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July 1, 2018

*We review different approaches for measuring solute retention.*

**In Part III of this series, we review different approaches for measuring solute retention in liquid chromatography. The origins of these parameters are described, as well as their significance and applications. **

In Part I, we established that chromatographic separations are governed by the distribution coefficient of solutes when they partition between mobile and stationary phases (1). The energetics of this process, described in Part II (2), were shown to depend upon both enthalpic and entropic changes that take place; either of which can dominate, depending upon the separation mechanism. Part III of this series reviews approaches for determining liquid chromatography (LC) retention characteristics of solutes, their derivations, and applications. In this manner, the relationship between solute retention and thermodynamic parameters can be established.

Chromatography is a dynamic process in which solutes are in a state of continuous sorption and desorption from the mobile phase to the stationary phase. Since the retention of solutes by the stationary phase is characteristic of their chemical and structural composition, approaches are needed to accurately measure elution time or volume.

Historically, retention parameters were required for sample identification and characterizing adsorbents and solvents used for elution. These parameters were also deemed useful for studying the separation process, including peak broadening. Different retention parameters have been introduced, depending upon the LC technique employed; each of these approaches are reviewed in detail.

**Retention Factor, k**

During elution, solute molecules are in continuous motion, traveling longitudinally through a column, and dancing laterally between stationary and mobile phases. The ratio of the average time a solute molecule spends in the stationary phase (*t*_{s})_{i} to the time it spends in the mobile phase (*t _{m}*)

Equation 1 represents a snapshot of the activity of a solute at a given instance of time at equilibrium. By integrating these time increments, we can now formulate a measurable quantity:

where *t*_{s} is the total time a solute spends in the stationary phase, and *t*_{m} is the total time in the mobile phase.

In practice, the measured elution time of a solute *t*_{r} is equal to the time spent in the stationary phase plus the time spent in the mobile phase: *t*_{r} = *t*_{s} + *t*_{m}. Furthermore, the time spent in the mobile phase is identical to the elution time of an unretained peak *t*_{0}. Equation 2 becomes

This term was formerly called the capacity factor *k*', but has been renamed retention factor *k*. By subtracting unity, as shown in the last part of Equation 3, we uncouple the time or volume a solute spends in the mobile phase. Thus *k *represents the retention properties of a solute in the stationary phase. Since *t _{r}* is normalized with respect to

**Retention Factor, R_{f}**

Retention is also used in both planar (paper and thin layer) and classical, open-column chromatography, and is still popular among aficionados of these techniques. This definition has the form

where *d*_{0} is the migration distance of the solvent front measured from the bottom of the plate in contact with developing solvent, and *d*_{r} is the migration distance of the solute spot on the plate or on paper. In open-column chromatography, the retention is ascertained in the opposite direction, going from column top to bottom, with *d*_{r} typically being the length of the packed bed, and *d*_{0} the distance from the top of the bed to the solute band or zone after a set elution time.

*R*_{f} values from thin-layer chromatography (TLC) are used for peak identification and evaluating adsorbent–solvent systems. The retention factor is used less frequently for open-column LC because of poor reproducibility. This mode of LC, however, is used mainly for preparative, not analytical, separations.

**Retention, R**

The next retention parameter *R* is defined as the ratio of the time a solute spends in the mobile phase *t*_{0} with respect to the total time it remains in the column *t*_{r},

By inference, the ratio is also equivalent to the amount of solute in the mobile phase at any given time to the total amount injected, *w*_{t} = *w*_{m} + *w*_{s}, where *w*_{m} and *w*_{s} are the solute weight in the mobile and stationary phases, respectively. Thus

**Retention Volume, V_{r}**

Equation 6 allows us to develop a relationship between retention and the distribution coefficient *K*. This transformation is accomplished by replacing solute weights with corresponding concentrations and volumes (*w*_{m} = *c*_{m}*V*_{m} and *w*_{s} = *c*_{s}*V*_{s}),

Dividing numerator and denominator by *c*_{m}, and letting *K* = *c _{s}*/

We return to Equation 5, and multiply both numerator and denominator by flow rate *F*,

Equation 8 is set equal to equation 9, and we solve for V_{r}, obtaining

Equation 10 is one of the more important relationships in chromatography. It tells us that the retention a solute is simply the sum of two volumes: the mobile phase *V*_{m}, plus some fraction or multiple of the stationary phase, *KV*_{s}. The separation mechanism is defined by *K* and column properties by *V*_{s} and *V*_{m}.

Equation 10 is the general chromatographic equation valid for every mode of chromatography, with modification as required. For example, if the separation is based on adsorption, the surface area of the packing is used in place of *V*_{s}. If the separation is controlled by several mechanisms (an undesirable situation, as described in Part I), additional distribution coefficients must be added, along with the stationary-phase volume (or amount) associated with each distribution coefficient:

In actual practice, however, it is preferable to optimize the separation, such that a single mechanism dominates to ensure reproducible, high resolution results. If not, mixed-mechanism separations can lead to a metastable LC system.

SEC Elution Volume, *V*_{e}

When using size-exclusion chromatography (SEC), Equation 10 must be modified, since the entire SEC separation occurs within the pore volume *V _{i}* of the packing (3). Thus,

Equation 12 is used when constructing SEC calibration plots (log molecular weight versus *V _{e}*). Furthermore,

**Distribution Coefficient, K**

We can now establish a relationship between the retention factor *k*, which can be readily measured, and the distribution coefficient *K*, a therodynamic parameter. Elution times in Equation 3 are converted to elution volumes by multiplying the numerator and denominator by the flow rate. After rearranging, Equation 3 becomes

Setting Equations 13 and 10 equal to one another and rearranging terms, we arrive at

The *V*_{s}/*V*_{m} term, called the phase ratio, controls the elution properties of a solute. In effect, the elution properties of a given separation can be adjusted through the phase ratio, provided that the distribution coefficient is kept constant.

It is important to note that *k* can be used in place of *K* to establish thermodynamic relationships or trends among compounds, or used to characterize the thermodynamic properties of chromatographic systems.

All possible LC retention parameters are summarized in Table I. These retention parameters are based on the distribution coefficient, which drives the separation process (see Part I).

Historically, the parameter *R* was used to study chromatographic properties of adsorbents and to establish a system of identifying and categorizing elution behavior of compounds. The *R*_{f} retention factor was used for identification of solutes with planar chromatographic methods. With the introduction of high performance liquid chromatography (HPLC) more than 50 years ago, the capacity factor *k*', subsequently renamed the retention factor *k*, is the most widely used HPLC parameter for characterizing elution properties of solutes, validating LC methods, identifying compounds, and accessing thermodynamic behavior of solutes.

Subsequent articles will discuss the origin, significance, and measurement of theoretical plates and peak broadening in LC.

(1) H.G. Barth, *LCGC North Am. ***36**(3), 200–202 (2018).

(2) H.G. Barth, *LCGC North Am. ***36**(6), 394–396, 405 (2018).

(3) H.G. Barth, *J. Chem. Educ. (Online early access). http://pubs.acs.org/doi/abs/10.1021/acs.jchemed.8b00171*

(4) P.A. Bristow, *Liquid Chromatography in Practice* (hetp, Cheshire, UK, 1976).

(5) E. Heftmann, *Chromatography, 3rd Ed.* (Reinhold, New York, 1975).

(6) B.L. Karger, L.R. Snyder, and C. Horvath, *An Introduction to Separation Science *(Wiley-Intersciences, NY, 1973).

(7) L.R. Snyder, J.J. Kirkland, and J.W. Dolan, I*ntroduction to Modern Liquid Chromatography, *3rd Ed. (Wiley, New York, 2010).

**Howard G. Barth **is with Analytical Chemistry Consultants, Ltd., in Wilmington, Deleware. Direct correspondence to: howardbarth@gmail.com