The Thermodynamics of Chromatography

The retention of a solute in a chromatographic system is determined firstly, by the magnitude of the distribution coefficient of the solute between the two phases and secondly, by the amount of stationary phase available to the solute for interaction. This is fully discussed in Plate Theory and Extensions of this series. In addition, the mechanism of distribution has been considered exclusively on the basis of molecular interactions in The Mechanism of Chromatographic Retention . However, the distribution coefficient in chromatography is an equilibrium constant and, consequently, it can be treated rationally by conventional thermodynamics.

It follows, that the distribution coefficient can be expressed in terms of the standard energy of solute exchange between the phases employing the traditional and well established Arrhenious relationship,

RT ln (K) = -?G^o (1)

where (R)	is the gas constant,
(T)	is the absolute temperature,
and (G^o)	is the standard energy.

Now, classical thermodynamics gives another expression for the standard energy which separates it into two parts, the standard enthalpy and the standard entropy.

Thus, ?G^o = ?H^o - T?S^o (2)

where (H^o)	is the standard enthalpy,
and (S^o)	is the standard entropy.

The standard enthalpy and standard entropy represent two distinctly different portions of the energy associated with distribution and are related to quite different parts of the distribution processes.

The enthalpy term represents the energy involved when the solute molecules break their interactions with the mobile phase and interact with, and enter, the stationary phase. These interactions result from intermolecular forces that are electrical in nature (see book 7 for details) and are accompanied by the absorption or evolution of heat.

However, when the solute interacts with the stationary phase, because the interactive forces between the solute and the stationary phase molecules are stronger than those between the solute molecules and the mobile phase, the solute molecules are held more tightly and, consequently, are more restricted. This motion restriction, reduced freedom of movement or loss of randomness, is measured as the entropy change. The nature of the entropy change can be illustrated in GC, when the distribution of a solute between a gas and a liquid is considered. In the gas, the solute molecules have high velocities and can travel in any direction. However, when in the liquid phase, they are held strongly by interacting molecular forces to the molecules of stationary phase and can no longer travel through the phase at high velocities or with the same directional freedom of movement.

Thus, the standard energy change is made up of an actual energy or standard enthalpy change resulting from the intermolecular forces between solute and stationary phase and a standard entropy change that reflects the resulting restricted movement, or loss in randomness, of the solute while preferentially interacting with the stationary phase.

From, equations (1) and (2), substituting for (?G^o )

RT ln (K) = -?G^o = -?H^o+ T?S^o

Thus, (3)

or (4)

It might be assumed from equation (3) that it would be relatively easy to calculate the retention volume of a solute from the distribution coefficient, which, in turn, could be calculated from a knowledge of the standard enthalpy and standard entropy of the specific distribution. However, the thermodynamic properties of a distribution system are bulk properties, and as such, they represent, in a single measurement, the net effect of a number of different types interactions which, at the present time (except for dispersive interactions) are almost impossible to separate, identify and assess quantitatively.

Although the thermodynamic functions can be measured for a given distribution system, they can not, at present, be predicted before the fact. However, as will be seen in due course, the possibilities might be more hopeful for the future.

Despite these difficulties, the thermodynamic properties of a distribution system can help explain the characteristics of the distribution (i.e., the standard enthalpy and standard entropy involved) and predict, quite accurately, the effect of temperature on the separation.

Now, V'_r = KV_s and

where (V'_r)	is the retention volume of the solute,
(V_s)	is the volume of stationary phase in the column,
(V_m)	is the volume of mobile phase in the column,
(a)	is the phase ratio of the column,
and (k')	is the capacity ratio of the solute.

Then, from equation (3),

(5)

and (6)

Furthermore, If (V_r) is the retention volume per ml of stationary phase (i.e., V'_r = V'_o(r)) then , ln(V_s) = 0 and,

(7)

It is clear that a graph of ln(V'_r) or ln(k') against 1/T must give a straight line. From the linear curve, the standard enthalpy (?H^o) can be calculated from the slope and the standard entropy (?S^o), from the intercept. Curves relating ln(K), ln(V'_r) or ln(k') against 1/T are called van't Hoff curves. However, to ensure linearity, the distribution system must remain unchanged throughout the temperature range examined. If the distribution system does change, then a graph relating ln(V'_r) against (1/T) will not be linear and the curves will no longer be van't Hoff curves. The conditions, where curves relating ln(V'_r) or ln(k') against 1/T are not linear, will be considered later.

There are two basic forms that the van't Hoff curve can take. These two types of curve relate to the two basic types of chromatography. The first interactive chromatography where the major retentive mechanism results from solute phase interactions and the second exclusion chromatography where the major retention mechanism depends on the amount of stationary phase available to each solute. As stated in book 7, neither form of chromatography can be exclusive, but one can be predominant. An example of the type of linear curve that is produced by interactive chromatography is shown in figure 1.

It is seen the van't Hoff curve indicates a very large enthalpy value (the slope of the curve is steep), but conversely a very low entropy contribution(the intercept is relatively small).

Figure 1. Interactive Chromatography Using Energy Driven Distribution

The large value of means that the distribution in favor of the stationary phase is dominated by molecular forces. That is to say, the solute is retained in the stationary phase as a result of molecular interactions and that the forces between the solute molecules with those of the stationary phase are much greater than the forces between the solute molecules and those of the mobile phase. Thus, the change in standard enthalpy is the major contribution to the change in standard energy and it can be said, In thermodynamic terms, the distribution is "energy driven".

An example of the type of linear curve that is produced by exclusion chromatography is shown, in an exaggerated form, in figure 2

It is seen that the van't Hoff curve shown in figure 2, is a completely different type to that in figure 1. In this distribution system, there is only a very small enthalpy change , but in this case, a very high entropy contribution.

Figure 2. Exclusion Chromatography Using Entropically Driven Distribution

This distribution system is not dominated by molecular forces. The relatively large entropy change is a measure of the loss of randomness or freedom that happens when the solute molecule transfers from one phase to the other. The more random and 'more free' the solute molecule is in a particular environment, the greater its entropy in that environment. The large entropy change shown in system (B) (figure 2), indicates that the solute molecules are more constrained in the stationary phase (e.g., confined in the pores of the exclusion media) than they were in the mobile phase. This restriction is responsible for the greater distribution of the solute in the stationary phase and its greater retention. Because the change in entropy is the major contribution to the change in standard energy, In thermodynamic terms, the distribution is "entropically driven".

Chiral separations and separations that are dominated by size exclusion are examples of entropically driven separations.

It is important to understand that chromatographic separations cannot be exclusively "energetically driven" or "entropically driven"; both components will always be present to a greater or lesser extent. It is by the careful adjustment of both the "energetic" and "entropic" components of a distribution that very difficult and subtle separations can be accomplished.

There are some reports in the literature that feign to show nonlinear van't Hoff curves. Nonlinear van't Hoff curves are, in fact, a contradiction in terms. To give an example, a curve relating log(V'_r) against 1/T for solutes eluted from a reverse bonded phase by a mixed solvent will sometimes be nonlinear. Graphs relating log(V'_r) against 1/T, however, can only be termed van't Hoff curves if they apply to an established and constant equilibrium system, where the interactive character of the system does not change with temperature. In a reversed phase system, the solvents themselves are also differentially distributed between the two phases in addition to the solute. As a consequence, (depending on the actual concentration of solvent) if the temperature changes, so will the relative amount of solvent adsorbed on the stationary phase surface alter, and so the interactive character of the stationary phase will also change. It follows, that the curves relating log(V'_r) against 1/T will not be linear and, more importantly, as the distribution system itself is varying, the curves will not constitute van't Hoff curves. This effect is well known, an early example is afforded by work carried out by Scott and Lawrence (1).

Scott and Lawrence examined the effect of water vapor as a moderator on the surface of alumina in the gas/solid separations of some n-alkanes. Examples of the results obtained by those authors are shown in figure 3. The alumina column was moderated by a constant concentration of water vapor (constant partial pressure of water) contained in the carrier gas.

J. Chromatogr. Sci.7(1969)65

Figure 3. Graphs of Log(V'r) against 1/T for Some n-Alkanes Separated on Water-Vapor-Moderated Alumina

When the temperature of the column is increased, some of the water moderator is desorbed from the surface and the alumina became more active. As a consequence, as the temperature is initially raised, the retention volume of each solute is increased. However, when all the water has been removed from the alumina then the surface takes on a constant interactive character. Subsequent increases in temperature will cause the retention volume to begin to fall in the expected manner. The net result of these two effects are shown in the curves depicted in figure 3. The curves are not linear because the interacting surface changes as the temperature is raised and thus the distribution system also changes. As a consequence of the distribution system not remaining constant, the curves are not van't Hoff curves.

Under certain circumstan

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The Thermodynamics of Chromatography

Thermodynamics Basics