1.7: Sorption and Chromatography
- Page ID
- 76364
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\[\frac{1}{k_{ci,tot}}=\frac{1}{k_{c,i}}+\frac{R_p}{5\epsilon_{p,i}^*D_{e,i}}+\frac{3}{R_pk_{a,i}\epsilon_{p,i}^*}\left[\frac{K_{d,i}}{1+K_{d,i}}\right]^2 \tag{32.11}\]
\[s_i^2=\frac{\overline t_iH_i}{\omega_iu} \tag{32.12}\]
\[R_{1,2}=\frac{\textrm {abs}(\overline t_1-\overline t_2)}{2(s_1+s_2)} \tag{32.13}\]
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?