −54.4 eV and E₂ = −13.6 eV, the same as hydrogen's 1s. In an atom with more than one electron, each electron is partly shielded from the nucleus by the others, and the energy depends on both n and l. A penetrating orbital feels a larger effective nuclear charge, so within one shell s < p < d < f; in lithium 2s lies below 2p. The three orbitals of a p subshell (and the five of a d subshell) stay degenerate in a free atom. The subshells fill in order of increasing n + l, the lower n first when n + l is equal: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p. That is why potassium and calcium place their outer electrons in 4s rather than 3d, and why titanium (drawn) is 1s² 2s² 2p⁶ 3s² 3p⁶ 3d² 4s².

What to carry forward

n sets the shell (1, 2, 3, …), l sets the subshell (0 to n − 1; s, p, d, f), mℓ sets the orbital (−l to +l) and ms the spin (±½). A subshell has 2l + 1 orbitals, a shell n² orbitals and 2n² electrons, and each orbital holds two electrons of opposite spin. An orbital has l angular nodes and n − l − 1 radial nodes. In one-electron species the energy depends only on n (−13.6 eV × Z²/n²); in many-electron atoms shielding and penetration split the subshells so that s < p < d < f within a shell, and subshells fill in order of increasing n + l.

Problemas de prática (40)

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Preveja o produto 5

Quantum numbers

  • Allowed l values for n = 3List the allowed values of the angular momentum quantum number for a given shell.
  • Quantum numbers of sodium's valence electronAssign quantum numbers to the valence electron of a ground-state alkali-metal atom.

Orbital energies

  • Where does potassium's 19th electron go?Predict whether 4s or 3d receives the outer electron of potassium and justify it from orbital energies.
  • Aufbau filling orderOrder subshells by the n + l rule for filling in neutral many-electron atoms.

Shells and subshells

  • Subshells of the fourth shellList the subshells that make up a given shell.

Classificar 4

Orbital shapes and nodes

  • Angular nodes of a d orbitalUse the rule angular nodes = l to count the angular nodes of a d orbital.
  • Shape of a 2p orbitalRecognise the boundary-surface shape of a p orbital.

Orbital energies

  • What sets orbital energy in hydrogen?Identify the quantum number that alone determines orbital energy in the hydrogen atom.

Quantum numbers

  • Quantum numbers of a 3d electronMatch a set of four quantum numbers to an electron in the 3d subshell.

Comparar 4

Orbital energies

  • 2s and 2p in a carbon atomCompare the energies of s and p subshells of the same shell in a many-electron atom.
  • Rank one-electron energy levelsRank energy levels of hydrogen and He⁺ using Eₙ = −13.6 eV × Z²/n².

Shells and subshells

  • Rank by number of orbitalsRank subshells and shells by the number of orbitals they contain.

Orbital shapes and nodes

  • Rank orbitals by radial nodesRank orbitals by their number of radial nodes using n − l − 1.

Calcular 15

Shells and subshells

  • Electrons with l = 1 in argonCount the electrons with a given l value in a noble-gas atom from its configuration.
  • Capacity of the 4f subshellCalculate the electron capacity of a subshell from 2(2l + 1).
  • Capacity of the 5g subshellExtend the 2(2l + 1) rule to a subshell beyond f.
  • Electrons with l = 2 in ironCount the electrons with l = 2 in a transition-metal atom from its ground-state configuration.
  • Electrons with mℓ = 0 in neonCount the electrons with mℓ = 0 in an atom by checking every occupied subshell.
  • Orbitals in the n = 5 shellCalculate the number of orbitals in a shell from n².
  • Capacity of the n = 4 shellCalculate the maximum number of electrons in a shell from 2n².

Quantum numbers

  • Electrons with n = 3, l = 1, mℓ = 0Use the Pauli exclusion principle to find how many electrons can share three fixed quantum numbers.
  • Number of mℓ values when l = 3Count the allowed magnetic quantum numbers for a given value of l.
  • Orbitals with n = 4 and mℓ = −1Count the orbitals in a shell that share a given value of mℓ.
  • Electrons with n = 4 and ms = −½Count the electrons in one atom that can share a principal quantum number and a spin value.

Orbital energies

  • Hydrogen energy levels crowd togetherCompare successive energy gaps in hydrogen using Eₙ = −13.6 eV/n².
  • Orbitals degenerate with 3s in hydrogenCount the degenerate orbitals of a shell in the hydrogen atom.

Orbital shapes and nodes

  • Radial nodes in a 5d orbitalCalculate the number of radial nodes of an orbital from n − l − 1.
  • Total nodes in a 5f orbitalCalculate the total number of nodes in an orbital and split it into radial and angular nodes.

Encontre o erro 4

Quantum numbers

  • Error in beryllium's quantum numbersRecognise a violation of the Pauli exclusion principle in a list of electron quantum numbers.
  • Find the forbidden set of quantum numbersIdentify the set of quantum numbers that breaks the rule l ≤ n − 1.

Orbital energies

  • Find the error: shielding in hydrogenIdentify why a shielding argument for subshell splitting fails for the hydrogen atom.

Shells and subshells

  • Which subshell cannot exist?Recognise subshell notation that breaks the rule l ≤ n − 1.

Interprete as evidências 4

Orbital shapes and nodes

  • Nodal planes of d_xyIdentify the nodal planes of a d orbital whose lobes lie between the axes.
  • The odd one out: 3d_z²Describe the shape and nodes of the d_z² orbital and how it differs from the other d orbitals.
  • The nodal plane of 2pₓLocate the nodal plane of a p orbital from its axis label.

Quantum numbers

  • Subshell from n and lTranslate a pair of n and l values into subshell notation.

Trabalhe de trás para frente 2

Orbital energies

  • Li²⁺ level matching hydrogen's 1sFind the level of a hydrogen-like ion whose energy equals a given hydrogen level.

Orbital shapes and nodes

  • Identify an orbital from its nodesWork back from the numbers of radial and angular nodes to the orbital's n and l.

Explique por quê 2

Orbital shapes and nodes

  • Electron density at the nucleusExplain why only s orbitals have non-zero electron density at the nucleus.

Orbital energies

  • Why lithium uses 2s, not 2pExplain the splitting of 2s and 2p in a many-electron atom by penetration and shielding.

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