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Study Guide - Practical 2
Intramolecular Binding Forces
Introduction
At the end of the 19th century, while investigating the behavior of real gases, particularly their deviation from the ideal behavior described by the well-known equation P V = nRT , the Dutch physicist J.D. van der Waals began considering the influence of intermolecular forces to explain the discrepancies observed in experimental data.
Over the following decades, extensive research by numerous scientists established that inter- molecular forces are far from simple. Early in the 20th century, the growing field of statistical thermodynamics provided the theoretical foundation to describe these interactions with in- creasing analytical precision.
Intermolecular forces can be classified in various ways, for example, as attractive or repulsive, or as short-range or long-range. In the sections that follow, we will focus on those interactions that most significantly affect the systems relevant to this course and that are essential for solving the proposed exercises.
Attraction and Repulsion
At the most fundamental level, the interaction potential w(r) between two particles serves as a model to describe the key features of intermolecular interactions in systems composed of simple atoms or molecules. These interactions are characterized by a strong repulsion at very short distances, an attractive region at intermediate distances, and negligible interaction as the distance approaches infinity.
The first mathematical formulation of such a potential was proposed by Mie in 1903 and is commonly expressed as:
w(r) = Cε
hσ r
n −
σ r
mi
With:
C =
n n − m
(^) n m
m n − m
In this expression, ε [J] represents the dispersion energy, and σ [m] denotes the distance at which w(σ) = 0, often referred to as the collision radius. The parameters n and m are adjustable and define the steepness of the repulsive and attractive contributions, respectively. Typically, m is fixed at 6, while n can be tuned to better fit experimental observations.
This potential comprises two distinct terms: one accounting for the attractive interactions σ r
n , and the other for the repulsive interactions
σ
r
m .
A special case of the Mie potential, obtained by setting n = 12 and m = 6, is the widely recognized Lennard-Jones potential, which is the most frequently used pair potential to describe interactions between neutral, non-bonded particles. Its common form is:
wLJ (r) = 4ε
σ r
σ
r
Or even:
wLJ (r) = −
A
r^6
B
r^12
In which A = 4εσ^6 and B = 4εσ^12.
Both the Mie and Lennard-Jones potentials are categorized as pair potentials, meaning they only consider interactions between two individual particles. These models are not suitable for describing many-body interactions involving three or more particles directly. Nevertheless, the total potential energy of a multi-particle system can be approximated by summing the contributions from all pairwise interactions:
Vtot =
NXatoms
i̸=j
wij (r)
Polarization
Molecular polarization is a term that refers to the dipole moments induced in molecules by the electric fields emanating from nearby molecules.
In non-polar molecules, the polarizability arises from the displacement of the negatively charged electron cloud relative to the positively charged nucleus under the influence of an external
electric field. The polarizability of a molecule is generally expressed with α 0
C^2 m^2 J
and, for
a non-polar molecule, it can be calculated as:
α 0 = 4πε 0 S
Where ε 0 = 8. 854 × 10 −^12
C^2
N m^2
is the electric constant or vacuum permittivity, while S [m^3 ]
is the volume of the molecule.
In the case of polar molecules, the uneven distribution of electronic charge influences the overall polarizability of the system. This effect originates from the presence of a permanent dipole moment μ, which gives rise to an additional contribution to the total polarizability, known as the orientational polarizability, denoted as αorient. This contribution is defined as:
αorient =
μ^2 3 k T
Where μ [D] = [3. 336 × 10 −^30 C · m] is the permanent dipole moment of the molecule,
k = 1. 380649 × 10 −^23
J
K
is the Boltzmann constant, and T [K] is the absolute temperature.
The total polarizability of a polar molecule can thus be expressed through the Debye–Langevin equation, which combines the electronic contribution with the orientational one:
α = α 0 + αorient = 4πε 0 S +
μ^2 3 k T
Although tabulated values of the atomic volume S for certain simple molecules can be found in the literature, such data are often limited to small or well-characterized species. For more complex molecular systems, an effective approach to approximate the molecular volume consists
on the molecule, and can be expressed as:
μind = α · E = l · |e|
Where E
N
C
is the intensity of the electric field, l [m] is the displacement of the electron, and
e = − 1. 602 × 10 −^19 [C] is the elemental charge of an electron.
Dissipation
Unlike the interactions previously discussed, which arise from electrostatic effects involving both polar and non-polar molecules, dispersion forces (also known as London forces) act universally between all molecules, including those that are electrically neutral and non-polar.
Dispersion forces represent a fundamental contribution to the total van der Waals interac- tions, as they are always present, independent of a molecule’s polarity or permanent dipole moment. These forces play a crucial role in a wide range of physical phenomena, including adhe- sion, surface tension, physical adsorption, wetting, gas and liquid properties, thin film behavior, particle flocculation in liquids, and the structural organization of condensed macromolecular systems such as polymers.
Dispersion forces originate from quantum mechanical fluctuations in electron density. While their rigorous derivation lies beyond the scope of this course, the first and most straightfor- ward mathematical description of the interaction energy w(r) between two identical atoms or molecules was proposed by London in 1937 and is given by:
w(r) = −
α^20 I (4πε 0 )^2 r^6
For the case of two distinct atoms or molecules, the interaction can be approximated by:
w(r) = −
α 0 ,iα 0 ,j (4πε 0 )^2 r^6
IiIj Ii + Ij
In this expression, Ii is the first ionization potential of species i, and α 0 ,i is its molecular polarizability. It is noteworthy that the interaction energy varies as ∼ 1 /r^6 , which is the same distance dependence as the attractive term in the Lennard-Jones potential. This emphasizes the underlying connection between dispersion forces and the broader category of van der Waals interactions.
The significance of dispersion forces becomes evident even when comparing molecules with similar chemical compositions, such as alkanes. Small molecules like methane and ethane exist as gases at room temperature, whereas larger molecules such as hexane are liquid under the same conditions. This trend reflects the increasing strength of dispersion forces with molecular size, as larger molecules possess greater polarizability and thus stronger attractive interactions.
