Rotamer

is negligible in alkanes since the bond angles are all near the tetrahedral ideal. The energy profile is thus periodic with 2\pi/3 (120°) periodicity due to the threefold symmetry of sp3-hybridized carbon atoms. This suggests a sinusoidal potential energy function V(\theta, k), typically modelled using a Fourier series truncated to the dominant terms: V(\theta, k) = \sum_{k=0}^{\infty} \frac{V_k(n)}{2} [1 - \cos(k\theta)] Here: • \theta is the dihedral angle in degrees, • V_k(n) are coefficients representing the amplitude of the nth harmonic, corresponding to various energy barriers due to torsional influences and asymmetry in steric interactions. • The factor of \tfrac{1}{2} and the form [1 - \cos(k\theta)] ensure energy minima at staggered conformations and energy maxima at eclipsed conformations. For alkanes, the dominant term is usually k=3, reflecting the threefold rotational symmetry. Higher terms may be included for precision where steric effects vary. The primary contribution comes from torsional strain due to alkyl groups eclipsing, captured by the \cos(3\theta) term. Steric interactions rise with the size of substituents (H– for ethane, CH3– for butane, C2H5– for hexane, etc.), taken into account by the \cos(\theta) term (k=1). The number of carbon atoms clearly influences the size of substituents on the central C–C bond. In general, for unbranched linear alkanes with even-numbered chains, there will be two Cn-1H2n-1 group substituents. A parameterization using energy values derived from rotational spectroscopy data and theoretical calculations gives the following simplified equation: V(\theta, n) = 0.25 (n-1) [1 - \cos(\theta)] + [1.45 + 0.05 (n-1)] [1 - \cos(3\theta)] Here V(\theta, n) is given in kcal/mol and k=1,3. This function largely neglects angle strain and long-range interactions for the n members of the series. While simple molecules can be described by these types of conformations, more complex molecules require the use of the Klyne–Prelog system to describe the conformers. • Akamptisomerism – due to restricted inversion of a bond angle. ==Equilibrium of conformers==

Conformers generally exist in a dynamic equilibrium Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at various temperatures. At a free energy difference of 0 kcal/mol, this analysis gives an equilibrium constant of 1, meaning that two conformers exist in a 1:1 ratio. The two have equal free energy; neither is more stable, so neither predominates compared to the other. A negative difference in free energy means that a conformer interconverts to a thermodynamically more stable conformation, thus the equilibrium constant will always be greater than 1. For example, the ΔG° for the transformation of butane from the gauche conformer to the anti conformer is −0.47 kcal/mol at 298 K. This gives an equilibrium constant is about 2.2 in favor of the anti conformer, or a 31:69 mixture of gauche:anti conformers at equilibrium. Conversely, a positive difference in free energy means the conformer already is the more stable one, so the interconversion is an unfavorable equilibrium (K  \frac{N_i}{N_\text{total}} = \frac {e^{-E_i/RT}} {\sum_{k=1}^M e^{-E_k/RT}}. The left hand side is the proportion of conformer i in an equilibrating mixture of M conformers in thermodynamic equilibrium. On the right side, Ek (k = 1, 2, ..., M) is the energy of conformer k, R is the molar ideal gas constant (approximately equal to 8.314 J/(mol·K) or 1.987 cal/(mol·K)), and T is the absolute temperature. The denominator of the right side is the partition function. Factors contributing to the free energy of conformers The effects of electrostatic and steric interactions of the substituents as well as orbital interactions such as hyperconjugation are responsible for the relative stability of conformers and their transition states. The contributions of these factors vary depending on the nature of the substituents and may either contribute positively or negatively to the energy barrier. Computational studies of small molecules such as ethane suggest that electrostatic effects make the greatest contribution to the energy barrier; however, the barrier is traditionally attributed primarily to steric interactions. In the case of cyclic systems, the steric effect and contribution to the free energy can be approximated by A values, which measure the energy difference when a substituent on cyclohexane in the axial as compared to the equatorial position. In large (>14 atom) rings, there are many accessible low-energy conformations which correspond to the strain-free diamond lattice. ==Observation of conformers==

The short timescale of interconversion precludes the separation of conformer in most cases. Atropisomers are conformational isomers which can be separated due to restricted rotation. The equilibrium between conformational isomers can be observed using a variety of spectroscopic techniques. Protein folding also generates conformers which can be observed. The Karplus equation relates the dihedral angle of vicinal protons to their J-coupling constants as measured by NMR. The equation aids in the elucidation of protein folding as well as the conformations of other rigid aliphatic molecules. Protein side chains exhibit rotamers, whose distribution is determined by their steric interaction with different conformations of the backbone. This effect is evident from statistical analysis of the conformations of protein side chains in the Backbone-dependent rotamer library. Spectroscopy Conformational dynamics can be monitored by variable temperature NMR spectroscopy. The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called "fluxional". For example, in cyclohexane derivatives, the two chair conformers interconvert rapidly at room temperature. The ring-flip proceeds at a rates of approximately 105 ring-flips/sec, with an overall energy barrier of 10 kcal/mol (42 kJ/mol). This barrier precludes separation at ambient temperatures. Besides NMR spectroscopy, IR spectroscopy is used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, νCBr differs by almost 50 cm−1. ==Conformation-dependent reactions==

Reaction rates are highly dependent on the conformation of the reactants. In many cases the dominant product arises from the reaction of the less prevalent conformer, by virtue of the Curtin-Hammett principle. This is typical for situations where the conformational equilibration is much faster than reaction to form the product. The dependence of a reaction on the stereochemical orientation is therefore usually only visible in configurational analysis, in which a particular conformation is locked by substituents. Prediction of rates of many reactions involving the transition between sp2 and sp3 states, such as ketone reduction, alcohol oxidation or nucleophilic substitution is possible if all conformers and their relative stability ruled by their strain is taken into account. One example where the rotamers become significant is elimination reactions, which involve the simultaneous removal of a proton and a leaving group from vicinal or antiperiplanar positions under the influence of a base. The mechanism requires that the departing atoms or groups follow antiparallel trajectories. For open chain substrates this geometric prerequisite is met by at least one of the three staggered conformers. For some cyclic substrates such as cyclohexane, however, an antiparallel arrangement may not be attainable depending on the substituents which might set a conformational lock. Adjacent substituents on a cyclohexane ring can achieve antiperiplanarity only when they occupy trans diaxial positions (that is, both are in axial position, one going up and one going down). One consequence of this analysis is that trans-4-tert-butylcyclohexyl chloride cannot easily eliminate but instead undergoes substitution (see diagram below) because the most stable conformation has the bulky t-Bu group in the equatorial position, therefore the chloride group is not antiperiplanar with any vicinal hydrogen (it is gauche to all four). The thermodynamically unfavored conformation has the t-Bu group in the axial position, which is higher in energy by more than 5 kcal/mol (see A value). As a result, the t-Bu group "locks" the ring in the conformation where it is in the equatorial position and substitution reaction is observed. On the other hand, cis-4-tert-butylcyclohexyl chloride undergoes elimination because antiperiplanarity of Cl and H can be achieved when the t-Bu group is in the favorable equatorial position. The repulsion between an axial t-butyl group and hydrogen atoms in the 1,3-diaxial position is so strong that the cyclohexane ring will revert to a twisted boat conformation. The strain in cyclic structures is usually characterized by deviations from ideal bond angles (Baeyer strain), ideal torsional angles (Pitzer strain) or transannular (Prelog) interactions. ==Alkane stereochemistry==

Alkane conformers arise from rotation around sp3 hybridised carbon–carbon sigma bonds. The smallest alkane with such a chemical bond, ethane, exists as an infinite number of conformations with respect to rotation around the C–C bond. Two of these are recognised as energy minimum (staggered conformation) and energy maximum (eclipsed conformation) forms. The existence of specific conformations is due to hindered rotation around sigma bonds, although a role for hyperconjugation is proposed by a competing theory. The importance of energy minima and energy maxima is seen by extension of these concepts to more complex molecules for which stable conformations may be predicted as minimum-energy forms. The determination of stable conformations has also played a large role in the establishment of the concept of asymmetric induction and the ability to predict the stereochemistry of reactions controlled by steric effects. In the example of staggered ethane in Newman projection, a hydrogen atom on one carbon atom has a 60° torsional angle or torsion angle • Gauche, conformer – 3.8 kJ/mol • Eclipsed H and CH3 – 16 kJ/mol • Eclipsed CH3 and CH3 – 19 kJ/mol. The eclipsed methyl groups exert a greater steric strain because of their greater electron density compared to lone hydrogen atoms. The textbook explanation for the existence of the energy maximum for an eclipsed conformation in ethane is steric hindrance, but, with a C-C bond length of 154 pm and a Van der Waals radius for hydrogen of 120 pm, the hydrogen atoms in ethane are never in each other's way. The question of whether steric hindrance is responsible for the eclipsed energy maximum is a topic of debate to this day. One alternative to the steric hindrance explanation is based on hyperconjugation as analyzed within the Natural Bond Orbital framework. In the staggered conformation, one C-H sigma bonding orbital donates electron density to the antibonding orbital of the other C-H bond. The energetic stabilization of this effect is maximized when the two orbitals have maximal overlap, occurring in the staggered conformation. There is no overlap in the eclipsed conformation, leading to a disfavored energy maximum. On the other hand, an analysis within quantitative molecular orbital theory shows that 2-orbital-4-electron (steric) repulsions are dominant over hyperconjugation. A valence bond theory study also emphasizes the importance of steric effects. Nomenclature Naming alkanes per standards listed in the IUPAC Gold Book is done according to the Klyne–Prelog system for specifying angles (called either torsional or dihedral angles) between substituents around a single bond: • a torsion angle between 0° and ±90° is called syn (s) • a torsion angle between ±90° and 180° is called anti (a) • a torsion angle between 30° and 150° or between −30° and −150° is called clinal (c) • a torsion angle between 0° and ±30° or ±150° and 180° is called periplanar (p) • a torsion angle between 0° and ±30° is called synperiplanar (sp), also called syn- or cis- conformation • a torsion angle between 30° to 90° and −30° to −90° is called synclinal (sc), also called gauche or skew • a torsion angle between 90° and 150° or −90° and −150° is called anticlinal (ac) • a torsion angle between ±150° and 180° is called antiperiplanar (ap), also called anti- or trans- conformation Torsional strain or "Pitzer strain" refers to resistance to twisting about a bond. Special cases In n-pentane, the terminal methyl groups experience additional pentane interference. Replacing hydrogen by fluorine in polytetrafluoroethylene changes the stereochemistry from the zigzag geometry to that of a helix due to electrostatic repulsion of the fluorine atoms in the 1,3 positions. Evidence for the helix structure in the crystalline state is derived from X-ray crystallography and from NMR spectroscopy and circular dichroism in solution. ==See also==