Limites operacionales

The thermodynamic properties of a system are typically affected by the presence of an interface, which exhibits fundamental differences in molecular structure compared to bulk matter phases. An ideal interface can be described as the boundary separating two distinct, adjoining bulk phases. In bulk phases, cohesive forces between molecules exist, whereby molecules attract surrounding like molecules through a variety of intermolecular interactions including van der Waals forces and hydrogen bonding. However, at interfaces, molecules are only partially surrounded by like molecules and therefore interact with a smaller number of adjacent like molecules than those in the bulk. These arrangements are energetically unfavourable compared to the bulk matter case. There are three common types of interfaces that exist between different types of phases: solid-liquid, solid-vapour and liquid-vapour. For a liquid-vapour interface, the unbalanced intermolecular forces at the interface give rise to a net force which acts to minimise the interfacial area. As a result, work is required in order to relocate a molecule from the bulk to the interface and increase the interfacial area. The surface

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tension, 𝛾, is a fundamental quantity which is used to define the work required to increase the area of a liquid-vapour interface (Equation 2.3):

𝛾 =𝑑𝑊 𝑑𝐴 2.3 Where: 𝛾 = Surface tension (N m-1) 𝑊 = Work (N m) 𝐴 = Interfacial area (m2)

Similarly, the surface energy of a solid surface is defined in terms of the work per unit area required to create two separate, flat surfaces from a bulk solid. The work of cohesion is equal to the cohesive energy per unit area, and is related to the surface energy of the newly created surface by Equation 2.4:

𝑊 = 2𝛾_𝑠 2.4

Where:

𝛾_𝑠 = Surface energy (J m-2)

𝑊 = Work of cohesion per unit area (N m-1)

Thus, the surface tension parameter for a liquid is equivalent to the surface energy parameter for a solid.

Although ideal interfaces are useful in defining certain thermodynamic properties such as the surface tension, further examination of thermodynamic properties is possible when the real molecular structure of interfaces is considered. Real interfaces do not

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exist as infinitesimally thin borders between two bulk phases, but instead appear as a region with a finite thickness of perhaps a few molecules (Figure 2.2). When proceeding across this region in a direction normal to the interface, the concentration of one phase decreases whilst the concentration of the adjacent phase increases until it reaches the bulk concentration. Consequently, real interfaces form an important part of multi-phase systems and cannot be considered in isolation.

Figure 2.2: Schematic showing the differences between ideal and real interfaces. Ideal interfaces represent the boundary between two distinct phases. Conversely, real interfaces

appear as a region with a finite thickness of a few molecules.

Thus, a system containing an interface can divided into three areas for the purposes of studying the thermodynamic properties of a system; two bulk phases 𝛼 and 𝛽, with volumes 𝑉𝛼 and 𝑉𝛽, and the interface, 𝜎. Here, the interface is treated using the Gibbs convention. With the Gibbs convention, the interface is modelled as an infinitesimally thin geometric surface, known as the Gibbs dividing plane. The Gibbs dividing plane is used to define the ideal volume of the bulk phases and to calculate surface excess properties for the system. Excess thermodynamic properties quantify the difference between the value measured for a real system and an equivalent reference system. In

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this instance, the surface excess of component 𝑖 is defined as the additional amount of the 𝑖th

material in the system due to the presence of the interfacial region. This value is calculated with respect to an equivalent reference system with an ideal interface, in which the bulk concentrations of the two phases remain uniform throughout 𝑉𝛼 and 𝑉𝛽 (Equation 2.5)

𝑁_𝑖𝜎 _{= 𝑁}

𝑖− 𝑐𝑖𝛼𝑉𝛼− 𝑐𝑖𝛽𝑉𝛽 2.5

Where:

𝑁_𝑖𝜎 = The excess of species 𝑖 in the phase 𝜎 which would have remained in 𝜎 if the bulk composition of 𝛼 and 𝛽 phases had extended to the Gibbs dividing plane (molecules)

𝑁𝑖 = Total quantity of species 𝑖 in system (molecules)

𝑐_𝑖𝛼_{, 𝑐}

𝑖𝛽 = Concentration of species 𝑖 in phase 𝛼,𝛽 (molecules/m2)

𝑉𝛼, 𝑉𝛽 = Volume of species 𝑖 in phase 𝛼,𝛽 (m2)

The surface excess concentration can then be defined as the concentration of species 𝑖 at an interface with the area, 𝐴 through Equation 2.6:

Γ𝑖 =

𝑁_𝑖𝜎

𝐴

2.6

Where:

Γ𝑖 = Surface excess concentration of component 𝑖 (molecules m-2)

Thus, the surface excess and surface excess concentration relies on the appropriate placement of Gibbs dividing plane. The Gibbs convention states that the correct

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location of the dividing plane is such that the surface excess concentration of the bulk phases, Γ_𝛼= Γ_𝛽 = 0. An accurate definition of the interface is required in order to account for adsorption phenomena.