# Distribution Isotherms

Distribution Isotherms (Isotherm means constant temperature):

We have used the distribution and partition coefficients as ways to express the distribution of a molecule between the mobile and stationary phase.  Note that these equations take the form of equilibrium expressions, and that KC is a constant for the distribution of a solute compound between two particular phases.  (KX depends on the relative volumes of the two phases.)

Suppose that we plotted CS versus CM, as shown on the coordinate system in Figure 10.

Figure 10. Coordinate system with the concentration of analyte in the stationary phase (CS) shown on the y-axis and the concentration of analyte in the mobile phase (CM) shown on the x-axis.

Notice that CM plus CS would give the total amount of the solute injected.  So as you move to the right on the CM axis, it means that more sample is being injected into the chromatograph.  If we have the expression for the distribution coefficient:

$\mathrm{K_C = \dfrac{C_S}{C_M} = constant}$

The idealized form of the plot of the distribution isotherm is shown in Figure 11.  If we inject more sample into the column, it distributes according to a set ratio of the distribution coefficient.  The result would be a straight line, with the slope being the distribution coefficient.

Figure 11. Idealized plot of the distribution isotherm.

If we examine this in more detail, though, we will realize that the volume of stationary phase is some fixed quantity, and is usually substantially less than the volume of the mobile phase.  It is possible to saturate the stationary phase with solute, such that no more can dissolve.  In that case, the curve would show the behavior shown in Figure 12, a result known as the Langmuir isotherm (Langmuir was a renowned surface scientist and a journal of the American Chemical Society on surface science is named in his honor).  If we get a region of Langmuir behavior, we have saturated the stationary phase or overloaded the capacity of the column.

Figure 12. Plot of the Langmuir isotherm.

We might also ask whether you could ever get the following plot, which is called anti-Langmuir behavior.

Figure 13. Plot of the anti-Langmuir isotherm.

There are actually two ways this could happen.  One is if the solute dissolves in the stationary phase, creating a mixed phase that then allows a higher solubility of the solute.  This behavior is not commonly observed.  Another way that this can occur is in gas chromatography if too large or concentrated a sample of solute is injected.  At a fixed temperature, a volatile compound has a specific vapor pressure.  This vapor pressure can never be exceeded.  If the vapor pressure is not exceeded, all of the compound can evaporate.  If too high a concentration of compound is injected such that it would exceed the vapor pressure if all evaporates, some evaporates into the gas phase but the rest remains condensed as a liquid.  If this happens, it is an example of anti-Langmuir behavior because it appears as if more is in the stationary phase (the condensed droplets of sample would seem to be in the stationary phase because they are not moving).

The last question we need to consider is what these forms of non-ideal behavior would do to the shape of a chromatographic peak.  From laws of diffusion, it is possible to derive that an “ideal” chromatographic peak will have a symmetrical (Gaussian) shape.  Either form of overloading will lead to asymmetry in the peaks.  This can cause either fronting or tailing as shown in Figure 14.  The Langmuir isotherm, which results from overloading of the stationary phase, leads to peak tailing.  Anti-Laugmuir behavior leads to fronting.

Figure 14. Representation of a chromatographic peak exhibiting ideal peak shape, fronting, and tailing.