Bond Length - Bond Valence Relationships for Carbon - Carbon and Carbon - Oxygen Bonds

In the present study, relationships are developed for determining bond orders (also referred to as bond valences or bond numbers) from published bond lengths for carbon-carbon (C-C) and carbon-oxygen (C-O) bonds. The relationships are based on Pauling’s empirical formula s = exp((Ro-R)/b)), where s is the bond order, R is the corresponding bond length, Ro is the unit valence bond length, and b is a fitting parameter. We use a recently derived relationship for the b parameter in terms of the bonding atoms’ published atomic orbital exponents. The resulting equations were checked against published x-ray diffraction (XRD) data for 176 carbon systems with 540 published C-C bond lengths, and 50 oxygen systems having 72 published C-O bond lengths. The C-C and C-O bond length-valence relationships are shown to have sufficient applicability and accuracy for use in any bonding environment, regardless of physical state or oxidation number

Theoretical Justification for Bond Valence -- Bond Length Empirical Correlations

Bond valence – bond length empirical correlations are of great interest in chemistry, biology, geology and materials science because they offer a quick and convenient way of checking and evaluating molecular structures. Linus Pauling’s relationship is the most commonly used, but is a two-parameter fit where R0 and b must be optimized. In this study, a simplified quantum-mechanical approach was used to derive Pauling’s empirical bond valence – bond length relationship. A covalency factor was also introduced to account for the difference in “softness” between cation and anion (resulting in increased orbital overlap). An expression for the b parameter was determined that yields values that are in agreement with experimental data. The derived relationship for the b parameter allows an independent determination of b using orbital exponents and electronegativity values for the cation and anion

Titanium-Oxygen Bond Length -Bond Valence Relationship

A bond length–bond valence correlation is a simple method of checking and evaluating molecular structures and is of great interest in chemistry, biology, geology, and material science. Recently, we used quantum-mechanical arguments to derive Pauling’s bond length-valence relationship and to define the adjustable fitting parameter b in terms of atomic-orbital exponents. Improved orbital exponents were generated for elements 1-103 using published atomic radii and single-bond covalent radii as well as a continuous function for effective principal quantum number. In this study, we use orbital exponents for titanium (Ti) and oxygen (O) to generate a bond length-valence relationship for Ti-O bonds. Recent crystallographic Ti-O bond lengths from 32 environments were collected and converted to Ti-O bond valences to check the reliability of the bond length-valence relationship where Ro was found (bond length of unit valence). This relationship is expected to apply to any Ti-O bond regardless of environment, physical state, or oxidation number

Valence-Length Correlations for Chemical Bonds from Atomic Orbital Exponents

Pauling’s empirical bond valence-length correlation has proven valuable because it offers a quick and convenient way of checking and evaluating molecular structures and determining oxidation states from measured bond lengths. In this study, a simplified quantum-mechanical approach was used to derive Pauling’s empirical bond valence-length relationship by considering overlap of hydrogen-like orbitals. An expression for the b “empirical” fitting parameter was derived in terms of atomic-orbital exponents. A new set of orbital exponents is presented using published atomic/covalent radii and a continuous function for the effective principal quantum. The b parameters calculated from the orbital exponents are consistent with bond valence-length data from crystallographic data. In general, atomic-orbital exponents may be used to determine bond valence- length relationships for any chemical bond regardless of state, oxidation number, or environment

Raman Spectroscopy of Titania (TiO\u3csub\u3e2\u3c/sub\u3e) Nanotubular Water-Splitting Catalysts

The phase composition of nanotubular TiO2 films was correlated with photoelectrochemical activity as a function of O2-annealing temperature. TiO2 nanotubes have been shown to be more efficient than polycrystalline TiO2 for the photocatalytic splitting of water. Raman spectroscopy was used to identify and quantify the amorphous and crystalline TiO2 phases. The amorphous TiO2 nanotubular array was found to consist of TiO6 8- octahedra having the same average structure as those present in the anatase and rutile phases of TiO2. Results show that the anatase-to-rutile transformation on Ti metal initiates at much lower temperatures compared to polycrystalline TiO2 and this is attributed to oxygen vacancies located at the metal/oxide interface and is likely responsible for increasing the photocurrent density

A Bond Length – Bond Valence Relationship for Carbon – Nitrogen Bonds

In a recent study, Pauling’s relationship between bond length and valence was derived along with a definition for his fitting parameter b that incorporates the orbital exponents for each atom contributing to the bond of interest. The values of b for various bonds, including C-N bonds, were calculated using the orbital exponent data. In this study, Pauling’s correlation between bond length and bond valence, as well as his valence sum rule, were used with the recently-derived definition for b in order to produce a relationship specifically applicable to C-N bonds. The resulting equation was checked against published x-ray diffraction data for 430 C-N bonds. It is expected, and shown by the data presented in this study, that these equations relating the bond length and bond valence of C-N bonds have sufficient applicability and accuracy for use in any bonding environment, regardless of physical state or oxidation number

Bond Length and Bond Valence for Tungsten-Oxygen and Tungsten-Sulfur Bonds

In 1947, Linus Pauling presented an “empirical” dependence of bond valence (s, also referred to as bond order) and bond length R: s = exp[(R0-R)/b], where R0 is bond length of unit valence and “b” is a fitting parameter. Recently, an expression was derived for relating the b fitting parameter to theoretically derived atomic orbital exponents. With a method to calculate b, both R0 and atomic orbital exponents can be experimentally determined through optimized fitting for W-O and W-S bonds. In the present study, bond length – valence relationships are found for W-O and W-S chemical bonds using published crystallographic data. The atomic orbital exponent for tungsten was found to be zW = 1.534. Unit valence (single bond) bond lengths were found to be Ro(W-O) = 1.901 Å and Ro(W-S) = 2.307 Å

Bond Length and Bond Valence Relationships for Chromium Oxides, Chromium Sulfides, Molybdenum Oxides, and Molybdenum Sulfides

Pauling determined an empirical logarithmic dependence of bond order (bond valence), s, to bond length, R, s = exp(R0 – R/ b), where R0 is unit bond length and b is a fitting parameter. Recently, an expression was derived for relating the b fitting parameter to theoretically derived atomic orbital exponents. With a method to calculate b, both R0 and atomic orbital exponents can be experimentally determined through optimized fitting for Cr-O, Cr-S, Mo-O, and Mo-S. In the present study, bond length – valence relationships are found for Cr-O, Cr-S, Mo-O, and Mo-S chemical bonds using published crystallographic data. In addition, atomic orbital exponents were found for chromium and molybdenum: z­­Cr = 1.247 and zMo = 1.381. Finally, bond lengths of unit bond valence, or true single bonds, were found using the bond valence model: Ro(Cr-O) = 1.770 Å, Ro(Cr-S) = 2.159 Å, Ro(Mo-O) = 1.893 Å, and Ro(Mo-S) = 2.264 Å

A Chandra study of particle acceleration in the multiple hotspots of nearby radio galaxies

We present Chandra observations of a small sample of nearby classical double radio galaxies which have more than one radio hotspot in at least one of their lobes. The X-ray emission from the hotspots of these comparatively low-power objects is expected to be synchrotron in origin, and therefore to provide information about the locations of high-energy particle acceleration. In some models of the relationship between the jet and hotspot the hotspots that are not the current jet termination point should be detached from the energy supply from the active nucleus and therefore not capable of accelerating particles to high energies. We find that in fact some secondary hotspots are X-ray sources, and thus probably locations for high-energy particle acceleration after the initial jet termination shock. In detail, though, we show that the spatial structures seen in X-ray are not consistent with naive expectations from a simple shock model: the current locations of the acceleration of the highest-energy observable particles in powerful radio galaxies need not be coincident with the peaks of radio or even optical emission.Comment: Accepted for ApJ. 33 pages, 8 figures inc. 2 in colo

A General Valence-Length Correlation for Determining Bond Orders: Application to Carbon-Carbon and Carbon-Hydrogen Chemical Bonds

A quantum-mechanical LCAO approach was used to derive Pauling’s popular empirical bond valencelength relationship s = exp((R₀-R)/b), where s is the bond order or bond valence associated with bond length R, and R₀ and b are fitting parameters. An expression for the b “empirical” fitting parameter is derived in terms of atomic orbital exponents. The b parameters calculated from the atomic orbital exponents are consistent with optimized b parameters. In general, atomic orbital exponents may be used to determine bond valence-length relationships for any chemical bond regardless of valence state, oxidation number, physical or chemical environment. In this study, almost two-thousand carbon-carbon and carbon-hydrogen bond lengths were evaluated from over 40 compounds to yield reliable bond valence – bond length relationships for C-C and C-H bonds. The atomic orbital exponent for carbon was found to be ζC = 1.651. Unit valence bond lengths (R₀ where s = 1) were found to be 1.493 Å for the carbon-carbon bond and 1.061 Å for the carbon-hydrogen bond