Linear molecular geometry

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Linear molecular geometry

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Linear molecular geometry

Examples	Carbon dioxide CO₂ Xenon difluoride XeF₂
Point group	D_∞h
Coordination number	2
Bond angle(s)	180°
μ (Polarity)	0

Structure of beryllium fluoride (BeF₂), a compound with a linear geometry at the beryllium atom.

The linear molecular geometry describes the geometry around a central atom bonded to two other atoms (or ligands) placed at a bond angle of 180°. Linear organic molecules, such as acetylene (HC≡CH), are often described by invoking sp orbital hybridization for their carbon centers.

According to the VSEPR model (Valence Shell Electron Pair Repulsion model), linear geometry occurs at central atoms with two bonded atoms and zero or three lone pairs (AX₂ or AX₂E₃) in the AXE notation. Neutral AX₂ molecules with linear geometry include beryllium fluoride (F−Be−F) with two single bonds,^[1] carbon dioxide (O=C=O) with two double bonds, hydrogen cyanide (H−C≡N) with one single and one triple bond. The most important linear molecule with more than three atoms is acetylene (H−C≡C−H), in which each of its carbon atoms is considered to be a central atom with a single bond to one hydrogen and a triple bond to the other carbon atom. Linear anions include azide (N⁻=N⁺=N⁻) and thiocyanate (S=C=N⁻), and a linear cation is the nitronium ion (O=N⁺=O).^[2]

Linear geometry also occurs in AX₂E₃ molecules, such as xenon difluoride (XeF₂)^[3] and the triiodide ion (I−3) with one iodide bonded to the two others. As described by the VSEPR model, the five valence electron pairs on the central atom form a trigonal bipyramid in which the three lone pairs occupy the less crowded equatorial positions and the two bonded atoms occupy the two axial positions at the opposite ends of an axis, forming a linear molecule.

References

[edit]

^ Housecroft, C. E.; Sharpe, A. G. (2004). Inorganic Chemistry (2nd ed.). Prentice Hall. p. 43. ISBN 978-0-13-039913-7.
^ Greenwood, N. N.; Earnshaw, A. (1997). Chemistry of the Elements (2nd ed.). Oxford: Butterworth-Heinemann. ISBN 0-7506-3365-4.
^ Housecroft, C. E.; Sharpe, A. G. (2004). Inorganic Chemistry (2nd ed.). Prentice Hall. p. 47. ISBN 978-0-13-039913-7.

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v t e Molecular geometry
VSEPR theory Coordination number
Coordination number 2	Linear Bent
Coordination number 3	Trigonal planar Trigonal pyramidal T-shaped
Coordination number 4	Tetrahedral Seesaw Square planar
Coordination number 5	Trigonal bipyramidal Square pyramidal Pentagonal planar
Coordination number 6	Octahedral Trigonal prismatic Pentagonal pyramidal
Coordination number 7	Pentagonal bipyramidal Capped octahedral Capped trigonal prismatic
Coordination number 8	Square antiprismatic Dodecahedral Bicapped trigonal prismatic Cubic
Coordination number 9	Tricapped trigonal prismatic Capped square antiprismatic
Category

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Linear molecular geometry describes the spatial arrangement of atoms in a molecule where the central atom forms bonds with two surrounding atoms, resulting in a straight-line configuration with a bond angle of exactly 180 degrees.^[1] This geometry arises primarily from the valence shell electron pair repulsion (VSEPR) theory, which posits that electron pairs around the central atom repel each other and adopt positions that minimize electrostatic repulsion.^[2] Developed by Ronald J. Gillespie and Ronald S. Nyholm in the 1950s, VSEPR theory predicts linear molecular shapes for molecules with a steric number of two—meaning two regions of high electron density (typically bonding pairs) and no lone pairs on the central atom—or in cases with five electron domains where three are lone pairs, such as in trigonal bipyramidal electron geometry yielding a linear arrangement.^[3]^[4] Common examples of linear molecules with steric number two include carbon dioxide (CO₂), where the central carbon atom is double-bonded to two oxygen atoms, and beryllium fluoride (BeF₂), featuring a beryllium central atom with two single bonds to fluorine atoms.^[2]^[5] Other notable instances are hydrogen cyanide (HCN) and acetylene (C₂H₂), both exhibiting the characteristic 180-degree bond angle due to sp hybridization of the central atoms.^[6] In contrast, linear geometry with lone pairs, such as in the triiodide ion (I₃⁻) or xenon difluoride (XeF₂), occurs when the central atom has three equatorial lone pairs in a trigonal bipyramidal arrangement, positioning the two axial bonds linearly opposite each other.^[7]^[8] These structures are fundamental in understanding molecular polarity, reactivity, and properties like dipole moments, as linear symmetric molecules like CO₂ are nonpolar despite polar bonds.^[1]

Fundamentals

Definition

Linear molecular geometry describes the arrangement of atoms in a molecule where the constituent atoms lie in a straight line, with the bond angle between the bonds to the central atom measuring exactly 180 degrees. This shape arises in molecules featuring a central atom bonded to exactly two other atoms, with no lone pairs of electrons on the central atom to distort the alignment. The central atom serves as the point around which the terminal atoms are positioned symmetrically, a fundamental concept in understanding covalent bonding structures beyond simple diatomic molecules.^[8] The notion of linear geometry as a distinct molecular shape was first systematically explored in early 20th-century structural chemistry, building on advances in valence theory. In 1940, Nevil V. Sidgwick and Herbert M. Powell presented foundational ideas in their Bakerian Lecture, proposing that the spatial distribution of electron pairs in the valence shell of the central atom governs the overall stereochemistry of polyatomic molecules, including linear configurations for two-pair arrangements.^[9] These insights predated the full formulation of VSEPR theory, which later formalized linear geometry as a predicted outcome when electron pair repulsions position bonds oppositely.^[8] Distinguished from angular or planar geometries, linear molecular geometry stands as the simplest non-deviated form for multi-atom species, avoiding the deviations seen in bent structures (bond angles <180°) or trigonal arrangements (bond angles ~120°), where additional electron domains or lone pairs introduce asymmetry.^[8] This straight-line configuration ensures maximal separation of bonded atoms, reflecting an idealized balance in electron repulsion without the complexities of higher coordination numbers.

Key Characteristics

Linear molecular geometry is characterized by its high symmetry, belonging to the point group

D_{\infty h}

for ideal cases, which includes an infinite-fold principal rotation axis (

C_{\infty}

) along the internuclear axis, an infinite number of twofold rotation axes (

C_2

) perpendicular to

C_{\infty}

, infinite vertical mirror planes (

\sigma_v

) containing the

C_{\infty}

axis, and a horizontal mirror plane (

\sigma_h

) perpendicular to it.^[10] This symmetry arises in homonuclear diatomic molecules and triatomic AX

_2

species like CO

_2

, where the atoms are symmetrically arranged.^[10] The electronic prerequisite for linear geometry involves a central atom with exactly two electron domains, both of which are bonding pairs and no lone pairs, corresponding to the AX

_2

notation.^[11] This configuration ensures that the valence electrons occupy regions that maximize separation. Energetically, the collinear arrangement minimizes repulsion between these electron domains, positioning the bonds at a 180° angle to achieve the lowest energy state.^[2] In spatial terms, linear molecules exhibit a one-dimensional projection, with all atoms aligned collinearly along a single axis and no angular deviations in the ideal structure.^[11] This hallmark bond angle of 180° underscores the geometry's simplicity and stability.^[12]

Theoretical Basis

VSEPR Theory

The Valence Shell Electron Pair Repulsion (VSEPR) theory serves as a foundational model for predicting the geometry of molecules, including linear arrangements, by considering the repulsive interactions among electron pairs in the valence shell of the central atom. Developed by Ronald J. Gillespie and Ronald S. Nyholm in 1957, the theory posits that these electron pairs—encompassing both bonding pairs and lone pairs—arrange themselves in space to achieve the lowest possible energy configuration, thereby minimizing mutual repulsion.^[13] This qualitative approach focuses on the spatial distribution of electron density around the central atom, providing a simple yet effective tool for estimating molecular shapes without delving into detailed quantum mechanical calculations. To apply VSEPR theory, one first determines the steric number of the central atom, which represents the total number of electron domains (bonding pairs and lone pairs) surrounding it. The steric number is calculated by considering the Lewis structure: it equals half the sum of the central atom's valence electrons, the electrons contributed by surrounding atoms (typically one per monovalent ligand), plus any additional electrons from a negative charge or minus electrons for a positive charge. For a steric number of 2, the electron domains adopt a linear arrangement with a bond angle of 180°, as this geometry maximizes separation between the pairs. The theory assumes that bonding pairs, often involving multiple bonds treated as single domains, behave similarly to lone pairs in terms of repulsion, though actual bond lengths may vary. Linear molecular geometry also arises for a steric number of 5 in the AX₂E₃ configuration, where the electron geometry is trigonal bipyramidal; the three lone pairs occupy the equatorial positions to minimize repulsion, positioning the two bonding pairs in the axial positions opposite each other at 180°. A key aspect of VSEPR is the hierarchy of repulsions among electron pairs: lone pair-lone pair interactions are the strongest, followed by lone pair-bonding pair repulsions, with bonding pair-bonding pair interactions being the weakest.^[13] This ordering arises because lone pairs occupy more space due to their higher electron density concentrated near the central atom, influencing distortions in geometries with lone pairs but resulting in ideal linear shapes when absent. In cases with only bonding pairs and no lone pairs, such as a steric number of 2, all repulsions are equivalent (bonding pair-bonding pair), leading to symmetric linear geometry. For AX₂E₃, the lone pairs' stronger repulsions favor their placement in the less crowded equatorial plane, preserving the linear arrangement of the bonds. The VSEPR model employs a standardized notation, AX_nE_m, to classify molecular geometries, where A denotes the central atom, X represents each bonding pair to a ligand, and E indicates each lone pair, with n + m equaling the steric number.^[14] Linear molecular geometry corresponds to the AX₂E₀ configuration (steric number 2) or AX₂E₃ (steric number 5), with the ligands positioned at opposite ends of the central atom for maximal repulsion minimization in both cases. This notation systematically links the electron pair arrangement to the observed shape, offering a predictive framework distinct from, yet complementary to, the hybridization model that rationalizes bonding through atomic orbital overlap.

Hybridization Model

The hybridization model, developed by Linus Pauling in 1931, describes how atomic orbitals on a central atom combine linearly to form equivalent hybrid orbitals that better accommodate the geometry of molecular bonds.^[15] This mixing of orbitals equalizes their energies and shapes, enabling directed bonding that aligns with observed molecular structures. For linear molecular geometry with steric number 2, the relevant process is sp hybridization, in which the central atom's valence s orbital and one p orbital merge to generate two sp hybrid orbitals.^[16] These hybrids are collinear, separated by a bond angle of 180°, pointing in opposite directions to maximize separation and minimize repulsion.^[16] This configuration is exemplified in molecules like BeCl₂, where the beryllium atom undergoes sp hybridization to form two sigma bonds.^[17] In sp-hybridized systems, the linear arrangement arises from the end-to-end overlap of these sp hybrid orbitals with ligand atomic orbitals, creating strong sigma (σ) bonds along the molecular axis.^[16] Such overlap ensures efficient sharing of electrons and stability in the linear form, consistent with the geometric predictions of VSEPR theory.^[16] While effective for explaining bonding in many cases, the hybridization model primarily applies to main-group elements and has limited utility for transition metals, where d-orbital involvement complicates simple s-p mixing.

Examples

Triatomic Molecules

Triatomic molecules exemplify the simplest cases of linear molecular geometry, where a central atom is bonded to two terminal atoms with no lone pairs on the central atom, resulting in an AX₂ VSEPR classification and bond angles of 180°.^[18] This arrangement arises from the repulsion minimization of two bonding pairs, often associated with sp hybridization of the central atom's orbitals.^[18] Carbon dioxide (CO₂) is a classic AX₂ linear molecule with the structure O=C=O.^[19] The central carbon atom utilizes its four valence electrons to form two double bonds with oxygen atoms, achieving a stable octet while maintaining linear symmetry.^[19] This symmetry renders CO₂ nonpolar, as the dipole moments of the C=O bonds cancel out.^[20] In the gas phase, beryllium chloride (BeCl₂) adopts an AX₂ linear configuration, Cl–Be–Cl. Beryllium's electron deficiency, with only two valence electrons, leads to this structure where it forms two single bonds without lone pairs, violating the octet rule but stabilizing through linear arrangement.^[18] Nitrous oxide (N₂O) exhibits linear geometry but is asymmetric, best represented by resonance structures such as N≡N⁺–O⁻ and ⁻N=N⁺=O, which delocalize electrons and contribute to its stability.^[21] The central nitrogen serves as the AX₂ central atom in this AX₂E₀ classification, though the unequal bonding lengths reflect the resonance influence.^[22] The linear structures of these triatomic molecules have been experimentally confirmed through techniques such as X-ray diffraction for crystalline analogs and infrared (IR) spectroscopy, which reveals characteristic vibrational modes consistent with 180° bond angles—for instance, the asymmetric stretch in CO₂ at approximately 2349 cm⁻¹ indicates high symmetry.^[23] Gas-phase electron diffraction for BeCl₂ yields a Be–Cl bond length of 1.791 Å, affirming linearity, while microwave spectroscopy for N₂O provides rotational constants matching a linear asymmetric model.^[22]

Polyatomic Molecules

Polyatomic molecules exhibiting linear geometry often feature central atoms with sp hybridization or hypervalent electron configurations that favor collinear arrangements to minimize repulsion and maximize bond overlap. A classic example is acetylene (HC≡CH), where the carbon-carbon triple bond consists of one σ bond and two π bonds, resulting from sp hybridization on each carbon atom; this hybridization directs the bonds along a straight line with a 180° bond angle, extending linearity across the entire molecule.^[24]^[25] Xenon difluoride (XeF₂) represents a hypervalent case, classified under VSEPR theory as AX₂E₃, where the central xenon atom has five electron pairs in a trigonal bipyramidal arrangement; the three lone pairs occupy equatorial positions to minimize repulsion, leaving the two fluorine atoms in axial positions and yielding a linear molecular shape with Xe-F-Xe bond angles of 180°.^[3]^[26] This configuration underscores how lone pair placement can enforce linearity in expanded octet systems. Allene (H₂C=C=CH₂) demonstrates linearity through cumulated double bonds, where the central carbon atom adopts sp hybridization, forming two perpendicular π bonds with the terminal CH₂ groups while aligning the overall carbon skeleton linearly; this arrangement stabilizes the molecule by optimizing orbital overlap in the cumulated system.^[27] The stability of such linear polyatomic molecules arises from multiple bonding, which strengthens connections through enhanced electron sharing as seen in acetylene's triple bond, and hypervalency, which allows elements like xenon to accommodate extra ligands without destabilizing the linear framework, as evidenced in XeF₂'s 3c-4e bonding interactions.^[25]^[28] These factors enable extended linear chains in larger systems by balancing steric and electronic repulsions.

Properties and Analysis

Bond Angles and Distances

In linear molecular geometry, the ideal bond angle is precisely 180° for symmetric molecules, arising from the arrangement that minimizes electron repulsion along the molecular axis. This collinear configuration ensures that atoms lie on a straight line, with no deviation in the equilibrium structure for prototypical cases like carbon dioxide (CO₂). Bond lengths in linear molecules vary based on atomic radii, electronegativity differences, and bond order, leading to shorter distances for higher bond orders and smaller atoms. For instance, the C–O bond in CO₂ measures 116 pm, reflecting the double-bond character and comparable atomic sizes of carbon and oxygen.^[29] In contrast, single bonds in linear species like beryllium hydride (BeH₂) exhibit longer H–Be distances around 133 pm, influenced by the larger effective radius of beryllium.^[30] These variations directly impact the overall molecular dimensions and stability. Although the equilibrium geometry is strictly linear, real-world measurements reveal slight deviations from the 180° bond angle in the gas phase, primarily due to vibrational modes that introduce transient bends and anharmonicity in the potential energy well. These effects broaden the bond angle distribution slightly, but the time-averaged angle remains 180° in dilute conditions, with deviations on the order of a few degrees in high-temperature or dense phases. Microwave spectroscopy provides high-precision measurements of bond angles and distances for polar linear molecules by analyzing rotational transitions, which yield moments of inertia convertible to geometric parameters via the relation

I = \sum m_i r_i^2

, where

I

is the moment of inertia,

m_i

are atomic masses, and

r_i

are distances from the center of mass.^[31] This technique achieves accuracy to within 0.01 Å for bond lengths and confirms linearity through the characteristic spacing of spectral lines.^[32]

Spectroscopic Features

Infrared (IR) spectroscopy provides key insights into the vibrational modes of linear molecules, particularly those with D_{\infty h} symmetry, such as CO₂. The symmetric stretching mode in CO₂ is IR inactive because it does not change the molecule's dipole moment. In contrast, the asymmetric stretching mode is IR active and appears as a strong absorption band at approximately 2350 cm⁻¹.^[33]^[34] Raman spectroscopy complements IR by detecting modes that are Raman active due to changes in molecular polarizability. For linear molecules like CO₂, the symmetric stretching mode is Raman active and observed around 1330 cm⁻¹.^[34] The degenerate bending mode is also Raman active, appearing near 667 cm⁻¹, highlighting the symmetry selection rules that distinguish Raman from IR activity.^[33] Rotational spectroscopy of linear molecules reveals simple spectra characterized by a single rotational constant

B

, which governs the spacing between rotational energy levels. This constant is given by

B = \frac{h}{8\pi^2 c I},

where

h

is Planck's constant,

c

is the speed of light, and

I

is the moment of inertia about the molecular axis.^[35]^[36] The resulting microwave spectra show evenly spaced lines separated by

2B

, facilitating precise determination of bond lengths from

I

.^[36] Nuclear magnetic resonance (NMR) spectroscopy in linear molecules with high symmetry often results in simplified spectra due to magnetically equivalent nuclei. For instance, in symmetric linear molecules like acetylene (HC≡CH), the terminal hydrogen nuclei are chemically and magnetically equivalent, producing a single signal in ¹H NMR.^[37]^[38] This equivalence arises from the molecule's inversion symmetry, reducing the number of distinct resonances and aiding structural confirmation.^[38]

Advanced Topics

Linear Geometry in Ions

Linear molecular geometry in ions arises from the valence shell electron pair repulsion (VSEPR) model, where the net charge modifies the total valence electron count and thus the steric number around the central atom, often resulting in two electron domains for linearity. In polyatomic ions, positive charges reduce electron density, favoring compact arrangements like linear to enhance bond strength through reduced repulsion and better orbital alignment. The cyanide ion (CN⁻) adopts a linear geometry with a steric number of 2, featuring a triple bond between carbon and nitrogen, and belongs to the C∞v point group as an isolated species.^[39] When serving as a ligand in metal complexes, CN⁻ retains its linear C≡N arrangement, coordinating through the carbon atom to maintain overall symmetry in the coordination sphere. Mercury(II) complexes, such as HgCl₂, exhibit linear geometry around the central Hg²⁺ ion due to its d¹⁰ electron configuration, which promotes sp hybridization and minimizes steric interactions with two coordinating ligands.^[40] This linearity is characteristic of two-coordinate d¹⁰ metal centers, where the empty valence orbitals align collinearly for optimal bonding. The nitronium ion (NO₂⁺) is a classic example of an AX₂ species with linear O=N=O geometry, where the positive charge on nitrogen results in 16 valence electrons and no lone pairs, enforcing a 180° bond angle.^[41] This ion serves as the electrophile in electrophilic aromatic substitution reactions, such as nitration, where its linear structure facilitates direct attack on π-electron systems.^[42]

Deviations and Exceptions

While the ideal linear geometry for triatomic molecules like BeCl₂ is predicted by VSEPR theory for the gas phase, significant deviations occur in the solid state due to intermolecular interactions. In solid BeCl₂, the monomeric linear structure polymerizes into a one-dimensional chain where each beryllium atom adopts a tetrahedral coordination through edge-sharing BeCl₄ tetrahedra, with Be–Cl bridge bond lengths of approximately 2.02 Å and terminal bonds of 1.82 Å. This polymeric arrangement arises from the electron-deficient character of beryllium, enabling dative bonding with chloride ligands from neighboring units, thus stabilizing the solid phase over the isolated linear monomer.^[43] Relativistic effects play a crucial role in maintaining near-linear geometries in heavy-element compounds, such as those of mercury, but can introduce subtle bond compressions that deviate from non-relativistic predictions. In HgCl₂, the solid-state structure features nearly linear Cl–Hg–Cl units with a bond angle of 180° and Hg–Cl distances of 2.38 Å, shorter than expected without relativity due to the contraction of the 6s orbital, which enhances s-p hybridization and favors two-coordinate linearity while inert-pairing reduces higher coordination. This relativistic stabilization results in slightly compressed bonds compared to lighter analogs like ZnCl₂ (~2.30 Å), contributing to the molecule's preference for linear over bent configurations in both gas and solid phases.^[44] Quantum mechanical treatments using molecular orbital theory reveal exceptions where linear geometries are pseudo-linear due to Jahn-Teller distortions in systems with degenerate electronic states. For certain open-shell linear molecules, the pseudo Jahn-Teller effect couples the ground state with excited states via bending vibrations, leading to instability and small distortions from linearity, as the theorem excludes perfect linearity only for closed-shell cases but allows pseudo-linear equilibria in degenerate configurations. In MO theory, this manifests as mixing of σ_g and π_u orbitals under D_{∞h} symmetry, resulting in a shallow bending potential that keeps the geometry nearly linear but with dynamic distortions, as seen in some transient species or ions.^[45]

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