Thermodynamics of HPLC retention

There are two basic approaches for thermodynamic description of the HPLC retention phenomena, one is based on the partitioning theory and another is based on adsorption.

Partition is a concentrational changes in the system due to the distribution of the components between two (or more) phases.

Adsorption is the concentrational changes in the system in presence of interface with another phase and due to the surface forces.

Phase is a form of matter that is uniform throughout in chemical composition and physical state.

In HPLC we have two main phases:
mobile phase (eluent) is a liquid solvent or mixture of solvents which is moving through the chromatographic column and carrying analytes:
Stationary phase (adsorbent) is solid porous media which consists of the rigid porous particles, usually silica based, with the specific surface properties (surface chemistry).

Adsorbent particles are considered to be nonpermeable and nonsoluble for the eluent and analyte molecules. It only introduces surface forces in the system.

Consideration of the HPLC process based on the partitioning theory was transferred from GC (gas chromatography) theory, where we usually have a mobile gas phase and stationary liquid phase, and where a true partitioning occurs.

Usual description of liquid chromatography on the partitioning basis consider the assumption of the existence of the separate liquid phase which is close to the adsorbent surface. Chemically bonded phases are usually considered by this manner. The most popular bonded phase is octadecylsilica, where relatively long (21 ) alkyl chains are chemically bonded to the silica surface. The main partitioning concept is that analyte molecules can penetrate between these alkyl chains. This process thermodynamically considered as dissolving of the analyte molecules in the surface alkyl phase.

Two restrictions exists for application of this approach in thermodynamic description:
the bonded chains have a certain thickness (about 3.8 in diameter) and depending on the bonding density the layer may not behave as a liquid;

monomolecular layer would not be considered a phase in classical thermodynamics.

Actually an application of the partitioning theory leads to the certain controversy in the description of some HPLC data.

Here, an adsorption thermodynamic theory of HPLC retention will be considered , and we start with the brief description of adsorption from solutions.

Adsorption from solutions

A classic description of the adsorption process is based on the Gibbs excess adsorption theory, which basically considers two similar hypothetical adsorption systems with the same volume, temperature, pressure, and adsorbent surface area. The only difference is that the first system does not show any adsorption on the surface (no surface forces), and the second does.

Simple adsorption experiment

After achieving an equilibrium in both systems a concentration of the components in the bulk solution over the adsorbent are measured. The first system obviously will have an original concentration (co) of component A (considering a binary solution), and in the second system different concentration, ce of the same component will be observed.

Excess adsorption is defined as an excess amount of the component concentrated on the adsorbent per unit of the surface area:

          (13)
 

The dependence of the Gibbs excess adsorption on the equilibrium analyte concentration is usually called adsorption isotherm.

The slope of the adsorption isotherm of the analyte at low concentrations represents a power of the surface molecular interactions. For the simple binary mixture an adsorption isotherm may be described by the equation:

          (14)
 

where x is an analyte equilibrium concentration in mole fractions, K is the thermodynamic equilibrium constant. K is a measure of interaction energy difference of eluent and analyte molecules with the adsorbent surface, and it may be expressed in the form:

          (15)
 

where DG is the difference of the free Gibbs energy of the analyte and eluent, R is the gas constant, and T is an absolute temperature.

Basic retention equation

It is assumed that the column is in equilibrium. This means that at any moment and in any part of the column, the conditions are infinitely close to thermodynamic equilibrium.

By considering the analyte mass dynamics in the small cross-sectional part of the column we can write the mass balance equation for the dx part of the column on x distance from the inlet. This mass balance equation has a differential form, and has an exact solution only for binary system. This solution establishes a connection between the analyte retention volume and its excess adsorption isotherm.

In the mass balance equation section the logic of the derivation of the mass balance equation is shown. This information is optional, but as a result we will get:

basic retention equation

For the binary system the exact solution of the mass balance differential equation may be written in form:

          (16)
 

This equation is the basic retention equation in adsorption chromatography. It could be used for the thermodynamically based explanation of most of the chromatographic effects.

In general it says, that component retention volume is a sum of the dead volume and product of the adsorbent surface area and the derivative of its excess adsorption isotherm