# 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=10s

e

-2

-1

The three governing equations for the concentrations of the species are

c

i

A

i

∂c

i

∂t

i

∂c

2

i

∂x

2

i

∂c

i

∂x

e

i

e

j

e

m

where } are three permutations of .

{i,j,m

{1,2,3}

The boundary and initial conditions are and .

lim

x±∞

c(x,t)=0

j

c(x,0)=δ(x)

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 , , and .

D=D=D=10cm/s.

1

2

3

-5

2

A

1

A

2

A

3

v=0.02cm/s

1

v=-0.03cm/s

2

v=0.04cm/s

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(x,t)=1+μ8X-12X/62μ/

j

1

3

-X

2

j

3j

3

j

j

2j

1.5

2πμ

2j

where .

X=

j

x-μ

'

1j

2μ

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=kt.

e

τ

τ

A

1

A

2

A

3

τ

τ

A

1

A

2

A

3

The cyan peak corresponds to c (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

i

τ