N ll expaiKfrC0

where t is the extraction time, and a is a time constant, representing how fast an equilibrium can be reached.

When the extraction time is long, Eq. (1.6) becomes Eq. (1.3), characterizing equilibrium extraction. If the extraction equilibrium is not reached, Eq. (1.6) indicates that there is still a linear relationship between the amount (n) of analyte extracted onto the fibre and the analyte concentration (C0) in the sample matrix, if the agitation, the extraction time, and the extraction temperature remain constant.

1.2.4 Calibration based on first-order reaction rate constant

The main challenge in organic analysis is polar compounds. They are difficult to extract from environmental and biological matrices and difficult to separate on the chromatographic column. Derivatization approaches are frequently used to address these challenges. Figure 1.5 summarizes various derivatization techniques that can be implemented in combination with SPME [15]. Some of the techniques, such as direct derivatization in the sample matrix, are analogous to well-established approaches used in solvent extraction. With the direct technique, the derivatizing agent is first added to the sample vial. The derivatives are then extracted by SPME and introduced into the analytical instrument.

Because of the availability of polar coatings, the extraction efficiency for polar underivatized compounds is frequently sufficient to reach the sensitivity required. Occasionally, however, there are problems associated with the separation of these analytes. Good chromatographic performance and detection can be facilitated by in-coating derivatization following extraction. In addition, selective derivatization to analogues containing high detector response groups will result in enhancement of the sensitivity and selectivity of detection. Derivatization in the GC injector is an analogous approach, but it is performed at high injection port temperatures.

Direct deriv the samp

Direct deriv the samp

Derivatization in the GC injector port

Derivatization in the SPME fiber coating

Derivatization in the SPME fiber coating

Fig. 1.5. SPME derivatization techniques.

Simultaneous derivatization and extraction

Derivatization following extraction

Fig. 1.5. SPME derivatization techniques.

Doping the SPME

fiber with the derivatizing reagent

Placing the doped fiber into Fiber desorption, gaseous phase or headspace above separation, and aqueous phase in reaction vial for and quantitation in-fiber derivatization/SPME

Fig. 1.6. In-coating derivatization technique with fibre doping method.

The most interesting and potentially very useful technique is simultaneous derivatization and extraction, performed directly in the coating. This approach allows for the high efficiencies and can be used in remote field applications. The simplest way to execute the process is to dope the fibre with a derivatization reagent and subsequently expose it to the sample (Fig. 1.6). The analytes are then extracted and simultaneously converted to analogues that possess a high affinity for the coating. This is no longer an equilibrium process, since derivatized analytes are collected in the coating as long as the extraction continues.

It is emphasized that if the sorbent is almost completely coated with a derivatizing reagent before its exposure to the analyte, a reaction between the approaching gaseous analyte and the sorbed derivatizing reagent is more likely to occur. This is especially true for short exposure times. When the reaction is the rate-limiting step, the reaction rate v (weight/time) is proportional to the concentration of gaseous analyte (C0) and the rate constant of the reaction between the derivatization reagent and the analyte [16]:

Therefore, quantitative analyses of an unknown analyte concentration (C0) is possible using an empirically determined constant K* and Eq. (1.7).

This simple and efficient approach is limited to low-volatility de-rivatizing reagents. The approach can be made more general by chemically attaching the reagent directly to the coating. The chemically bound product can then be released from the coating, either by a high temperature in the injector, light illumination, or a change of the applied potential. The feasibility of this approach was recently demonstrated by synthesizing standards bonded to silica gel, and which were then released during heating. This approach allowed for solvent-free calibration of the instrument [17].

In addition to using a chemical reagent, electrons can be supplied to produce redox processes in the coating and convert analytes to more favourable derivatives. In this application, the rod and the polymeric film must have good electrical conductivity. A similar principle has been used to extract amines onto a pencil 'lead' electrode [18]. The use of conductive polymers, such as polypyrrole, will introduce additional selectivity of the electrochemical processes associated with the coating properties [19].

1.2.5 Calibration based on diffusion

1.2.5.1 Diffusion

Diffusion is the random movement of a chemical substance in a material system consisting of two or more components, from an area of higher chemical potential of the diffusing substance in the given phase towards an area of lower chemical potential [20]. Two mathematical methods are often used to formulate the transport by diffusion [21,22]. The first, referred to as a mass transfer model, relates the net flux J to the occupation density difference between two adjacent subsystems, A and B:

Fluxes J are usually expressed as mass per unit area and per time (ng cm-2 s-1), and the concentration C as mass per volume (ng cm-3). Then the constant (mass transfer coefficient h) in the flux expression must have the dimension of a velocity (cm s-1). The second model, the gradient-flux law, is considered to be more fundamental. One well-known example of the gradient-flux law is Fick's first law, which relates the diffusive flux of a chemical to its concentration gradient and to the molecular diffusion coefficient:

dz where D is molecular diffusivity, and dC/dz is the spatial gradient of C along the Z direction. The molecular diffusivity (or molecular diffusion coefficient) D has the dimension (cm2 s-1), and depends on the properties of the diffusing chemical as well as on those of the medium through which it moves.

Equation (1.10) is often referred to as Fick's second law of diffusion, and can be readily derived from Fick's first law of diffusion:

dt dz2

Fick's second law states that the local concentration change with time, due to a diffusive transport process, is proportional to the second spatial derivative of the concentration. As a special case, consider a linear concentration profile along the z-direction C(z) — a0+a1z. Since the second derivative of C(z) of such a profile is zero, diffusion leaves the concentrations along the z-direction unchanged. In other words, a linear profile is a steady-state solution of Eq. (1.10).

The relationship between the flux of a property and the spatial gradient of a related property called a gradient-flux law is typical for an entire class of physical processes, in which some physical quantity such as mass or energy or momentum or electrical charge is transported from one region of a system to another. For example, heat flows through the bar from the high-temperature reservoir to the low-temperature reservoir. Another example is the transport of the electrical charge through a conductor by the application of an electrical potential difference between the ends of the conductor. Table 1.1 lists some physical processes obeying the gradient-flux law [23].

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