Anal. Chem. 1996, 68, 2782-2789
Kinetic Model of Membrane Extraction with a Sorbent Interface Min J. Yang, Marc Adams, and Janusz Pawliszyn*
The Guelph-Waterloo Center for Graduate Work in Chemistry and the Waterloo Center for Groundwater Research, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Membrane extraction with a sorbent interface (MESI) is an unique sample preparation alternative for trace organic analysis. The main features of MESI include its solventfree nature, the rugged and simple design with no moving parts for long-term reliable performance, the fact that it is a single-step process which ensures good precision, its easy automation, and its feasibility for on-site operation. Among the available membrane extraction modules designed for the MESI system, the headspace configuration has continued to show its superior durability and versatility in membrane applications. The headspace membrane extraction configuration effectively eliminates the need for a sampling pump and flow metering and hence prevents the extraction system from plugging and greatly simplifies the extraction process. The module can be used for extraction of VOCs from gaseous, aqueous, or solid samples. A mathematical model has been developed for headspace membrane extraction of an aqueous sample, based on the assumption that the aqueous phase is perfectly stirred. The model is in good agreement with the experimental benzene extraction results obtained with an efficient agitation method such as high-speed magnetic stirring or sonication. The model has also been used to study the effects of various extraction parameters with respect to the sensitivity and response time of the MESI system. Sample agitation facilities analyte mass transport and hence improves both the system sensitivity and the response time. The sensitivity of the extraction method also increases with an increase of the extraction temperature. A large portion of environmental contamination problems, especially in water, are attributed to volatile organic compounds (VOCs). Many of the VOCs, such as chlorinated solvents, are known to be human carcinogens. Monitoring for the presence of VOCs in the environment is necessary in order to protect the public against chronic exposure to these contaminants. Membrane extraction with a sorbent interface (MESI) is a recently developed solvent-free technique for the analysis of VOCs.1,2 The MESI method offers many advantages, including simplicity, durability, elimination of solvents, and easy automation. Extraction of VOCs using a hollow fiber membrane can be performed either directly in the sample or in its headspace. The choice of membrane module and its extraction efficiency depend on the application and the sample matrix. The flow-through (1) Yang, M. J.; Harms, S.; Luo, Y. Z.; Pawliszyn, J. Anal. Chem. 1994, 66, 1339-1346. (2) Yang, M. J.; Luo, Y. Z.; Pawliszyn, J. CHEMTECH 1994, 24, 31-37.
2782 Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
configuration has been used by other researchers and during the early stages of the MESI method development.3-6 Mathematical models for the flow-through module based on the diffusion theories have previously been created to study the properties of this extraction configuration.7,8 Placing the membrane in contact with the sample works well for gaseous samples and relatively clean water samples but is less effective for solid matrices or wastewater samples, which contain high molecular weight nonvolatiles. Alternatively, VOCs can be effectively extracted from the headspace of a sample.9-13 The advantages of the headspace membrane method include its superior selectivity, its durability, its versatility, and the easy attainment of analytes from the matrix. The headspace membrane extraction module is the simplest and least problematic configuration, because it eliminates the need for a sampling pump and flow monitoring. The headspace method also prevents the extraction system from plugging by placing the membrane above the sample without physical contact of the sample. The headspace membrane module can be made rugged and suitable for long-term field monitoring applications. The main objective of this paper is to develop and apply a mathematical model to optimize the membrane extraction parameters and determine the properties of the MESI system using a headspace membrane module for the analysis of VOCs from water. The mathematical model allows the user to theoretically determine the analyte concentration at any given position and time during the extraction process. The effects of agitation and temperature have been experimentally investigated for the extraction method. THEORY Extraction Geometry. The experimental setup for the headspace hollow fiber membrane extraction is illustrated in Figure 1a. An aqueous sample contaminated with VOCs is transferred to a glass vial, and a hollow fiber membrane probe is placed into the headspace of the vial. Analytes then undergo a series of transport processes: first they move from the water into the gas (3) Melcher, R. G.; Morabito, P. L. Anal. Chem. 1990, 62, 2183-2188. (4) Melcher, R. G.; Bakke, D. W.; Hughes, G. H. Anal. Chem. 1992, 64, 22582262. (5) Pratt, K. F.; Pawliszyn, J. Anal. Chem. 1992, 64, 2107-2110. (6) Yang, M. J.; Pawliszyn, J. Anal. Chem. 1993, 65, 1758-1763. (7) Pratt, K. F.; Pawliszyn, J. Anal. Chem. 1992, 64, 2101-2106. (8) Yang, M. J.; Pawliszyn, J. Anal. Chem. 1993, 65, 2538-2541. (9) Charalambous, G. Analysis of Food and Beverages, Headspace Technique; Academic Press: New York, 1978. (10) Kolb, B. Applied Headspace Gas Chromatography; Heyden: London, 1980. (11) Poole, C. F.; Schuette, S. A. J. High Resolut. Chromatogr. 1983, 6, 526549. (12) Ioffe, B. V.; Vitenberg, A. G. Headspace Analysis and Related Methods in Gas Chromatography; Wiley: New York, 1984. (13) Zhang, Z.; Pawliszyn, J. Anal. Chem. 1993, 65, 1843-1852. S0003-2700(95)01175-9 CCC: $12.00
© 1996 American Chemical Society
At x ) L, C3 ) KmC2 at any time by definition of partition coefficient, and
∂C3(x,t) ∂C2(x,t) FDm ) Dh ∂x ∂x
(3)
At x ) 0, C2 ) KhC1 at any time by definition of partition coefficient, and
∂C2(x,t) ∂C1(x,t) Dh ) Ds ∂x ∂x
Figure 1. Headspace hollow fiber membrane extraction module for the analysis of volatile organic compounds from water.
phase, and then they eventually penetrate the membrane wall. The diffusion process occurs not only in the axial direction but also in the radial direction. To obtain a reasonable solution to this diffusion problem, the extraction system can be described by a simple one-dimensional diffusion model, as shown in Figure 1b. In the simplified model, diffusion occurs only in the x direction. The diffusion process can be modeled as the mass transfer of an analyte through three phases. Km and Kh are the membrane/ gas and gas/water partition coefficients, respectively; C1, C2, and C3 are the concentrations of the analyte in water, the headspace, and the membrane, respectively; Dm, Dh, and Ds are the diffusion coefficients of the analyte in the membrane, the headspace, and water, respectively; and M, L, and S are the thicknesses of the membrane, the headspace, and the aqueous phase, respectively. The membrane is treated as a flat membrane at one end of the vial. The vial is symmetric about its axis, and it is assumed that there is no adsorption or absorption of the analyte to the vial surface or the vial wall. The aqueous phase is assumed to be perfectly stirred, and both the membrane and the headspace are held stationary. The diffusion process in the membrane and headspace is governed by Fick’s second law:
∂2C(x,t) ∂C(x,t) )D ∂t ∂x2
FDm∂C3(x,t) 2R ∂x
Equation 3 and 4 are based on Fick’s first law, and the flux of the analyte must continuously flow through the boundaries. The following initial states (at t ) 0) are also considered in the mathematical model: the concentration of the analyte in the membrane is zero, C3(x,0) ) 0; the concentration of the analyte in the headspace is zero, C2(x,0) ) 0; and the initial analyte concentration in the aqueous solution before it is transferred to the sample vial is C0. The follow relation can be derived from mass balance:
C0S )
∫
M+L
L
C3(x,t) dx +
∫ C (x,t) dx + C (t)S + L
2
0
1
total amount extracted (5)
The diffusion problem for headspace membrane extraction can be solved in two ways. Many have chosen the finite difference method13-15 to obtain numerical solutions and avoid lengthy algebraic derivation. An analytical solution can be derived from the membrane system on the basis of the Laplace transformation method.16 The derivation steps are listed in the Appendix. The analyte concentration in the headspace at position x and time t can be calculated from the following equation:
(1) C2(x,t) )
where C(x,t) is the concentration of the analyte at position x and time t. Modeling of the Diffusion Process. There are a few steps in mathematical modeling. First, the boundary conditions and initial states must be defined for the extraction system. A solution to eq 1 is then derived to describe the diffusion process during the extraction. Finally, a computer program is developed to obtain graphical representations of the mathematical model. The following boundary conditions can be applied to the headspace membrane extraction system. At x ) M + L, the membrane inner wall is swept by a carrier gas, and the analyte concentration on the inner wall of the membrane can be estimated by Km multiplied by the average analyte concentration in the carrier gas,
C3(t) ) -Km
(4)
(2)
where F is the surface area of the membrane divided by the crosssectional area of the vial, and R is the flow rate of the carrier gas.
∑e zi
2R KmxDh
∂
[
2FRxDm Dh
z Λ(z) ∂z
cos(zL) cos(zm) -
sin(zL) sin(zm) - FzxDm sin(zL) cos(zm) -
DmKmF2z
xDh
1
-Dhz2t
2KhC0Dh
]
cos(zL) sin(zm) [Sz sin(xz) - Kh cos(xz)] (6)
The analyte concentration in the membrane at position x and time t can be calculated from the following equation: (14) Riggs, J. B. An Introduction to Numerical Methods for Chemical Engineers; Texas Tech University Press: Lubbock, TX, 1988. (15) Crank, J. The Mathematical of Diffusion, 2nd ed.; Clarendon Press: Oxford, U.K., 1989. (16) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Clarendon Press: Oxford, U.K., 1986; Sect. 12.8.
Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
2783
∑
C3(x,t) ) 2SKhC0
zi
(
{2R sin[z(x - L - M)xDh/Dm] -
/
zKmFDmxDh/Dm cos[z(x - L - M)xDh/Dm]} ∂
)
(8)
Λ(z) ) FxDm(2RKh - SDhz2) cos(zm) cos(zL) -
) )
2RKh - KmF2SDmz2 sin(zm) sin(zL) Km
(
2SR + KmKhF2Dm sin(zm) cos(zL) Km zFxDm(2RS + KhDh) cos(zm) sin(zL) (9)
∂ Λ(z) ) [2F2SK2mDmzxDh sin(zm) sin(zL) ∂z 2mRKhxDh cos(zm) sin(zL) + 2F2SK2mDmz2xDh × cos(zm) sin(zL) - 2LRKhxDh sin(zm) cos(zL) + LF2SK2mDmz2xDh sin(zm) cos(zL) - 2FKmRSxDm × cos(zm) sin(zL) - FKmKhDhxDm cos(zm) sin(zL) + 2FzmKmRSxDm sin(zm) sin(zL) + FzmKmKhDhxDm × sin(zm) sin(zL) - 2FzLKmRSxDm cos(zm) cos(zL) FzLKmKhDhxDm cos(zm) cos(zL) - 2FzKmSDhxDm × cos(zm) cos(zL) - 2FmKmKhRxDm sin(zm) cos(zL) + FmKmSDhz2xDm sin(zm) cos(zL) - 2FLKmKhRxDm × cos(zm) sin(zL) + FLKmSDhz2xDm cos(zm) sin(zL) 2RSxDh sin(zm) cos(zL) - K2mKhF2DmxDh sin(zm) × cos(zL) - 2zmRSxDh cos(zm) cos(zL) zmK2mKhF2DmxDh cos(zm) cos(zL) + 2zLRSxDh × sin(zm) sin(zL) + zLK2mKhF2DmxDh sin(zm) sin(zL)]/Km (10)
The total amount extracted, T, at time, t, can be calculated as a flux of the analyte at the position M + L integrated over time,
∑
T(t) ) 4RFSKhC0xDm
zi
1
2
(e-Dhz t - 1)
(11)
∂
z Λ(z) ∂z
The amount extracted, E, at time, t, can be calculated by taking the derivative of eq 11 with respect to time, 2784
Λ(z)
2
m ) MxDh/Dm
zxDh
(12)
∂z
In the above equations, m is represented by eq 8; the z1, z2, z3, ..., zi are the nonnegative roots of Λ(z); the function Λ(z) is expressed by eq 9; and the partial derivative of Λ(z) is given in eq 10.
(
zi
2
e-Dhz t
Λ(z) e-Dhz t (7)
∂z
xDh
Dhz
∑∂
E(t) ) -4RFSKhC0xDm
Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
To obtain useful information from the algebraic expressions of the mathematical model, graphical representations for the equations are needed. A computer program was developed by using Microsoft Quick C for Windows. The program is provided with a set of parameters from the user so that plots of analyte concentration profiles and overall extraction profiles can be made with respect to time and space. To apply the model for benzene extraction, it was assumed that a benzene solution with a depth (S) of 2 cm and a concentration (C0) of 1000 (in arbitrary units) was extracted by the headspace membrane module shown in Figure 1. The thickness for the headspace (L) was 3 cm, and the thickness of the membrane (M) was 0.0165 cm. For a 4-cm-long membrane with an o.d. of 0.0635 cm, the area ratio (F) between the membrane surface and the cross section of the vial was 0.139. The carrier gas flow rate (R) was set at 0.0333 mL/s, or 2 mL/ min. The diffusion coefficients (D) and partition coefficients (K) for the benzene, toluene, ethylbenzene, and xylene (BTEX) compounds13 are listed in Table 1. The time required for exhaustive extraction can be predicted by plotting the total amount extracted vs time according to eq 11. The theoretical extraction-time profile can be obtained by plotting the amount extracted vs time according to eq 12. This theoretical time profile can be verified experimentally. EXPERIMENTAL SECTION Apparatus and Reagents. The major components of the MESI system (Figure 2) include a membrane extraction module, a sorbent interface, an instrument such as a gas chromatograph (GC), and a microcomputer. The hollow silicone fiber membrane (Baxter Healthcare Corp., McGaw Park, IL) had an inner diameter of 0.305 mm and a wall thickness of 0.165 mm. The first step for construction of an extraction module was to connect a piece of 0.53-mm-i.d. uncoated fused silica tubing (Supelco Canada Ltd., Mississauga, ON, Canada) at each end of the membrane to facilitate the transfer of the stripping gas. The technique for connecting the membrane to the capillary tubing has been previously described.1,2 One end of the silicone fiber was first submerged in toluene for 10 s, and after the fiber had swollen, 1 cm of the capillary tubing was inserted into the hollow fiber. When the toluene had evaporated, the membrane shrank, and a tight seal was formed between the membrane and the tubing at the junction. A 40-mL glass vial was used as sample container. The septum on the vial cap was pierced by the capillary tubing, which was attached to the membrane, and provided a seal for the extraction module. The active length of membrane in the module was 4 cm. The sorbent interface, which provided a direct connection between the extraction module and the GC, consisted of a sorbent tube, a heating coil, and an automated heater switch. In this study, the sorbent tube consisted of an 1-cm fused silica fiber coated with 100 µm of poly(dimethylsiloxane), which was immobilized within a 0.53-mm-i.d. deactivated fused silica tubing. A piece of Ni-Cr wire, 0.1 mm in diameter, 55 Ω total resistance (Johnson
Table 1. Diffusion Parameters for BTEX Compounds at Room Temperature and Pressure compounds
Km
Kh
Dm (cm2/s)
Dh (cm2/s)
benzene toluene ethylbenzene m-xylene o-xylene p-xylene
493 1332 3266 3507 4417 3507
0.26 0.26 0.16 0.24 0.15 0.24
2.8 × 10-6
0.077 0.0709 0.0658 0.059 0.062 0.056
Figure 2. Experimental setup of a MESI system using the headspace membrane module.
Matthey Metals, Ltd.), was used as the heating coil for thermal desorption. A three-stage solid-state cooler was used to maintain the trapping temperature at -40 °C. The cooler was operated with a 12-V dc source. A “black box”, also referred to as the MESI box, included a variable dc power supply and the automated heater switch. A Varian 3500 GC (Varian Canada Inc., Georgetown, ON, Canada) equipped with a flame ionization detector (FID) and a modified on-column injector was used for analyte separation and detection. The separation column was removed from the GC. The outlet of the sorbent interface was directly connected to the FID with a deactivated fused silica tubing with an i.d. of 0.2 mm (Polymicro Technologies, Phoenix, AZ). The temperature for both the GC oven and the injector was set at 50 °C. The detector temperature was set at 250 °C. Ultrahigh-purity grade hydrogen was used as the carrier gas, at a flow rate of 2.0 mL/min. For process automation, a desktop computer was used to control the overall trapping process on the MESI system. A Microsoft Windows-based MESI program was developed using Microsoft Quick C to provide desorption commands to the MESI box and to collect FID signals. Like most of the other Windows applications, the MESI program offered many features, including drop-down menus, dialog boxes, scrollbars, virtual control buttons, and data exchange between applications through cut and paste. The user directly interacted with the program through a graphical display, a keyboard, and a mouse. A DAS1401 data acquisition board (Keithley Data Acquisition, Taunton, MA) was used. Transistor-transistor logic (TTL) signals were sent to the MESI box to trigger thermal desorption at the sorbent interface via a digital output channel on the data acquisition board, while the
FID signals were collected and saved via an analog-to-digital converter on the data acquisition board. Deionized water was used for all aqueous sample preparations. Procedure. A 10-mL water sample containing 5 µg of benzene was placed in the 40-mL extraction module. The exterior of the membrane was exposed to the headspace of the sample. The carrier gas flow through the lumen of the hollow fiber membrane and transported the permeated analyte into the sorbent interface, which connected the module to the GC. The water sample was agitated with either a magnetic bar at a constant rate or sonicated to facilitate the mass transport process and improve extraction efficiency. A fresh sample was used for each extraction experiment. The MESI program was loaded into the computer memory to control the thermal desorption and detector data collection during the extraction. A trapping experiment was started by selecting a menu option in the MESI program. During the experiment, the MESI program generated a binary sequence corresponding to the predefined desorption time and intervals. The binary digits in the sequence were sequentially sent to the MESI box in the form of TTL signals via a digital channel on the DAS 1401 data acquisition board at a fixed frequency of 2 Hz. Upon receiving a positive TTL pulse from the computer, the MESI box transformed a regular ac power source to dc at a predetermined level, typically 14 V, and delivered it to the heating coil. When electric current passed through the coil, heat was generated, and analytes were desorbed at the sorbent interface. When the electrical power was turned off, the temperature of the sorbent tube dropped rapidly (within 5 s) to the ambient level, and the interface began to trap the analytes for the next desorption pulse. The desorbed analytes formed a sample concentration pulse, which was eventually measured with the FID. The pulse interval was set at 40 s. The desorption pulse width was 2 s. The experimental headspace membrane extraction profiles of benzene were obtained by applying 30 trapping-desorbing cycles. Each cycle produced one benzene peak. RESULTS AND DISCUSSION Theoretical Predictions. The mathematical model can be used to draw theoretical predictions and can help us to understand and optimize the headspace membrane extraction system. All theoretical studies were based on benzene extraction, because a complete set of the diffusion parameters is available only for benzene, as shown in Table 1. Diffusion coefficients in the membrane, Dm, for other BTEX compounds were not found in the literature. Once they become available, the model can also be applied for these compounds. All predictions made are based on a well-agitated extraction system. The extracted amount vs time or the extraction-time profile can be plotted by using eq 12. Figure 4 illustrates theoretical extraction-time profiles obtained with different membrane dimensions. The maximum in an extraction-time profile indicates the time (tmax) required for the flux of benzene exiting the extraction system to reach its top level (Emax). The tmax is directly related to the time required for the analyte concentration to reach a maximum in the headspace and is affected by the analyte diffusion rates through the three phases in the extraction system. It can be used as an indicator for the comparison of two extraction-time profiles. It also provides an indication of system response time toward a change of the analyte concentration in the aqueous phase. A shorter tmax means a quicker system response time. The Emax, Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
2785
a
b
Figure 3. Theoretical extraction-time profiles for the benzene extraction using the headspace membrane extraction system calculated with (a) different membrane thickness and (b) different membrane lengths.
on the other hand, provides an indication of system sensitivity. A higher Emax means a greater sensitivity. According to the model, it requires 68 min to extract 95% of benzene using a membrane with a thickness of 0.0165 cm. If the thickness of the membrane is halved, the time required for 95% benzene removal would be reduced to 47 min. If the thickness of the membrane is doubled, the time required for 95% benzene removal would increase to 115 min. Similarly, the tmax in the theoretical extraction-time profile increases as the membrane thickness increases (Figure 3a). For membrane thickness of 0.00825, 0.0165, and 0.033 cm, the tmax values are 130, 190, and 340 s, respectively. The Emax decreases as the membrane thickness increases. As a result, the extraction system has a higher sensitivity and faster response time with a thinner membrane. Experimentally, the membrane thickness can be reduced by stretching the membrane. The change in the membrane length will directly affect the area ratio between the membrane surface and the vial cross section. Figure 3b illustrates three theoretical extraction-time profiles obtained with different lengths of membrane. The time required for 95% benzene removal increases as the length of membrane decreases. For membrane lengths of 8, 4, and 2 cm, the times required for 95% benzene removal are 50, 68, and 100 min, respectively. The tmax values remain unchanged in this case. The Emax values increases with the membrane length. In other words, an increase in the length of membrane leads to an increase in system sensitivity, but it has no influence on the system response time. Using the setting given in the previous section, it requires 68 min to extract 95% of benzene using a headspace membrane 2786
Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
module with a headspace thickness of 3 cm. If the thickness of the headspace is halved, the time required for 95% benzene removal would be reduced to 52 min. If the thickness of the headspace is doubled, the time required for 95% benzene removal would increase to 100 min. The tmax in the theoretical extractiontime profile also increases as the thickness of the headspace increases. For headspace thickness of 1.5, 3, and 6 cm, the tmax values are 110, 190, and 410 s, respectively. The Emax decreases as the headspace thickness increases. Similar to the effect of reduction in membrane thickness, the extraction system has a greater sensitivity and faster response time with a thinner headspace. Experimentally, the headspace thickness can be reduced by lowering the membrane toward the sample. In the current model, the aqueous sample is assumed to be perfectly stirred at all times. The analyte concentration is uniform throughout the aqueous phase. It requires 68 min to extract 95% of benzene using a headspace membrane module with a sample thickness of 2 cm. If the thickness of the aqueous sample were halved, the time required for 95% benzene removal would be reduced to 47 min. If the thickness of the aqueous sample is doubled, the time required for 95% benzene removal would increase to 128 min. Both the tmax and Emax in the theoretical extraction-time profile also increase as the thickness of the sample increases. For sample thicknesses of 1, 2, and 4 cm, the tmax values are 160, 190, and 220 s, respectively. The Emax decreases as the sample thickness increases. In general, the extraction system has a greater sensitivity and faster response time with a thinner aqueous phase. The improvement in response time by reducing the sample volume, however, is relatively small. The change of sample concentration has no effect on exhaustive extraction time and system response time. The system requires 68 min to remove 95% of benzene, regardless of the sample concentration. The tmax values also remain unchanged with respect to changing concentration. The change in the carrier gas flow rate will affect the average analyte concentration in the carrier gas and hence will affect the analyte concentration at the inner wall of the membrane, as described by eq 2. The time required for 95% benzene removal increases as the carrier gas flow rate decreases. For the carrier gas flow rates of 4, 2, and 1 mL/s, the times required for 95% benzene removal are 65, 68, and 70 min, respectively. An increase in the carrier gas flow leads to a slight increase in system sensitivity and has no influence on the system response time. The relatively small effect of the carrier gas flow rate on the overall extraction kinetics is expected because of the high linear gas flow rate. The volumetric flow rate of 1-4 mL/min can be translated into 23-92 cm/s in linear flow rate in this extraction system. If the length of the membrane is 4 cm, the time required for the gas to pass through the membrane is in the order of 0.1 s. A similar trend has been observed experimentally by comparing the extraction-time profiles obtained under different carrier gas flow rates. Effects of Agitation and Temperature. Based on the theory, the dynamics of mass transport in the headspace membrane extraction process is controlled by the diffusion coefficient of the analyte in the aqueous phase. For a rapid extraction, the solution must be well agitated. Under the well-agitated conditions, the dynamics of mass transport is limited only by the diffusion of analyte through the membrane. Figure 4 shows the benzene extraction-time profiles obtained under different agitation condi-
Figure 4. Headspace hollow fiber membrane extraction-time profiles for a 10-mL aqueous sample containing 5 µg of benzene, acquired by using the GC/FID system in the trapping mode under different agitation conditions. The solid curves represent the theoretical prediction for the extraction under perfect stirring.
tions at room temperature. The extraction systems with 300rpm stirring and 1800-rpm stirring had tmax values of 9 and 3 min, respectively. The theory predicts a tmax of 3 min, which is in reasonable agreement with the extraction-time profile obtained with 1800-rpm stirring (Figure 4b). The pattern of the experimental results shown in Figure 4a did not match the theoretical profile, because the mathematical model assumed that the aqueous phase is perfectly stirred. The experimental results clear show that the extraction-time profiles are dramatically affected by the stirring rate. The system sensitivity also increases with the rate of stirring, because sample agitation facilitates the mass transport of the analyte. Less conventional agitation methods were also considered during this investigation. Figure 4c,d illustrates the effects of sonication. The use of an ultrasonic glassware cleaner or sonic bath (Crest Ultrasonics Corp., Trenton, NJ) improved the mass transport of the analyte but was not as efficient as stirring at the rate of 1800 rpm. Experimentally, the extraction vial was inserted at the center of the water bath of the glassware cleaner. The water bath was temperature controlled. The ultrasound was turned on for the entire extraction period. The extraction-time profile obtained with the sonic bath (Figure 4c) is similar to the 300RPM stirring case (Figure 4a). It was expected that sonication generated by using a high-power coupling horn or sonic disruptor would improve the efficiency of sample agitation. In the experiment, the sonic disruptor was inserted on one side of the water bath of the glassware cleaner. The ultrasound generated by the disruptor was applied during the entire extraction period. When
the two sonication methods were combined, the extraction efficiency improved dramatically. The theoretical extraction-time profile represented by a curve is superimposed on the experimental result. Figure 4d shows the good agreement between the experimental extraction-time profile and the theoretical model, which assumes perfect sample agitation in the aqueous phase. The ultrasonic power level cannot be adjusted for the sonic bath, but it can be regulated with the output control on the disruptor. The extraction profiles shown in Figure 4d were obtained with the disruptor set at 100% power. The percent power corresponds to the position on the dial of the disruptor output control. Changing the disruptor power from 50% to 100% has little effect on the extraction-time profile. When the power level was reduced below 50%, the tmax in the extraction-time profile was also decreased, but the Emax remained almost the same. As a result, the optimal power level for the disruptor is 50%. In general, diffusion coefficients (D) increase with temperature, and analytes in a hot sample diffuse more quickly than those in a cool sample for a given concentration gradient. For diffusion in a fixed-volume gaseous phase, D is proportional to the square root of the temperature.17 To test the temperature effect, the extraction vial was inserted in a temperature-controlled water bath. The magnetic stirrer was placed directly under the bath. The extraction-time profile obtained at 46 °C under the two magnetic stirring conditions showed an increase in amount extracted of up to 30% compared to the ones obtained at room temperature. In (17) Atkins, P. W. Physical Chemistry, 4th ed.; W. H. Freeman and Co.: New York, 1990.
Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
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other words, the sensitivity of the headspace membrane extraction system increases with the extraction temperature. However, the tmax remains almost the same for different extraction temperatures. Clearly, the increase in extraction temperature increases the diffusion rates of the analyte in all three phases of the system, and the time required for the analyte concentration to reach a maximum in the headspace is not affected. CONCLUSIONS A mathematical model has been developed for headspace membrane extraction of an aqueous sample, based on an assumption that the aqueous phase is perfectly stirred. The model is in good agreement with the experimental results obtained with efficient agitation methods such as high-speed magnetic stirring and combined sonication. The model has been used to predict the effects of various extraction parameters within a well-agitated system. Improvements in extraction system response time and system sensitivity are expected when the membrane thickness decreases, the membrane length increases, the headspace thickness decreases, or the aqueous sample thickness decreases. Although this paper has demonstrated only the application of the mathematical model for headspace benzene extraction with a silicone membrane, the analytical solution to the diffusion problem can also be used to describe extraction of other analytes with other membrane material by choosing the appropriate diffusion coefficients and partition constants. Based on the experimental results, application of agitation significantly facilitates the mass transport of the analyte and improves both the system response time and the system sensitivity for headspace membrane extraction. The system sensitivity can be further improved by applying sample heating. For quantitative determination using this extraction method, good system precisiion can be obtained only by carefully controlling the agitation conditions and extraction temperature. ACKNOWLEDGMENT The project has been financially supported by the Natural Sciences and Engineering Research Council of Canada. APPENDIX The mathematical solution for the diffusion problem that describes the headspace membrane extraction system follows the method outlined in section 12.8 of the book by Carslaw.16 By applying the Laplace transform the letting c2 and c3 be the transforms of C2 and C3, eq 1 becomes
∂2c2 2
∂x
2
- q2 c2 ) 0
∂2c3
and
∂x
2
2
- q3 c3 ) 0
(13)
where qi ) (p/Di)1/2 for i ) 2, 3, p ) Dhq22 ) Dmq32, and q3 ) q2(Dh/Dm)1/2. Likewise, the boundary conditions will be transformed to
∂c2 ∂c3 at x ) L Kmc2 ) c3 and Dh ) FDm ∂x ∂x c2 C0 Dh ∂c2 + ) Kh P Sp ∂x KmFD3 ∂c3 c3 ) 2R ∂x
at
x)0
(14)
(15)
for i ) 2, 3. To satisfy eq 15,
(
)
Kh Dhq2 A C + P 0 S 2
B2 )
with
at
x)L+M
(16)
Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
a2 ) b2 ) 0 (18)
To satisfy eq 16,
B3 ) -
KmFDm A3q3 with a3 ) b3 ) -L - M (19) 2R
To satisfy eq 14, we use eqs 18 and 19 to solve for A2 and A3:
(
)
Kh Dhq2 A cosh(q2L) ) C0 + P S 2 FDm A3 A q cosh[q3(-M)] (20) sinh[q3(-M)] Km 2R 3 3
A2 sinh(q2L) +
and
A2q2 cosh(q2L) +
(
)
Kh Dhq2 q C + A sinh(q2L) ) P 2 0 S 2
Dm KmF2Dm2 A3q3 cosh[q3(-M)] A q 2 sinh[q3(-M)] Dh 2RDh 3 3 (21)
F
giving
A2 )
-SKhC0 q∆(p)
[
xDm2FR cosh(qL) cosh(qφM) + Dh
2R KmxDh
sinh(qL) sinh(qφM) +
xDmFq sinh(qL) cosh(qφM) +
]
DmKm 2 F q cosh(qL) sinh(qφM) (22) xDh and
A3 )
-2RSKhC0
(23)
xDhq∆(p)
where
(
)
Kh ∆(p) ) xDh 2R + KmF2SDmq2 sinh(qm) sinh(qL) + Km FxDm(2RS + KhDh)q cosh(qm) sinh(qL) + FxDm(2RKh + SDhq2) cosh(qm) cosh(qL) +
(
xDh
)
1 2SR + KmKhF2Dm q sinh(qm) cosh(qL) (24) Km
where we let q ) q2, φ ) (Dh/Dm)1/2 and m ) φM. To evaluate C2 and C3, we use the inversion theorem. The expressions for c2 and c3 do not appear to have branch discontinuities; therefore,
C2(x,t) )
∑[residues of e
λt
c2(x,λ)]
C2(x,t) )
The general solution for eq 13 is 2788
ci(x) ) Ai sinh[qi(x + ai)] + Bi cosh[qi(x + bi)] (17)
and
∑[residues of e
λt
c2(x,λ)]
To be real and bounded for arbitrary t, all poles must be at negative real values of λ. Poles appear to be simple. At negative real values of λ, q is imaginary, q ) iz. ∆(λ) becomes equation 9.
Therefore, the solutions are
C2(x,t) )
Likewise,
∑{residue of e
λt
c2(x)}
C3(x,t) )
∑{residue of e
λt
where
where
C2(x) ) A2 sinh(q2x) + B2 cosh(q2x) )
C3(x) ) [A3 sinh[q3(x - L - M)] +
{
-SKhC0 KhC0 cos(zx) + P z∆(p) 2R KmxDh
xDm2FR cos(zL) cos(zm) -
B3 cosh[q3(x - L - M)]]
Dh
sin(zL) sin(zm) - xDm Fz sin (zL) cos(zm) +
]
-2RSKhC0
xDhz∆(p)
2R KmxDh
[
xDm2FR cos(zL) cos(zm) -
hence eq 7.
Dh
sin(zL) sin(zm) - xDm Fz sin(zL) cos(zm) +
]
DmKm 2 F z cos(zL) sin(zm) cos(zx) xDh hence eq 6.
sin[zφ(x - L - M)] + KmFDm 2RSKhC0 zφ cos[zφ(x - L - M)]} 2R xDhq∆(p)
DmKm 2 F z cos(zL) sin(zm) sin(zx) + xDh Kh Dhz -SKhC0 P S z∆(p)
C3(x)}
Received for review December 4, 1995. Accepted June 17, 1996.X AC9511758
X
Abstract published in Advance ACS Abstracts, August 1, 1996.
Analytical Chemistry, Vol. 68, No. 17, September 1, 1996
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