= Chromatographic partition coefficient
{title2=$K=c_s/c_m$}
For a dilute <solute> distributed between two liquid regions at local <chemical equilibrium>, define the <chromatographic partition coefficient> as
$$
K=\frac{c_s}{c_m},\qquad
n=c_mV_m+c_sV_s=c_m(V_m+KV_s),\qquad
f_m=\frac{c_mV_m}{n}=\frac{V_m}{V_m+KV_s}.
$$
Here $c_m,c_s$ are <concentrations> expressed in the same units, $V_m,V_s$ specify accessible <volume (thermodynamics)> in the <chromatographic mobile phase> and immobilized liquid <chromatographic stationary phase>, respectively, $n$ is total <solute> <amount of substance>, and $f_m$ is the fraction of that <amount of substance> in the <chromatographic mobile phase>. Assume a constant <chromatographic partition coefficient>, rapid exchange, unchanged phases and <temperature>, and no <chemical reaction>. A <chemical equilibrium> population still exchanges <molecules>; it is not permanently divided into two sets.
The <amount of substance> balance proves the expression for $f_m$. Increasing either $K$ or $V_s/V_m$ reduces the mobile fraction, illustrating why a <concentration> ratio alone cannot determine retention. For surface <adsorption>, specify an <adsorption> storage law instead of assigning an arbitrary liquid <volume (thermodynamics)> to the <solid>.
An original discrete model makes the separation visible. In each identical cell, first establish local <chemical equilibrium>, then advance only the mobile fraction by one cell; let the stationary fraction wait. Repeat twelve times, without any additional spreading. With $V_s=V_m$, stipulated <chromatographic partition coefficients> $K_A=1/3$ and $K_B=3$ give $f_A=3/4$ and $f_B=1/4$. Each cycle contributes mean displacement $f_m$ cells, so the mean positions after twelve cycles are nine cells and three cells. Redistribution itself broadens each band: the cell positions have a <binomial distribution> with parameters $12,f_m$. These dimensionless chosen values illustrate transport, not measurements of any particular substances.
\Image[/chromatography-phase-exchange-separation.png]
{title=Reversible phase exchange turns different storage fractions into separated bands}
{id=image-chromatography-phase-exchange-separation}
{description=Original local <chemical equilibrium> model: the same accessible <volume (thermodynamics)> in each phase, constant <chromatographic partition coefficients>, and twelve discrete transport cycles. Both bands start at cell zero. The plotted fractions sum to one for each component; dashed lines mark mean positions. This is a partition model, not a depiction of <SEC pore access> or an actual column measurement.}
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