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1.7: Sorption and Chromatography

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    76364
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    Adsorption, Ion Exchange, and Chromatography

    \(c_i\) = concentration of species i in the mobile phase (mass volume-1) or (mole volume-1)

    \(k_i\) = empirical constant for species i for isotherms (units vary)

    \(K_i\) = adsorption equilibrium constant for species i

    \(n_i\)= internal parameter for isotherms (units vary)

    \(p_i\)= partial pressure of species i (pressure)

    \(q_i\)= amount of species i adsorbed per unit mass of adsorbent at equilibrium (mass mass-1) or (mole mass-1)

    \(q_{m_i}\) = amount of species i adsorbed per unit mass of adsorbent at maximum loading, where maximum loading corresponds to complete surface coverage (mass mass-1) or (mole mass-1)

    linear isotherm:

    \[q_i=k_ip_i \tag{31.1}\]

    Freundlich isotherm:

    \[q_i=k_ip_i^{1/n_i} \tag{31.2}\]

    Langmuir isotherm:

    \[q_i=\frac{K_iq_{m_i}p_i}{1+K_ip_i} \tag{31.3}\]

    chromatography equilibrium:

    \[K_i=\frac{q_i}{c_i} \tag{31.4}\]

    Watch a video from LearnChemE for an explanation about the concept of adsorption: Adsorption Introduction (8:49)

    Modeling Differential Chromatography

    \(\alpha_i\) = average partitioning of species i between the bulk fluid and sorbent (unitless)

    \(\epsilon_b\) = sorbent porosity, ranges from 0 to 1 (unitless)

    \(\epsilon^*_{p,i}\) = inclusion porosity, accounts for accessibility of sorbent pores to species i (unitless)

    \(\tau_f\) = sorbent tortuosity factor, usually approximately 1.4 (unitless)

    \(\omega_i\) = fraction of solute in the mobile phase, relative to sorbed solute, at equilibrium (unitless)

    \(A\) = cross-sectional area of the column (area)

    \(c_{f,i}\) = concentration of species i in the mobile phase (mass volume-1) or (mol volume-1)

    \(D_{e,i}\) = effective diffusivity of species i within the sorbent pores (length2 time-1)

    \(E_i\) = coefficient that accounts for axial diffusion of species i and non-uniformities of flow (length2 time-1)

    \(H_i\) = height of theoretical chromatographic plate for species i (length)

    \(k_{a,i}\) = kinetic rate constant of adsorption of species i to the sorbent (time-1)

    \(k_{c,i}\) = mass transfer coefficient of species i in the mobile phase (length time-1)

    \(k_{c,i,tot}\) = overall mass transfer coefficient of species i (length time-1)

    \(K_{d,i}\) = equilibrium distribution coefficient of species i between the mobile phase and sorbent (unitless)

    \(L\) = length of column (length)

    \(m_{0_i}\) = amount of solute i fed to column (mass) or (mol)

    \(R_{1,2}\) = resolution of species 1 and 2 in the proposed operating condition (unitless)

    \(R_p\) = radius of sorbent particles (length)

    \(s_i\) = variance of the Gaussian peak of the distribution of species i along the column length (time)

    \(t\) = elapsed time since loading of the column (time)

    \({\overline t}_i\) = mean residence time of species i in the column (time)

    \(u\) = actual fluid velocity through the bed (length time-1)

    \(u_s\) = superficial fluid velocity through the bed (length time-1)

    \(z\) = position along the length of the column, in the direction of flow (length)

    \(z_{0,i}\) = mean position of species i along the length of the column as a function of time (length)

    \[z_{0,i}(t)=\omega_iut \tag{32.1}\]

    \[\omega_i=\frac{1}{1+\frac{1-\epsilon_b}{\epsilon_b\alpha_i}} \tag{32.2}\]

    \[\alpha_i=\frac{1}{\epsilon_{p,i}^*(1+K_{d,i})} \tag{32.3}\]

    \[u=u_s/\epsilon_b \tag{32.4}\]

    \[\overline t_i=\frac{L}{\omega_iu} \tag{32.5}\]

    \[c_{f,i}(z,t)=\frac{m_{0_i}\omega_i}{A\epsilon_b(2\pi H_iz_0)^{0.5}}\textrm {exp}\left(\frac{-(z-z_0)^2}{2H_iz_0}\right) \tag{32.6}\]

    \[H_i=2\left[\frac{E_i}{u}+\frac{\omega_i(1-\omega_i)R_pu}{3\alpha_ik_{ci,tot}}\right] \tag{32.7}\]

    \[N_{{\rm Pe},i}=N_{\rm Re}N_{{\rm Sc},i}=\frac{2R_pu\epsilon_b}{D_i} \tag{32.8}\]

    if \(N_{{\rm Pe},i}<<1\)

    \[E_i=\frac{D_i}{\tau_f} \tag{32.9}\]

    else

    \[E_i=\dfrac{2R_pu \epsilon_b}{N_{ Pe,E,i}} \tag{32.10}\]

    \(N_{{\rm Pe},E,i}\) calculated by 15-61 or 15-62, Seader

    \[\frac{1}{k_{ci,tot}}=\frac{1}{k_{c,i}}+\frac{R_p}{5\epsilon_{p,i}^*D_{e,i}}+\frac{3}{R_pk_{a,i}\epsilon_{p,i}^*}\left[\frac{K_{d,i}}{1+K_{d,i}}\right]^2 \tag{32.11}\]

    \[s_i^2=\frac{\overline t_iH_i}{\omega_iu} \tag{32.12}\]

    \[R_{1,2}=\frac{\textrm {abs}(\overline t_1-\overline t_2)}{2(s_1+s_2)} \tag{32.13}\]

    Example

    1.0 g of species A is added to a chromatography column of cross-sectional area 1.0 m2 and length 1.0 m. Mobile phase is added at a flowrate of \(4.0 \times 10^{-3}\) m3/s. Species A has a mass transfer coefficient of \(2.0 \times 10^{-5}\) m/s in this solvent. The selected sorbent has a porosity of 0.40 m and average particle radius of \(5.0\times 10^{-6}\) m. For species A in this sorbent, the inclusion porosity is 0.80, \(K_d = 50\), \(E = 2.0\times 10^{-8}\) m2/s, \(k_a = 100\) s-1 and the effective diffusivity is \(3.5\times 10^{-12}\) m2/s.

    (a) When is mean expected elution time for species A?

    (b) Plot the concentration profile for species A at 0.05 m increments along the column length in 10-minute increments, until all of the solute has eluted.

    (c) Find the variance of the peak for species A in the proposed operating condition.

    (d) The column feed also contains 1.0 g of species B. Species B has a mass transfer coefficient of \(1.0\times 10^{-5}\) m/s in the mobile phase, inclusion porosity of 0.50, \(K_d = 60\), \(E = 3.0\times 10^{-8}\) m2/s, effective diffusivity of \(4\times 10^{-12}\) m2/s and \(k_a = 200\) s-1. What is the resolution of these two species in the proposed operating condition?


    This page titled 1.7: Sorption and Chromatography is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Monica Lamm and Laura Jarboe (Iowa State University Digital Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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