PCSI 2 – Lycée Carnot - Dijon

Facteur de Boltzmann

Facteur de Boltzmann (approche documentaire) Il s’agit dans ce document de généraliser l’intérêt de facteur de Boltzmann, vu jusqu’ici dans le cas particulier de la répartition des molécules de l’atmosphère isotherme et déterminant l’évolution de la pression en fonction de l’altitude. On considère un ensemble de particules n’interagissant pas entre elles et en équilibre à la température T. D’après la statistique de Maxwell-Boltzmann, la probabilité de trouver une de ces particules dans un état d’énergie E est proportionnelle au facteur de −

E

Boltzmann : e kT , où k est la constante de Boltzmann. Nous allons envisager un cas d’application concrète de ce résultat.

Conformational isomerism (http://en.wikipedia.org/wiki/Conformational_isomerism)

€

In chemistry, conformational isomerism is a form of stereoisomerism in which the isomers can be interconverted exclusively by rotations about formally single bonds (refer to figure on single bond rotation). Such isomers are generally referred to as conformational isomers or conformers and, specifically, as rotamers. Rotations about single bonds are restricted by a rotational energy barrier which must be overcome to interconvert one conformer to another. Conformational isomerism arises when the rotation about a single bond is relatively unhindered. That is, the energy barrier must be small enough for the interconversion to occur. Conformational isomers are thus distinct from the other classes of stereoisomers (i. e. configurational isomers) where interconversion necessarily involves breaking and reforming of chemical bonds. For example, L & D and R & S configurations of organic molecules have different handedness and optical activities, and can only be interconverted by breaking one or more bonds connected to the chiral atom and reforming a similar bond in a different direction or spatial orientation. The study of the energetics between different rotamers is referred to as conformational analysis. It is useful for understanding the stability of different isomers, for example, by taking into account the spatial orientation and through-space interactions of substituents. In addition, conformational analysis can be used to predict and explain product(s) selectivity, mechanisms, and rates of reactions.

Rotation about single bond of butane to interconvert one conformer to another Above : Newman projection Below : depiction of spatial orientation

Free energy diagram of butane as a function of dihedral angle

Types of corformational isomerism The types of conformational isomers are related to the spatial orientations of the substituents between two vicinal atoms. These are eclipsed and staggered. The staggered conformation includes the gauche (±60°) and anti (180°) conformations, depending on the spatial orientations of the two substituents. For example, butane has three rotamers relating to its two methyl (CH3) groups : two gauche conformers, which have the methyls ±60° apart and are enantiomeric, and an anti conformer, where the four carbon centres are coplanar and the substituents are 180° apart (refer to free energy diagram of butane). The energy difference between gauche and anti is 0.9 kcal/mol associated with the strain energy of the gauche conformer. The anti conformer is, therefore, the most stable ( ≈ 0 kcal/mol). The three eclipsed conformations with dihedral angles of 0°,120° and 240° are not considered to be rotamers, but are instead transition states of higher energy. Note that the two eclipsed conformations have different energies : at 0° the two methyl groups are eclipsed, resulting in higher energy ( ≈ 5 kcal/mol) than at 120°, where the methyl groups are eclipsed with hydrogens ( ≈ 3.5 kcal/mol). € molecules require the use of the KlyneWhile simple molecules can be described by these types of conformations, more complex Prelog system to describe the different conformers. More specific examples of conformational isomerism are detailed elsewhere :

€

€

1/2

PCSI 2 – Lycée Carnot - Dijon

Facteur de Boltzmann

1. Ring conformation * Cyclohexane conformations with chair and boat conformers. * Carbohydrate conformation. 2. Allylic strain - energetics related to rotation about the single bond between sp2 and sp3 carbons. 3. Atropisomerism- due to restricted rotation about a bond, a molecule can become chiral. 4. Folding of molecules, where some shapes are stable and functional, but others are not. Free energy and equilibria of conformational isomers Equilibrium of conformers Conformational isomers exist in a dynamic equilibrium, where the relative free energies of isomers determines the population of each isomer and the energy barrier of rotation determines the rate of interconversion between isomers : −

ΔG

K = e RT where K is the equilibrium constant, ΔG is the change in free energy for the interconversion of two conformers in kcal/mol, R is the universal gas constant (0.002 kcal/mol K), and T is the system's temperature in Kelvin (K). Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at different temperatures. Note € that a 0 kcal/mol free energy change gives an equilibrium constant of 1, meaning that two conformers have equal stability and exist in a 1:1 ratio. A negative change in free energy means that a conformer interconverts to a thermodynamically favored conformation, thus the equilibrium constant will always be greater than 1. For example, the ΔG of butane from gauche to anti is 0.9 kcal/mol, therefore the equilibrium constant is 4.5, favoring the anti conformation. Also notice that at large positive ΔG (i.e. unlikely for interconversion to occur), the equilibrium constant between two conformers can be increased by increasing temperature.

Equilibrium distribution of two conformers at different temperatures given the free energy of their interconversion

Boltzmann distribution % of lowest energy conformation in a two component equilibrating system at various temperatures (°C) and energy difference in kcal/mol (x-axis)

Population distribution of conformers The fractional population distribution of different conformers follows a Boltzmann distribution : Ni e −E rel / RT = N total N total e −E k / RT

∑

k =1

The left hand side is the equilibrium ratio of conformer i to the total. Erel is the relative energy of the i-th conformer from the minimum energy conformer. Ek is the relative energy of the k-th conformer from the minimum energy conformer. R is the molar ideal gas constant equal to 8.31 J/(mol·K) and T is the temperature in kelvins (K). The denominator of the right side is the partition € function. Questions : Dans la stéréoisomérie, qu’est ce qui distingue l’isomérie de conformation de celle de configuration dans le passage d’un isomère à un autre ? Dans le cas du butane, et d’après les données fournies par le texte : * Donner, en kJ.mol-1, l’ordre de grandeur de la barrière énergétique à franchir pour passer d’un isomère à l’autre. La comparer à l’énergie de liaison carbone - carbone. Conclusion ? * Estimer la proportion à 25°C des isomères gauche et anti. * Comment évolue-t-elle avec la température ? * A partir de quelles considérations peut-on expliquer la différence de stabilité des différents isomères de conformation ? 2/2