Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Chromatography

Field: Separation Science

A laboratory technique for the separation of a mixture.

Sequence of Expressions

Definition

Mobile Phase

Let v(t)\mathbf{v}(t) be the velocity vector of the mobile phase flow through the column. The mobile phase is characterized by its flow rate FF (volume/time) and its composition CM={CM,j}j=1N\mathbf{C}_{M} = \{C_{M, j}\}_{j=1}^{N}. The transport of an analyte ii is governed by the mass balance equation:\n∂Ci∂t+∇⋅(vCi)=DM∇2Ci\frac{\partial C_i}{\partial t} + \nabla \cdot (\mathbf{v} C_i) = D_M \nabla^2 C_i \nwhere DMD_M is the hydrodynamic dispersion coefficient.
Define the stationary phase S\mathbf{S} as a fixed solid matrix with a defined pore structure and surface area AsurfA_{surf}. The interaction capacity of the phase for analyte ii is modeled by the adsorption equilibrium qi=f(Ci,T)q_i = f(C_i, T), where qiq_i is the adsorbed concentration. The phase resistance RSR_S is proportional to the effective surface area and the interaction strength χi\chi_{i}: \nRS∝Asurf⋅χiR_S \propto A_{surf} \cdot \chi_{i}
Define the retention time tRt_R for an analyte ii as the time required for the analyte to pass the detector, measured from the injection point. It is related to the dead time tMt_M (void time) and the retention factor ki′k'_i by:\ntR=tM(1+ki′)t_R = t_M (1 + k'_i) \nwhere ki′k'_i is the retention factor, defined as ki′=Vi,eqVMk'_i = \frac{V_{i, eq}}{V_M}, with Vi,eqV_{i, eq} being the equilibrium volume and VMV_M being the mobile phase volume.
Consider an analyte AA partitioning between two immiscible phases, Phase 1 (concentration C1C_1) and Phase 2 (concentration C2C_2). The partition coefficient KK is defined as the ratio of the equilibrium concentrations:\nK=C2C1K = \frac{C_{2}}{C_{1}} \nThis coefficient quantifies the relative affinity of AA for Phase 2 over Phase 1 at equilibrium.
For two adjacent peaks, ii and jj, the chromatographic resolution RsR_s is defined as:\nRs=2(tR,j−tR,i)wi+wjR_s = \frac{2(t_{R,j} - t_{R,i})}{w_i + w_j} \nWhere tR,kt_{R,k} is the retention time of peak kk, and wkw_k is the width of peak kk at the base (or at a specified height).
The Van't Hoff factor ii relates the observed effective concentration CobsC_{obs} to the ideal concentration CidealC_{ideal} for an electrolyte solution undergoing dissociation:\ni=CobsCideali = \frac{C_{obs}}{C_{ideal}} \nFor a solute dissociating into nn ions, the factor ii is theoretically approximated by the number of ions produced, i→ni \to n, accounting for ionic strength effects on retention.
Let CiC_i be the concentration of analyte ii in the mobile phase and qiq_i be the concentration of analyte ii adsorbed onto the stationary phase. The equilibrium relationship is described by the adsorption isotherm, which relates qiq_i and CiC_i at constant temperature TT: \nqiCi=Kads,i+1mqiCi\frac{q_i}{C_i} = K_{ads, i} + \frac{1}{m} \frac{q_i}{C_i} \nwhere Kads,iK_{ads, i} is the adsorption equilibrium constant and mm is the slope of the isotherm (e.g., Freundlich or Langmuir model parameters).
The plate height HH (a measure of band broadening) is defined by the flow rate uu and the physical parameters of the system via the Van Deemter equation:\nH(u)=A+Bu+CuH(u) = A + \frac{B}{u} + C u \nWhere AA represents the eddy diffusion term (geometry), BB represents the longitudinal diffusion term (molecular diffusion), and CC represents the mass transfer resistance term (kinetics).
Principle

Equilibration

Let Cm(t)C_{m}(t) and Cs(t)C_{s}(t) be the concentrations of an analyte in the mobile and stationary phases, respectively. Equilibration is achieved when the rate of change of the analyte distribution approaches zero, satisfying the steady-state condition:\ndCmdt=0 and dCsdt=0\frac{dC_{m}}{dt} = 0 \text{ and } \frac{dC_{s}}{dt} = 0 \nThis implies that the rate of transfer between phases, keq(Cm−Cs/Keq)k_{eq}(C_{m} - C_{s}/K_{eq}), balances the flow dynamics.
For flow through a cylindrical capillary column of radius rr and length LL, the volumetric flow rate QQ is governed by the Hagen-Poiseuille equation, assuming laminar flow and negligible entrance effects:\nQ=πr48ηΔPLQ = \frac{\pi r^4}{8\eta} \frac{\Delta P}{L} \nWhere η\eta is the dynamic viscosity of the mobile phase and ΔP\Delta P is the pressure drop across the column.