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WOLFRAM NOTEBOOK
Chromatographic Reactions of Three Components
Chromatographic Reactions of Three Components
Three species , , and move in an electrostatic, centrifugal, or velocity field and mix according to the cyclic reaction system: . All reaction rate constants are assumed to be equal to =.
A
1
A
2
A
3
A
1
k
e
⇆
k
e
A
2
k
e
⇆
k
e
A
3
k
e
⇆
k
e
A
1
k
e
-2
10
-1
s
The three governing equations for the concentrations of the species are
c
i
A
i
∂
c
i
∂t
D
i
2
∂
c
i
∂
2
x
v
i
∂
c
i
∂x
k
e
c
i
k
e
c
j
k
e
c
m
where } are three permutations of .
{i,j,m
{1,2,3}
The boundary and initial conditions are (x,t)=0 and (x,0)=δ(x).
lim
x±∞
c
j
c
j
1
3
All components are assumed to be sufficiently similar to have identical diffusion coefficients The velocities of species , , and are taken to be =0.02cm/s, =-0.03cm/s, and =0.04cm/s.
===s.
D
1
D
2
D
3
-5
10
2
cm
A
1
A
2
A
3
v
1
v
2
v
3
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).
μ
1j
μ
2j
μ
3j
These moment expressions are used to construct the concentration profile using the following truncated series of the Gram-Charlier expansion:
c
j
1
3
-
2
X
j
μ
3j
3
X
j
X
j
1.5
2
μ
2j
2π
μ
2j
where =.
X
j
x-
μ
1j
2
μ
2j
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.
A
1
A
2
A
3
τ=t.
k
e
τ
τ
A
1
A
2
A
3
τ
τ
A
1
A
2
A
3
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.
3
∑
i=1
c
i
τ