Analytical Solution of the Ideal Model of Chromatography for a Bi

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Analytical Solution of the Ideal Model of Chromatography for a BiLangmuir Adsorption Isotherm Fabrice Gritti and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, United States ABSTRACT: The (closed-form) analytical solution of the ideal model of chromatography was derived in the case of a bi-Langmuir adsorption isotherm model. This explicit solution provides the sample concentration as a function of the elution time as the unique real and positive root of a quartic polynomial equation. Using this analytical solution, we could calculate the parameters of the bi-Langmuir adsorption model for phenol on a column packed with Poroshell 120 C18 core− shell particles, eluted with a mixture of methanol and water, and operated under high inlet pressure using the retention time method. The method permitted a rapid, economical determination of the best adsorption isotherm parameters for phenol. It required less than 32 mg of sample, 50 mL of eluent, and less than 4 h to complete the measurement of 20 adsorption data points. The relative precisions of the isotherm parameters were 5% for the equilibrium constant and 9% for the saturation capacity of the strong adsorption sites and 24% for the equilibrium constant and 5% for the saturation capacity of the weak adsorption sites. Compared with determinations made with the standard HPLC technology and frontal analysis experiments, the working time, eluent volume consumed, and mass of compound used were reduced by factors of 4, 20, and 400, respectively.

T

adsorption behavior. In practice, whether the retention mode is RPLC, HILIC, ion-exchange, or chiral chromatography, which together account for more than 70% of separation methods used in industry in 2011,9 the bi-Langmuir isotherm model applies because the surface of chromatographic supports and of solid adsorbents are generally heterogeneous.10−12 This was confirmed recently by Monte Carlo simulation and the bimodal energy distribution of the equilibrium constant.13 In numerous cases, the experimental data fit well to a bi-Langmuir adsorption model, for which the solution is not as trivial as that of the homogeneous Langmuir model because the function f is relatively complex and the chromatogram cannot be obtained explicitly. The problem can be solved by numerical integration,8 although this is time-consuming and produces less accurate results. Instead, the goal of this work was to provide a closed-form analytical solution of the ideal model of chromatography for a heterogeneous bi-Langmuir adsorption isotherm and to derive an explicit expression for the ideal chromatogram under dispersionless conditions. The analytical solution was validated by comparing the derived ideal chromatograms to ones that were calculated assuming a very large efficiency. In practice, this analytical solution was used to predict the elution times of the front shocks of a series of overloaded band profiles resulting from a two-site adsorption process. The retention time method (RTM)14,15 was then extended to the determination of the biLangmuir isotherm parameters of phenol on a Poroshell C18

he overloaded band profiles observed in liquid chromatography result from the combination of the nonlinear adsorption equilibrium process that takes place at high solute concentrations1,2 and the nonequilibrium mass transfer phenomena3,4 or, in other words, from the convolution of a thermodynamic nonlinear equilibrium process and a kinetic contribution.5 The thermodynamic contribution causes the band asymmetry, whereas the kinetic contribution smoothes the edges of the so-called thermodynamic ideal chromatograms that would be expected in the absence or mass transfer phenomena (i.e., if the efficiency of the chromatographic column were infinite). It is important to recall that the total variance or second moment of the concentration distribution is not the mere sum of the variances of the thermodynamic and the kinetic contributions, as is often wrongly assumed in the literature.5,6 The knowledge of the adsorption isotherm, q = f(C), including the isotherm coefficients, is of fundamental importance in preparative chromatography at all solute concentrations. In the absence of sample dispersion along the column, the solution of the mass balance equation is provided by the theory of characteristic lines.3 A given wave of concentration C propagates along the column, with each concentration moving at a constant velocity that is directly related to the local slope of the adsorption isotherm (dq/dC). It exits the column after a finite time t = f(C). In the particular case of a Langmuir isotherm, the inverse function of f was derived mathematically and the chromatogram C = f−1(t) = g(t) was expressed in an explicit analytical form.7,8 Unfortunately, this model cannot be applied to most compounds because they do not follow strict Langmuir © 2013 American Chemical Society

Received: February 7, 2013 Accepted: July 28, 2013 Published: July 29, 2013 8552

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stationary phase from a methanol−water eluent mixture. This was done by minimizing the distance between the predicted and experimental elution times of the shocks without the need for calibration curves. The accuracy and precision of the best isotherm parameters, the experimental time necessary to collect all the data, and the amount of solvent and compound needed during application of the RTM method are discussed and compared to those required in classical frontal analysis (FA) experiments using a standard chromatograph and conventional columns.

The use of these intermediate symbols k1, k2, C1, C2, x1, and τ may seem arbitrary. They are used only to allow the writing of simpler equations, at the cost of a somewhat more complex presentation. According to eq 4 and the above definitions, the general expression for the reduced time variable τ(C) in eq 5 for a biLangmuir isotherm is then given by

THEORY First, we recall the analytical solution of the ideal chromatogram in the case of the simple homogeneous Langmuir adsorption isotherm. Next, the general problem for a biLangmuir isotherm model is presented and its solution discussed. Solution of the Ideal Model of Chromatography for a Langmuir Isotherm. For the Langmuir isotherm and an ideal column, the elution profile of the compound, C(τ), is given by the following analytical solution based on the theory of characteristic lines:7 ⎤ 1⎡ 1 C(τ ) = ⎢ − 1⎥ ⎦ b⎣ τ (1)

In contrast to the case of a Langmuir isotherm, the determination of the inverse solution C = f−1(τ) for a biLangmuir isotherm is not trivial, and its unique solution will be derived analytically in the next section. Solution of the Ideal Model for a Bi-Langmuir Isotherm. We may rewrite eq 6 as

τ=

■

=

[C2 + C ]2

(1 − x1)C22 x1C12 (C2 + C)2 + (C1 + C)2 τ τ

P2(C)2 = P1(C)2

(6)

(7)

(8)

This equation has two solutions: P2(C) = P1(C)

(9)

or

P2(C) = −P1(C)

(10)

Consider an arbitrary real number y. We may add y(C1 + C)(C2 + C) + (y2/4) to both the right- and left-hand sides of eq 7 without changing the solution. This provides the equation y ⎤2 ⎡ ⎢⎣(C1 + C)(C2 + C) + 2 ⎥⎦

(3)

where b1 > 0 and b2 > 0 are the strictly positive adsorption− desorption equilibrium constants for the sample distributions between the bulk phase and adsorption sites of types 1 and 2, respectively, and qS,1 > 0 and qS,2 > 0 are the strictly positive saturation capacities for adsorption sites of types 1 and 2, respectively. In the case of a bi-Langmuir isotherm model, the elution time t(C) (i.e., the time at which the concentration C reaches the column outlet) is given by the equation of the characteristic line that is the solution of the mass balance equation of the ideal model (no axial dispersion) for this concentration:2 ⎡ ⎤ k1 k2 ⎥ t ( C ) = t p + t 0 ⎢1 + + 2 2 (1 + b1C) (1 + b2C) ⎦ ⎣

(1 − x1)C22

The principle used to solve this quadratic equation consists of rewriting it in terms of the squares of a second-degree polynomial P2(C) and a first-degree polynomial P1(C) that are functions of the concentration:

where t is the time, tp is the injection time, t0 is the column hold-up time, and k0 is the retention factor under linear conditions. The Problem of the Ideal Model of Chromatography for a Bi-Langmuir Isotherm. The equation of the biLangmuir isotherm is b1C b2 C + qS,2 1 + b1C 1 + b2 C

[C1 + C ]2

+

(C1 + C)2 (C2 + C)2

where b is the equilibrium constant of the Langmuir isotherm and τ is the dimensionless reduced time, given by t − t p − t0 τ= k 0t0 (2)

q(C) = qS,1

x1C12

= y(C1 + C)(C2 + C) + +

y2 x C2 + 1 1 (C2 + C)2 4 τ

(1 − x1)C22 (C1 + C)2 τ

(11)

This can be rewritten as y ⎤2 ⎡ C C C C + + + ( )( ) 2 ⎢⎣ 1 2 ⎥⎦ ⎡ 2C1C2C̅ ⎤ C2 ⎤ 2 ⎡ ⎥C + ⎢y(C1 + C2) + = ⎢y + ⎥C ⎢⎣ ⎣ τ ⎥⎦ τ ⎦ ⎡ y2 C 2C 2 ⎤ + ⎢ + yC1C2 + 1 2 ⎥ τ ⎦ ⎣4

(4)

where k1 = FqS,1b1 and k2 = FqS,2b2, in which F is the phase ratio of the column, given by F = (1 − ε)/ε, where ε is the total porosity of the column. We define the following variables: C1 = 1/b1, C2 = 1/b2, and x1 = k1/(k1 + k2). Also, the reduced time variable τ is defined as t − t p − t0 τ= (k 1 + k 2 )t 0 (5) 8553

(12)

y ⎤2 ⎡ ( C C )( C C ) + + + = aC 2 + bC + c 2 ⎢⎣ 1 2 ⎥⎦

(13)

y ⎤2 ⎡ ( C C )( C C ) + + + = (eC + f )2 2 ⎢⎣ 1 2 ⎥⎦

(14)

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yr into the general expressions for e and f in terms of the second-degree polynomial coefficients a(y) and b(y) obtained from eqs 13 and 14:

where, for the sake of writing simpler equations, we have introduced in eq 12 the symbols C̅ and C 2 , which are arbitrarily defined as C̅ = x1C1 + (1 − x1)C2

(15)

C 2 = x1C12 + (1 − x1)C22

(16)

e=

as well as the symbols a, b, and c in eq 13, which are defined by the corresponding bracketed expressions in eq 12. The right-hand side of eq 12 can be written as the square of a first-degree polynomial P1(C) = eC + f (as shown in eq 14) if and only if the discriminant of the expression on the right-hand side of eq 13, given by Δ = b2 − 4ac, is equal to zero: 2 ⎡ 2C1C2C̅ ⎤ 0 = ⎢y(C1 + C2) + ⎥ ⎣ τ ⎦ 2 ⎡ C 2C 2 ⎤ C 2 ⎤⎡ y ⎥⎢ + yC1C2 + 1 2 ⎥ − 4⎢y + ⎢⎣ τ ⎥⎦⎣ 4 τ ⎦

f=

y + αy + β y + γ = 0

2e

(18)

Δ2 = (C1 + C2 + e)2 − 4C1C2 − 2yr − 4f

(30)

The real solution yr of eq 20 can be found using the method of Cardano.11 Accordingly, we define the real parameter q and write the discriminant ΔC as follows:

(22)

C(τ ) = −

α + 3

3

−

q + 2

ΔC +

3

−

q − 2

ΔC

C1 + C2 − e 1 + Δ1 2 2

(33)

EXPERIMENTAL SECTION Chemicals. The mobile phase used was a mixture of methanol and water (15/85 v/v); both were HPLC-grade solvents purchased from Fisher Scientific (Fair Lawn, NJ, USA). The samples were phenol and thiourea (>99%), both purchased from Sigma-Aldrich (Suwannee, MO, USA). Apparatus. The instrument used was a 1290 Infinity HPLC system (Agilent Technologies, Waldbroen, Germany). It included a 1290 Infinity binary pump with solvent selection valves and a programmable autosampler. The injection volume was drawn into one end of the 20 μL injection loop and injected into the mobile-phase stream following the FILO scheme. The reproducibility of the injection system was

(23)

⎞⎤ ⎟⎥ ⎟⎥ − ΔC ⎟⎠⎥⎦ q

−2 q2 4

(32)

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Second, for ΔC < 0, the exact real solution for y is directly given by ⎡ ⎛ 2 ⎢1 ⎜ q α − ΔC cos⎢ arccos⎜ yr = − + 2 6 3 4 ⎢3 ⎜ ⎝ ⎣

C1 + C2 + e 1 + Δ2 2 2

In summary, Δ1 and Δ2 are explicit functions of the variables yr and τ for fixed values of C1 = 1/b1, C2 = 1/b2, and x1 = k1/(k1 + k2). The variable yr is itself an explicit function of α, q, and ΔC, all three of which are expressed as functions of C1, C2, x1, and the reduced time τ. Therefore, this set of algebraic results provides the unique closed-form solution C(τ) = f−1(τ) to the general problem τ = f(C) set by eq 6.

There are two possible situations. First, for ΔC ≥ 0, the exact real solution for y is directly given by yr = −

(31)

Third, if both discriminants are non-negative with Δ2 > Δ1 ≥ 0, C is given by

(20)

⎡ α3 γ⎤ ΔC = γ ⎢ + ⎥ 4⎦ ⎣ 27

C1 + C2 − e 1 + Δ1 2 2

Second, for Δ2 ≥ 0 and Δ1 < 0, C is given by C(τ ) = −

(21)

(26) 2

(29)

C(τ ) = −

or

2 3 q=γ+ α 27

C2 τ

Δ1 = (C1 + C2 − e)2 − 4C1C2 − 2yr + 4f

(19)

y + αy + γ = 0

2 yr +

2C1C 2C̅ τ

There are only three possible situations, as the two discriminants cannot both be negative because the physical problem must have a real solution for C. First, for Δ1 ≥ 0 and Δ2 < 0, the unique exact real solution for C is directly given by

⎡ C2 ⎤ C 2C 2 − (C1 − C2)2 ⎥y 2 + 4 1 2 [ C 2 − C̅ 2] = 0 y3 + ⎢ ⎢⎣ τ ⎥⎦ τ

2

yr (C1 + C2) +

(25)

Two discriminants Δ1 and Δ2 are defined for the quadratic equations 27 and 28, respectively:

Each of the real roots yr of the cubic equation 18 (at least one exists) must be found and checked to see whether it satisfies the above condition. After development of the expression of the discriminant Δ, the coefficient β was found to be strictly equal to zero, and expressions for the coefficients α and γ were obtained, allowing eq 18 to be written as

3

=

C2 >0 τ

Finally, the equation P2(C) = P1(C) provides two quadratic equations: y C 2 + (C1 + C2 − e)C + C1C2 + r − f = 0 (27) 2 y C 2 + (C1 + C2 + e)C + C1C2 + r + f = 0 (28) 2

(17)

2

b(yr )

yr +

2

This condition translates into a third-degree polynomial in the real variable y: 3

a(yr ) =

(24)

Accordingly, the real coefficients e and f in the first-degree polynomial P1(C) are obtained by substituting the real solution 8554

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excellent as long as the sample volume exceeded 0.1 μL. The instrument was equipped with a two-compartment oven and a multidiode-array UV−vis detection system and was controlled using the Chemstation software. The sample trajectory in the equipment involved the successive passage of the injected bolus through a series of elements. First was a 20 μL injection loop attached to the injection needle. The design of the injection system was such that the volume of sample drawn into the loop was equal to the volume of sample injected into the column, ensuring excellent injection repeatability. Second was a smallvolume needle seat capillary (115 μm i.d., 100 mm long, volume ≈ 1.0 μL) located between the injection needle and the injection valve. The total volume of the grooves and connection ports in the valve was around 1.2 μL. Next was the column, preceded by an 85 μm × 22 cm inlet capillary tube (1.2 μL) and followed by a 130 μm × 25 cm long Viper outlet capillary tube (3.3 μL) offered by the manufacturer (Dionex, Germering, Germany). Finally, there was a small-volume detector cell (0.8 μL, 10 mm path). The extracolumn volume contribution of the instrument to the elution volumes of all recorded bands was 1.0 μL (needle seat capillary) + 1.2 μL (injection valve) + 1.2 μL (inlet capillary) + 3.3 μL (outlet capillary) + 0.8 μL (detection cell half-volume) = 7.5 μL. Columns. The 2.1 mm × 50 mm column packed with 2.7 μm Poroshell C18 particles was a generous gift from Agilent Technologies (Little Falls, DE, USA). The hold-up time of the column was measured to be t0 = 0.425 min on the basis of the elution time of thiourea eluted with the same mobile phase, and the hold-up volume was 85 μL. The Retention Time Method. The retention time method of isotherm determination presents several advantages over classical frontal analysis, the perturbation method, or the inverse method of chromatography.4 It is based on a few rapid and economical measurements of the retention times of the compound studied under linear conditions (k = FqSb) and requires only one set of overloaded conditions with a known mass injected into the column, without calibration curves. It is rapid and inexpensive. For the simple Langmuir isotherm, an analytical expression is known for C(τ) (eq 1), and its integration over time provides either the equilibrium constant (b) or the saturation capacity (qS) depending on whether the loading factor is expressed as a function of b or qS. The same approach applies for a bi-Langmuir isotherm. First, the elution time tR,0 is measured under linear conditions for an infinitesimally small loading factor (C0 → 0 and tp → 0). Accordingly, the experimental retention factor k′ is given by: k′ =

t R,0 − t0 − tex t0

= F(qS,1b1 + qS,2b2)

the mass balance equation, the elution times of these shocks are calculated from the analytical expression for the concentration C(τ) for a bi-Langmuir isotherm derived in this work. Minimization of the difference between the calculated and experimental elution times of the front shocks provides an estimate of the best adsorption isotherm parameters qS,1, qS,2, and b2, while b1 is given by eq 34. No calibration curve is needed; only one 1.5 mL sample in an HPLC vial is used, filled with a concentrated sample solution (this concentration C0 was close to the solubility of the compound in the eluent), and the UV wavelength is chosen so that the detection signal is saturated for the smallest loading factor. The increasing loading factors are generated by stepwise increases in the injection volume (Vp = Fvtp) of the prepared sample solution at concentration C0.

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RESULTS AND DISCUSSION The first part of this section reports on the calculation of the elution band profiles using the ideal model of chromatography assuming a bi-Langmuir adsorption isotherm. The second part illustrates how knowing the explicit expression of the elution time of the front shock of a series of overloaded band profiles permits the rapid and economical determination of the adsorption isotherm of compounds whose adsorption behavior is described by a two-site adsorption model (RPLC and chiral chromatography). Validation of the Unique Analytical Solution of the Ideal Model for a Bi-Langmuir Adsorption Isotherm. In this section we considered two different sets of bi-Langmuir adsorption isotherm parameters: (1) qS,1 = 100 g/L ≫ qS,2 = 1 g/L with b2 = 1 L/g ≫ b1 = 0.01 L/g and (2) qS,1 = 90 g/L > qS,2 = 10 g/L with b2 = 0.05 L/g > b1 = 0.01 L/g. The first situation is typical of the presence of a few very active sites in addition to the more abundant weak adsorption sites for the adsorption of ionizable compounds in RPLC.12 The second situation is typical of chiral chromatography with selective and nonselective adsorption sites.11 The dimensions of the column were 4.6 mm × 150 mm. The flow rate was set at 2.0 mL/min. The total porosity of the column was εt = 0.65. The injection was assumed to be rectangular with an injected concentration C0 = 20 g/L and an injection volume Vp = 100 μL. To validate the results of the analytical solution, we performed calculations of the band profiles using the equilibium-dispersive model of chromatography for the largest possible efficiency, N = 30 000 (beyond this value, the calculation could not be performed because the number of nodal points was too large). The method of calculation was the Rouchon method.4 Figure 1 compares the ideal and calculated chromatograms. The agreement is excellent because a large efficiency was deliberately chosen in the calculations. This validates the general analytical solution for the concentration as a function of the elution time. As expected, there are some small deviations between the two chromatograms due to the finite column efficiency in the calculations. Irrespective of the choice of the four isotherm parameters, the agreement between the ideal solution and the calculated chromatograms was always found to be excellent. This validates a posteriori the unique solution derived in this work for the ideal concentration profile expected for any bi-Langmuir adsorption isotherm. Application to the Rapid and Economical Determination of Bi-Langmuir Isotherm Parameters. Twenty overloaded band profiles of phenol were recorded by consecutively injecting 1, 2, ... 19, and 20 μL of a concentrated

(34)

where tex is the transit time through the extracolumn volume, including the volumes of the needle seat capillary, the groove of the injection valve, and the inlet and outlet capillaries plus half the volume of the detection cell. For a very high pressure LC instrument, this volume does not exceed 10 μL. Thus, knowledge of the analytical solution of the ideal model of chromatography for a bi-Langmuir isotherm model should allow the analyst to rapidly determine the four isotherm parameters. There are only three independent isotherm parameters (qS,1, qS,2, and b2). Therefore, at least three overloaded band profiles (hence three different elution times for the respective front shocks) are needed. By application of 8555

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Figure 2. Experimental overloaded band profiles of phenol on a 2.1 mm × 50 mm column packed with 2.7 μm Poroshell C18 core−shell particles from a 15/85 (v/v) methanol/water mixture. vHPLC system: 1290 Infinity system with an extracolumn volume of 7.5 μL. Flow rate: 0.2 mL/min. T = 295 K. Sample concentration: C0 = 150 g/L. The injection volumes increased stepwise (+1 μL) from 1 to 20 μL. The sharpness of the front shock of the bands should be noted.

Figure 1. Comparison between the ideal (black) and calculated (red) chromatograms on a 4.6 mm × 150 mm column (εt = 0.65). Flow rate: 2 mL/min. Rectangular injection profile: C0 = 20 g/L, Vp = 100 μL. The ideal profiles were obtained from the analytical solution of the ideal model of chromatography for a bi-Langmuir isotherm model using the values of the isotherm parameters (qS,1, b1, qS,2, and b2) indicated in the legends. The calculated profiles were obtained from the Rouchon method using the maximum efficiency N = 30 000.

solution of phenol (C0 = 150 g/L). A 5 min wash step (about 12 column hold-up volumes) with pure eluent was performed between each pair of successive injections in order to equilibrate the column with the neat mobile phase. A short, narrow-bore 2.1 mm × 50 mm column packed with 2.7 μm Poroshell 120 C18 core−shell particles was used with a flow rate of 0.20 mL/min. The eluent was a 15/85 (v/v) methanol/water mixture at ambient temperature (T = 295 K). Figure 2 shows the corresponding UV signal recorded at a wavelength of 220 nm. The detector was saturated for an absorbance of about 3200 mAU. The elution times of the shock were then estimated from the early uptake of the concentration in the shock layer. In practice, the time width of the front shocks was so thin when the sample volume was injected from the analytical injector (virtually no precolumn sample dispersion) that it was even smaller than the injection-to-injection repeatability of the elution times of the front shock. Precise measurements of the elution times of all front shocks were then made with an RSD smaller than 0.5%. Figure 3 shows the reduced experimental front-shock elution times (τshock) of these 20 overloaded band profiles [from the definition of τ (eq 5), τ = 1 under the linear condition (Lf = 0)]. In view of the level of precision of these experimental data, the sum of the 20 relative residual squares between the

Figure 3. Plot of the experimental (black ★) and best theoretical (red ○) reduced elution times of the front shock for the 20 overloaded band profiles of phenol shown in Figure 2. The best theoretical values were calculated from the analytical solution of the ideal model of chromatography for a bi-Langmuir isotherm model after minimization of the relative residual squares. The best isotherm parameter values and the Fisher test number are given in the legend.

theoretical and experimental elution times of the front shocks were low, being smaller than 1 × 10−3 for the following parameter intervals: 23.40 L/g < qS,2 < 29.80 g/L, 0.300 L/g < b2 < 0.340 L/g, 51.7 g/L < qS,1 < 57.5 g/L, and 0.0210 L/g < b1 < 0.0253 L/g. Beyond these intervals, either the quality of the agreement decreased rapidly or the best parameters made no physical sense, being negative or excessively large. The best agreement between the experimental (black ★) and the theoretical (red ○) elution times of the front shocks obtained using the above set of average isotherm parameter values is shown in Figure 3. These optimized isotherm parameters are consistent with those previously derived from adsorption data of phenol measured by the FA method using standard HPLC equipment with a series of 4.6 mm i.d. commercial columns.12,16 The ratio of the amount of weak adsorption sites to that of strong adsorption sites was typically on the order 8556

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of 2, and the difference in their adsorption energies, RT ln(b2/ b1), was close to 5 kJ/mol. This validates the results found in this work, which used the simpler RTM method with a very high pressure LC (vHPLC) instrument and column technologies. With this set of isotherm parameter values and the analytical solution of the ideal model of chromatography for a biLangmuir isotherm, 20 ideal chromatograms were calculated for injection of volumes of a phenol solution at 150 g/L increasing from Vp = 1 to 20 μL with stepwise increases of 1 μL. The elution times of their shocks are plotted versus the sample size in Figure 4. The agreement between the calculated and

conventional 400 bar HPLC technologies. The main advantage of the RTM is that it requires no calibration curve (because the width of the front shock is very narrow) and a small amount of sample during the experiments. In the RTM experiments reported earlier, only two 1.5 mL sample vials were used: one for the injection of a small sample and the determination of the retention factor k′ = 11.1 (with an infinitely diluted solution) and the other for the injection of 20 different volumes from Vp = 1 to 20 μL of a highly concentrated solution of phenol (C0 = 150 g/L). The total mass of sample injected into the Poroshell 120 C18 column was [(20 × 21)/2] × 10−3 cm3 × 0.15 g/cm3 = 31.5 mg. The volume of eluent consumed was 21 × 10 min × 0.2 mL/min = 42 mL. The time necessary to record all 21 retention data points (one linear injection + 20 shock elution times) was then 42 mL × 5 min/ mL = 210 min or 3.5 h. Overall, all the necessary data were recorded in about 4 h with a vHPLC instrument (1290 Infinity system) and a short, narrow-bore column. In contrast, in the classical FA method previously used for the determination of the adsorption isotherm of phenol using a HP1090 chromatograph with a standard 4.6 mm × 150 mm RPLC columns (equivalent hold-up volume V0 = 1.22 cm3).12,16,18 The flow rate was set at 1 mL/min. For the same retention factor (k′ = 11.1), the time width of the injected concentration plugs was 8 min (about half the elution time under linear conditions). Therefore, the injection of 20 different sample concentrations from C = 7.5 to 150 g/L (stepwise 7.5 g/L increase in concentration from one to the next injected plug) would require an amount of phenol of [(20 × 21)/2] × 7.5 × 10−3 g/cm3 × 8 min × 1 cm3/min = 12.6 g. The elution time of each breakthrough curve would require 1.22 cm3 × [12.1 + 12 (washing step)] × 1 min/cm3 + 8 min (injection plug) = 37.4 min. Therefore, the total time required to acquire the data would be 20 × 33.7 min = 748 min or 12.5 h, making this an overnight experiment. The total volume of eluent consumed would then be about 750 mL. In conclusion, when short, narrow-bore columns, a vHPLC instrument with minimum extracolumn volume, the RTM, and the solution of the ideal model of chromatography for the biLangmuir isotherm model are used, the experimental determination of the best adsorption isotherm parameters requires about 4 times less time, a 20 times smaller eluent volume, and a 400 times smaller mass of compound than a classical HPLC instrument and columns using the FA method.

Figure 4. Ideal chromatograms of phenol on a 2.1 mm × 50 mm column packed with 2.7 μm Poroshell C18 core−shell particles from a 15/85 (v/v) methanol/water mixture. These ideal profiles were obtained from the analytical solution of the ideal model of chromatography for a bi-Langmuir isotherm model using the best values of the isotherm parameters (qS,1, b1, qS,2, and b2) given in Figure 3. vHPLC system: 1290 Infinity system. Flow rate: 0.2 mL/min. T = 295 K. Sample concentration: C0 = 150 g/L. The injection volumes increased stepwise (+1 μL) from 1 to 20 μL.

theoretical values is similar to that illustrated in Figure 2. Because of (1) the excellent agreement between the experimental and the calculated elution times of the 20 shocks (see Figures 3 and 4) and (2) the small sample size increment (1 μL; see Figure 4), this also ensures an excellent agreement between the tail of the experimental and calculated overloaded band profiles, provided that the column efficiency is large enough. Therefore, it was not necessary to compare the experimental and recalibrated overloaded band profiles in this work for isotherm validation (i.e., the principle of inverse methods of extracting the best isotherm parameters from the best agreement between experimental and calculated data). The only uncertainty concerned the precision of each adsorption isotherm parameter from the fitting procedure used in any inverse method (see the results in the previous paragraph). The agreement between the overall adsorption isotherms determined by the direct FA method and by the inverse RTM method was shown to be good and even excellent for highly efficient columns.4,17 This was the case in this work, which used a column packed with sub-3 μm core−shell particles mounted on a low-dispersive UPLC instrument. Time, Eluent, and Sample Consumption during the Retention Time Method Using UPLC Technologies. The goal of this last section is to illustrate the advantage of using the retention time method with vHPLC technologies rather than the classical frontal analysis method previously used with

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CONCLUSION In this work, the analytical solution of the ideal model of chromatography (assuming no sample dispersion or infinite column efficiency) was derived by algebraic solution of a quartic polynomial equation for its unique, real, and positive root. The mathematical expression of this root is much more complex than that of the unique root of the simple homogeneous Langmuir model. However, this analytical solution can conveniently provide the exact ideal chromatograms for any bi-Langmuir adsorption isotherm without resorting to numerical solutions. This particular analytical solution was used to apply the retention time method and predict the elution times of the front shocks of a series of 20 overloaded band profiles of phenol using a short, narrow-bore column packed with 2.7 μm core−shell particles. The minimization of the difference between the predicted and experimental retention times of the shock allowed a rapid and economical solution to the 8557

dx.doi.org/10.1021/ac4015897 | Anal. Chem. 2013, 85, 8552−8558

Analytical Chemistry

Article

determination of the accurate and precise adsorption isotherm parameters of phenol onto the column used. The reductions in the working time, the eluent volume consumed, and the mass of compound needed are estimated to be factors of 4, 20, and 400, respectively, relative to those required to achieve the same determination using the classical FA method, standard HPLC equipment, and 4.6 mm i.d. analytical columns. The next step of this work will be to extend this approach to the determination of adsorption isotherms in solid/supercritical fluid systems and to apply the results to optimize the chromatographic separation and purification of enantiomeric compounds. More generally, the RTM could be used as an effective method for the determination of the adsorption isotherms of compounds following bi-Langmuir adsorption behavior. For compounds with a limited solubility in the mobile phase, the solution of the ideal model of chromatography for a bi-Langmuir isotherm can directly provide the largest injection volume required to maximize the column loading factor before the column is fully equilibrated.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: (865)974-2667. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the cooperative agreement between the University of Tennessee and Oak Ridge National Laboratory. We thank Ron Majors from Agilent Technologies (Little Falls, DE, USA) for the generous gift of the Poroshell 120 C18 column used in this work.

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