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The Mass Balance Equation



The Mass Balance Equation


The mass balance equation is the fundamental equation for chromatography. To understand chromatographic theory, a thorough understanding of the meaning of this equation is essential.


For a component, i, that migrates through the chromatographic column the mass balance equation can be written:


                  ( 1 )




S = surface area of the particles in the column (m2 / kg)


r = density of the solid particle ( kg / m3 )


e = column porosity ( dimensionless )


A = inner area of the column tube.


ni = analyte concentration on the stationary phase of component i ( mol / m2 )


v = mobile phase flow ( m3 / s )


ci = analyte concentration in the mobile phase of component i ( mol/m3)

Deff =  an effective diffusion constant ( m2 /s )


Deff is a measure of the dispersion or band broadening of the analyte zone that is obtained during the migration through the column. It is a result of a great number of random processes that the analyte experience  e.g. inhomogenous flow, diffusion in pores and the mobile phase etc.


It can often be assumed that the equilibrium for the analyte between the mobile and stationary phase is immediate, i.e.:


             ( 2 )


The mass balance equation for the ideal model of chromatography therefore becomes:


                ( 3 )


This equation is usually reformulated to the following form:


                 ( 4 )




 = phase ratio of the column ( m2 / m3 )



vl = v /eA = the linear velocity of the mobile phase in the column ( m/s )


When the relation between the concentration of component i in the mobile and stationary phase, respectively, is linear ( i.e. ni = K * ci ) the dni/dci - term is constant.


It can be shown that in this case the velocity of the zone, vi,  follows the relation


vi = v0 / ( 1 + F*K )             ( 5 )


where F*K is the retention factor, k, and v0 the linear velocity of the mobile phase through the column


This is the equation that normally is used in analytical applications of chromatography. This special case is called linear chromatography and only holds in the region where the adsorption isotherm is linear. Att best this is the case at low or moderatley high analyte concentrations.


In the general the adsorption isotherm is non-linear and in this case the local velocity of the zone becomes a function of ci. If e.g. the adsorption isotherm is of Langmuirian type we find that:


 and             ( 5 )


where Ki is the adsorption constant for component i ( m3/mol ) and n0i is the monolayer capacity of the stationary phase for component i ( mol/m2 ). This is called non-linear chromatography. In non-linear chromatography the shape of the eluted peak often is strongly affected by the non-linearity.


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