The six governing equations for the concentrations
of the species
are
,
where
} are the six permutations of
.
The boundary and initial conditions are
and
.
All components are assumed to be sufficiently similar to have identical diffusion coefficients
The velocities of species
,
, and
are taken to be
,
, and
.
An expression for the reduced first central moment,
, the second and third central moments (
and
) were derived in the paper by H. Binous and B. J. McCoy in 1992 (see Details section below).
These moment expressions are used to construct the concentration profile using the following truncated series of the Gram-Charlier expansion:
,
where
.
This Demonstration shows the concentration profiles of the species
,
, and
at different times. In accordance with the initial conditions, peaks are very high at small dimensionless time
For large values of
, all peaks merge into one. Thus, (1) separation of peaks occurs for small
; (2) the species
,
, and
behave as one component for large
. For intermediate values of
, higher-order terms may be needed to adequately represent the pulse. The concentrations of the species
,
, and
are shown as blue, magenta, and brown peaks, respectively.
The cyan peak corresponds to
(i.e., the concentration of the combined species). This combined concentration exhibits multiple peaks or a broad asymmetric peak for small
. This behavior is actually observed in reverse-phase liquid chromatographic separations where one protein can undergo conformational changes and be present in multiple states.
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