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Carbon held in organic matter, distinguished from carbonate and dissolved inorganic reservoirs.

An organometallic compound containing a carbon–aluminium bond.

An organic compound containing a carbon–boron bond.

An organolithium reagent contains a carbon–lithium interaction with substantial carbanionic carbon reactivity. Such reagents can act as bases or nucleophiles, and their solution aggregation affects both roles.

The chemistry of compounds containing metal–carbon bonds, including their structures, bonding and transformations in stoichiometric and catalytic reactions.

A four-membered titanium–oxygen–carbon–carbon intermediate in carbonyl olefination. Fragmentation yields a carbon–carbon double bond and a titanium–oxygen product.

electrophilic activation of an alkene by mercury(II), followed by nucleophilic opening of a bridged mercurinium ion. water usually attacks the more substituted carbon; subsequent demercuration replaces C–Hg with C–H.

The closed $1s^2$ and $2s^2$ subshells contribute zero total orbital angular momentum and spin, so only the two $2p$ electrons matter. The first of Hund's rules maximizes their total spin, giving $S=1$. The Pauli exclusion principle then requires an antisymmetric spatial wavefunction. Two parallel-spin electrons cannot occupy the same $m_l$ orbital: the greatest possible orbital projection is $1+0=1$, so the maximal allowed $L$ within this triplet is $L=1$.

The ground term is therefore ${}^{3}P$, in term symbol notation ${}^{2S+1}{L}_J$. Since the $p^2$ subshell contains fewer than half of its six possible electrons, the third of Hund's rules places the smallest $J$ lowest. Addition of angular momentum gives $J=0,1,2$, in that energy order within the ground term. ThusThese are the fine structure partners of the same electronic term, with magnetic multiplicities $1,3,5$, rather than different principal-shell excitations. The ordering agrees with the tabulated lowest neutral-carbon levels.^[ref]

The closed $1s^{2}2s^2$ shells have $L=S=0$, leaving the equivalent $2p^2$ electrons. Addition of angular momentum and the Pauli exclusion principle permit the atomic terms ${}^{3}P,{}^{1}D,{}^{1}S$. Their term symbols use ${}^{2S+1}{L}_J$. Hund's rules put the triplet state below the singlet states; for a less-than-half-filled $p$ shell the smallest $J$ in ${}^{3}P$ is lowest. ThusAll have even parity. These assignments agree with independent tabulated carbon energy levels.^[ref] The rounded source values should not be forced to satisfy an exact Landé interval rule; pure first-order LS coupling need not account for every measured fine structure correction.

For two decays from a common upper energy level, their photon wavenumber difference equals the separation of the two lower energy levels. Treating both quoted wavelengths as vacuum values initially giveswhose difference $11443.96\mathrm{c}{\mathrm{m}}^{-1}$ is close to $21648-10192=11456\mathrm{c}{\mathrm{m}}^{-1}$. The only matching lower pair among the supplied energy levels is ${}^1{D}_2,{}^1{S}_0$, with the shorter-wavelength photon ending on ${}^1{D}_2$. The two independent upper energy estimates areThusThe electric-dipole selection rules require odd parity, $S=0$, $L=1$ and $J=1$ to connect to both singlet lower energy levels. The apparent discrepancy has a concrete convention explanation. The NIST carbon line table lists $193.0906\mathrm{nm}$ under vacuum wavelengths and $247.8561\mathrm{nm}$ under air wavelengths, reproducing the source's rounded numbers.^[ref] The second number must therefore be converted to a vacuum wavelength before using $E=hc/\lambda$. For the tabulated upper $61981.82\mathrm{c}{\mathrm{m}}^{-1}$ and lower $21648.02\mathrm{c}{\mathrm{m}}^{-1}$ energy levels, its vacuum wavelength is $247.931\mathrm{nm}$, corresponding to an air refractive index about $1.00030$. This removes the spurious $12\mathrm{c}{\mathrm{m}}^{-1}$ discrepancy. The first calculation alone, which correctly uses a vacuum wavelength, already gives the intended upper energy level to the accuracy of the rounded data. The independent upper value is $61981.82\mathrm{c}{\mathrm{m}}^{-1}$.^[ref]

In the idealized isolated-carbon chemical atom interpretation, both beams carry approximately $2.68\mathrm{eV}$ per incident particle, close to the ${}^1{S}_0$ threshold. However equal incident energies do not imply equal excitation channels.

A single optical photon cannot strongly excite any of the listed even-parity $2p^2$ energy levels from the even-parity ${}^3{P}_0$ ground state through an electric dipole transition. In particular ${}^3{P}_0\to {}^1{S}_0$ also violates $\Delta S=0$ and the $J=0\leftrightarrow 0$ restriction. The lowest strong opposite-parity excitations lie well above $2.68\mathrm{eV}$. Thus one expects no strong resonant absorption or corresponding strong atomic fluorescence from these ground-state chemical atoms at this optical wavelength; weak higher-order processes and scattering are separate effects.

Incident electrons can undergo inelastic scattering and collisional excitation. Their orbital angular momentum and exchange with bound electrons allow transfers excluded for a single electric dipole photon. The ${}^1{D}_2$ energy level at $1.264\mathrm{eV}$ and the low fine structure excitations are energetically accessible. One therefore expects scattered electrons with discrete energy losses, possible weak forbidden spectral lines, and collisional or surface relaxation. For example ${}^1{D}_2\to {}^3{P}_2$ has wavelength approximately $10^7/(10192-43.4)=985\mathrm{nm}$; spin mixing supplies the weak spin-forbidden radiative channel.

There is a numerical threshold caveat. Literally,Neither quoted beam reaches ${}^1{S}_0$ from the exact ground state if these are taken as exact monoenergetic values. The electron value rounded to two decimal places could represent an actual energy above threshold; if so, ${}^1{S}_0$ is collisionally accessible and its decay to ${}^1{D}_2$ produces the weak electric quadrupole transition at $10^7/11456\simeq 873\mathrm{nm}$, followed by lower-level relaxation. These emissions require populations and sufficiently low collisional quenching of a metastable line.

An actual solid carbon foil has bands, collective excitations and surface losses, so it need not reproduce the isolated-chemical atom spectrum or selection rules. Its optical and electron-induced emission cannot be predicted quantitatively from the five atomic energy levels alone. This is the limitation of the foil wording in the source.

Lipid diversity allows a membrane to remain a selective, mechanically coherent barrier while retaining the molecular mobility required for function. Stability and fluidity are different properties: a lipid bilayer can be well ordered yet permit rapid lateral diffusion, or be mobile yet leaky and vulnerable to rupture. The effects depend on composition, temperature, leaflet organization and interactions among components rather than on one universal ranking of “rigid” and “fluid” lipids.

The basic stability comes from amphiphilic organization. Hydrophobic chains avoid exposure to water, while polar headgroups remain hydrated. A pore exposing chains has an energetic cost, encouraging local defects to close, but stress, chain disorder or incompatible molecular shapes can make defects easier to form. Useful measurements therefore include membrane permeability, leakage and mechanical failure as well as rotational mobility, lateral diffusion and acyl-chain order. Calling all of these “fluidity” conceals distinctions relevant to cell survival.

Acyl-chain length changes the contact area between neighboring chains. Other things equal, longer chains generally strengthen attractive contacts, increase bilayer thickness and raise the tendency to form an ordered phase. Shorter chains tend to lower the chain-melting transition and can increase permeability. A double bond between carbon atoms in the cis configuration restricts the chain geometry and disrupts close packing, generally lowering the chain-melting temperature relative to a comparable saturated chain. A double bond in the trans configuration has a different geometry and commonly permits closer packing than the corresponding cis configuration. Branching also changes packing. These comparisons hold the headgroup and other variables fixed; they are not predictions that all unsaturated lipids behave identically in a complex mixture.

An immediate consequence is that composition has to be interpreted relative to the growth temperature. A membrane made mostly of long saturated chains can become too ordered on cooling, slowing membrane protein conformational changes and molecular redistribution. Greater unsaturation can preserve mobility. On heating, the converse adjustments can limit excessive disorder and leakage. The original homeoviscous adaptation experiments found temperature-dependent changes in the saturated and long-chain fatty acid content of Escherichia coli, using spin-label electron paramagnetic resonance to compare membrane-lipid viscosity at the cells' growth temperatures.^[ref] This establishes an adaptive relationship rather than a claim that every organism maintains a single invariant value by the same biosynthetic route.

Headgroup diversity changes both local interactions and preferred geometry. Phosphatidylcholine is approximately cylindrical in many common bilayer compositions, whereas phosphatidylethanolamine has a smaller headgroup relative to its chains and can favor negative monolayer spontaneous curvature. A single-chain lysophospholipid often favors curvature of the opposite sign. Mixing these shapes can support bending and membrane-fusion intermediates, while too much of a nonbilayer-forming component can destabilize a planar lipid bilayer. The amphiphile packing parameter is a useful qualitative guide, but hydrated headgroup size and chain volume depend on conditions; a molecular cartoon does not supply a quantitative phase diagram.

Headgroup charge adds another control. phosphatidylserine and phosphatidylinositol 4,5-bisphosphate contribute negative inner-leaflet charge, recruiting basic membrane-binding regions of proteins and changing local electrostatics. Ionic strength, pH and divalent ions can screen, alter or bridge those charges. A lipid-induced change in domain organization could therefore arise from ion-mediated interactions rather than acyl-chain order alone. Sphingomyelin can pack tightly through its chain structure and headgroup/backbone interactions, influencing thickness and local order. A comparison of headgroups must specify the attached chains, because changing a headgroup while simultaneously changing chain saturation confounds the interpretation.

Cholesterol is particularly effective because its small polar head and rigid ring system fit between phospholipid chains. In a relatively disordered bilayer, it often increases chain order, condenses packing and reduces permeability to small solutes. In a tendency toward a tightly packed gel, it can disrupt regular packing and inhibit the cooperative freezing transition. Thus “cholesterol decreases fluidity” is incomplete: it can reduce some local motions while preventing an immobile gel and retaining lateral mobility. Its effects depend on concentration, temperature and the surrounding lipids; excessive amounts can exceed the mixture's capacity to accommodate it.

Mixtures of saturated lipids, unsaturated lipids and cholesterol can form coexisting liquid-ordered phase and liquid-disordered phase. The former combines greater acyl-chain order with lateral mobility; it is distinct from a solid gel. fluorescence microscopy of defined giant lipid vesicles directly demonstrated domains and mapped composition- and temperature-dependent coexistence.^[ref] The result proves that lipid interactions can organize a membrane without a protein scaffold. It does not establish that living plasma membranes contain large equilibrium domains of the same sizes. Cellular crowding, cytoskeleton coupling, asymmetry, trafficking and active turnover can change or suppress those patterns.

Diversity also works across the membrane. The outer and inner membrane leaflets differ in headgroups and chain distributions; equal total composition does not imply equal surface properties. Lipidomic and packing measurements found greater unsaturation and looser packing in the cytoplasmic leaflet of the examined mammalian plasma membranes.^[ref] Such asymmetry can couple to membrane protein shape and binding preferences. flippases and scramblases change transverse lipid distribution; because spontaneous translocation of many polar phospholipids is slow, this organization can be maintained and regulated. Leaflet imbalance creates curvature stress, but the result depends on lipid exchange and mechanical constraints rather than on headgroup charge alone.

Membrane composition also sets the energetic cost of deforming the surface around a membrane protein. Longer ordered chains may increase thickness and stiffness; mismatch with a transmembrane protein's hydrophobic span can favor local deformation, changed association or altered activity. Curvature-preferring lipids can lower the cost of particular conformational or membrane-shape transitions. Greater mechanical stiffness can resist bending but need not mean a larger rupture threshold. Oxidation of polyunsaturated chains can change molecular geometry and produce defects or reactive breakdown products, making chemical stability another tradeoff alongside mobility. Appropriate experiments should measure oxidation state and leakage rather than infer stability from saturation alone.

A good causal experiment uses compositionally defined lipid vesicles, independently varying chain saturation, chain length, headgroup and cholesterol. Keep vesicle size, temperature, ionic strength and osmotic conditions controlled. differential scanning calorimetry identifies cooperative transitions; spin-label electron paramagnetic resonance or calibrated fluorescence anisotropy probes local order and rotational freedom; fluorescence recovery after photobleaching measures lateral redistribution. Recovery also depends on geometry, immobile fractions and probe partitioning, so distinguish these from the diffusion coefficient. Test barrier stability by entrapped-dye leakage with dye-quenching and complete-release controls, and mechanical behavior with defined membrane-tension measurements. Repeat key comparisons with different probes because a probe can partition preferentially into one phase. In living biological cells, combine lipidomics with targeted biosynthetic perturbation, rescue and viability measurements to separate compositional effects from secondary stress responses.

Lipid diversity stabilizes useful combinations of properties: a sealed barrier, adaptable curvature, controlled local order and sufficient mobility. It also allows different compartments and leaflets to satisfy different physical demands; no single lipid composition maximizes all of those properties simultaneously.

Measure $x$ from the carbon nucleus toward the sulfur nucleus. The definition of the center of mass givesTherefore the requested distance isThe other distance is $r_2=r-r_1=m_{\mathrm{C}}r/(m_{\mathrm{C}}+m_{\mathrm{S}})$, consistent with $m_{\mathrm{C}}{r}_1=m_{\mathrm{S}}{r}_2$.

Place the carbon and sulfur nuclei on a line separated by $r$, with masses $m_{\mathrm{C}}$ and $m_{\mathrm{S}}$. The center of mass lies between them and closer to the heavier sulfur nucleus. In the linear rigid rotor approximation, $r$ remains fixed while the molecule rotates.

Closed subshells have $L=S=0$, so only the open $p$ or $d$ subshell determines each atomic term. Apply Hund's rules in order: maximize $S$, maximize $L$ at that $S$, then choose the lowest $J$ below half filling or the highest $J$ above half filling.

For carbon, the two $2p$ electrons occupy different magnetic atomic orbitals with parallel spins, giving $S=1$. Their largest allowed $M_L$ at that spin is $1+0=1$, so $L=1$, a $P$ term. The $p^2$ subshell is less than half full, giving $J=|1-1|=0$. The ground level is ${}^3{P}_0$.

For vanadium, three parallel-spin $3d$ electrons give $S=3/2$. The largest $M_L$ compatible with the Pauli exclusion principle is $2+1+0=3$, hence $L=3$, an $F$ term. The $d^3$ subshell is less than half full, so $J=3-3/2=3/2$. The ground level is ${}^4{F}_{3/2}$.

For cobalt, the $d^7$ subshell is equivalent to three $d$ holes for the allowed $L,S$ terms, so its maximum-spin, maximum-$L$ term is again ${}^{4}F$. Explicitly, five spin-up electrons fill all $m_l$ values and two spin-down electrons can occupy $m_l=2,1$, giving $M_S=3/2$ and $M_L=3$. Since $d^7$ is more than half full, $J=3+3/2=9/2$. The ground level is ${}^4{F}_{9/2}$.

For the $p_z$ atomic orbital basis, only orbitals whose centres remain fixed contribute to a character. Each such orbital contributes $+1$ if its $z$ direction is preserved and $-1$ if it is reversed. In the standard D4h molecular point group class order,No nonidentity operation other than ${\sigma }_h$ leaves any carbon site fixed. Reflection in the molecular plane fixes all eight sites but reverses every $p_z$ function.

Molecular character reduction givesThe dimensions sum to $1+1+1+1+2\cdot 2=8$. The two copies of $E_g$ are separate equivalent irreducible representations which may mix with one another; they do not force all four functions to have a common energy.

Each hydrogen lies on the radial line through its attached carbon. Consequently every molecular symmetry operation fixes or permutes the hydrogen sites exactly as it does the carbon sites, and the Cartesian trace is the same. Thus ${\Gamma }_H={\Gamma }_C$ and the complete benzene displacement group representation is ${\Gamma }_{3N}=2{\Gamma }_C$, of dimension $36$.

Subtract translations $A_{2u}+E_{1u}$ and rotations $A_{2g}+E_{1g}$. The full set of normal modes isThe one-dimensional components contribute $2+1+2+1+2+2=10$ coordinates. The two-dimensional components contribute $2(1+4+3+2)=20$, giving $30=3(12)-6$ vibrational degrees of freedom. There are twenty distinct symmetry-labelled fundamentals before accidental coincidences, because each doubly degenerate pair has one frequency.

Subtracting translations and rotations directly from the carbon-only result would instead give twelve carbon-skeleton coordinates. That is a six-site mechanical model, not all the normal modes of ${\mathrm{C}}_6{\mathrm{H}}_6$. The hydrogen contribution is required for the actual molecule.

For a displacement group representation, an atom contributes only if the operation leaves its site fixed. Its three Cartesian displacements then contribute the trace of the polar-vector transformation.

Take $z$ perpendicular to the ring and an in-plane $x$ axis through opposite carbon atoms. To avoid confusing the principal twofold rotation with the in-plane axes, write $C_2(z)$ for the former, $C_{2a}$ for an in-plane axis through atoms and $C_{2b}$ for one through opposite bond midpoints. The carbon displacement characters in the D6h molecular point group areA through-atom half-turn fixes two carbon sites and has Cartesian trace $-1$, giving $-2$. The molecular-plane reflection fixes all six carbons with trace $+1$, giving $6$. A through-atom vertical mirror fixes two sites with trace $+1$, giving $2$.

Using molecular character reduction, the 18-dimensional carbon-only result isThe six one-dimensional terms contribute six dimensions and the six copies of two-dimensional representations contribute twelve, totaling eighteen. The distinction between carbon-only and whole-molecule displacements is needed in part (ii).

The methoxy groups activate the aromatic ring by electron donation through chemical resonance and direct electrophilic substitution ortho and para. Their para positions are already occupied by one another, so the first alkylation occurs at an available ortho position; the four initially vacant positions are equivalent by symmetry.

After placing the first tert-amyl group at position 2, position 5 is favored for the second substitution. It is para to that bulky group and ortho to the other methoxy group. Position 3 is crowded between methoxy and tert-amyl substituents, while position 6 is near the first tert-amyl group across the intervening methoxy-bearing carbon. The 2,5 arrangement distributes the bulky groups on opposite parts of the ring and preserves the equivalence seen in the spectrum.

Alkyl groups do not electronically deactivate the ring, so the stopping point needs a steric explanation. After two such substitutions the remaining positions are much more congested, each adjacent to a bulky tert-amyl group and a methoxy group. Two large substituents are accommodated much more readily than a third, despite excess alcohol. The formula and NMR identify this sterically favored disubstitution; they do not imply that a Friedel-Crafts reaction always stops after two alkylations.

protonation of the alcohol converts its OH group into a water leaving group. Departure of water initially gives the secondary carbocation $\mathrm{C}{\mathrm{H}}_3-\mathrm{C}{\mathrm{H}}^{+}-\mathrm{CH}(\mathrm{C}{\mathrm{H}}_3{)}_2$. A 1,2-hydride shift from the neighboring methine group transfers its C–H bonding pair to the electron-deficient carbon:The resulting tertiary carbocation is the tert-amyl electrophile. An aromatic $\pi$ bond of the dimethoxybenzene attacks that carbon, producing a Wheland intermediate. Removal of the proton from the attacked ring carbon restores aromaticity and regenerates acid. Repeating this Friedel-Crafts alkylation at the symmetry-related second site gives the disubstituted product.

The hydride migration explains the new ethyl group and the absence of a benzylic methine group proton. Simply attaching the original alcohol skeleton at its OH-bearing carbon would not explain the twelve-proton NMR singlet.