Thermodynamics

SPME is a multiphase equilibration process. Frequently, the extraction system is complex, as in a sample consisting of an aqueous phase with suspended solid particles having various adsorption interactions with analytes, plus a gaseous headspace. In some cases specific factors have to be considered, such as analyte losses by biodegradation or adsorption on the walls of the sampling vessel. In the discussion below we will only consider three phases: the fibre coating, the gas phase or headspace, and a homogeneous matrix such as pure water or air. During extraction, analytes migrate between all three phases until equilibrium is reached. The following discussion is limited to partitioning equilibrium involving liquid polymeric phases such as poly(dimethylsiloxane). The method of analysis for solid sorbent coatings is analogous for low analyte concentration, since the total surface area available for adsorption is proportional to the coating volume if we assume constant porosity of the sorbent.

The mass of an analyte extracted by the polymeric coating is related to the overall equilibrium of the analyte in the three-phase system. Since the total mass of an analyte should remain constant during the extraction, we have:

where C0 is the initial concentration of the analyte in the matrix: C°°, C°° and C° are the equilibrium concentrations of the analyte in the coating, the head-space and the matrix, respectively; Vf, Vh and Vs are the volumes of the coating, the headspace and the matrix, respectively. If we define the coating/gas distribution constant as Kfh = C°°/C°°, and the gas/ sample matrix distribution constant as Khs = C°°/C°°, the mass of the analyte absorbed by the coating, n = C°°Vf, can be expressed as:

KfhKhs Vf C0 Vs

Also:

fgKgs

since the fibre/headspace distribution constant, Kfh can be approximated by the fibre/gas distribution constant Kfg, and the headspace/sample distribution constant, Khs, by the gas/sample distribution constant, Kgs, if the effect of moisture in the gaseous headspace can be neglected. Thus, eqn [2] can be written as:

KfsVfCnVs

The equation states, as expected from the equilibrium conditions, that the amount of analyte extracted is independent of the location of the fibre in the system. It may be placed in the headspace or directly in the sample as long as the volumes of the fibre coating, headspace and sample are kept constant. There are three terms in the denominator of eqn [4] which give measures of the analyte capacity of each of the three phases: fibre (KfsVf), headspace (KhsVh) and the sample itself (Vs). If we assume that the vial containing the sample is completely filled (no head-space), the term KhsVh in the denominator, which is related to the capacity (Q°Vh) of the headspace, can be eliminated, resulting in:

KfsVfCpVs KfsVf # Vs

Equation [5] describes the mass absorbed by the polymeric coating after equilibrium has been reached in the system. In most determinations, Kfs is relatively small compared to the phase ratio of sample matrix to coating volume (Vf « Vs). In this situation the capacity of the sample is much larger compared to capacity of the fibre, resulting in a very simple relationship:

n = KfsVfCo

The above equation emphasizes the field-sampling capability of the SPME technique. It is not necessary to sample a well-defined volume of the matrix since the amount of analyte extracted is independent of Vs as long as KfsVf « Vs. The SPME device can be placed directly in contact with the investigated system to allow quantitation.

Prediction of distribution constants In many cases, the distribution constants present in eqns [2]-[6] which determine the sensitivity of SPME extraction can be estimated from physicochemical data and chromatographic parameters. For example, distribution constants between a fibre coating and gaseous matrix (e.g. air) can be estimated from isothermal GC retention times on a column with a stationary phase identical to the fibre-coating material. This is possible because the partitioning process in gas chromatogra-phy is similar to the partitioning process in SPME, and there is a well-defined relationship between the distribution constant and the retention time. The nature of the gaseous phase does not affect the distribution constant, unless the components of the gas, such as moisture, swell the polymer, thus changing its properties. A most useful method for determining coating-to-gas distribution constants uses the linear temperature programmed retention index (LTPRI) system, which relates retention times relative to the retention times of w-alkanes. The logarithm of the coating-to-air distribution constants of w-alkanes can be expressed as a linear function of their LTPRI values. For poly(dimethylsiloxane) (PDMS), this relationship is log Kfs = 0.00415*LTPRI - 0.188. Thus, the LTPRI system permits interpolation of the Kfg values from the plot of log Kfg versus retention index. The LTPRI values for many compounds are available in the literature, hence this method allows estimation of Kfg values without experimentation. If the LTPRI value for a compound is not available from published sources, it can be determined from a GC run using a GC column coated with the same material as the fibre.

Estimation of the coating/water distribution constant can be performed using eqn [5]. The appropriate coating/gas distribution constant can be found by applying techniques discussed above, and the gas/water distribution constant (Henry's constant) can be obtained from physicochemical tables or can be estimated by the structural unit contribution method.

Some correlations can be used to anticipate trends in SPME coating/water distribution constants for analytes. For example, a number of investigators have reported correlation between the octanol/water distribution constant, Kow, and Kfw. This is to be expected, since Kow is a general measure of the affinity of compounds for the organic phase. It should be remembered, however, that the trends are valid only for compounds within homologous series, such as aliphatic hydrocarbons, aromatic hydrocarbons or phenols; they should not be used to make comparisons between different classes of compounds, because of different analyte activity coefficients in the polymer.

Effect of extraction parameters Thermodynamic theory predicts the effects of modifying certain n =

extraction conditions on partitioning and indicates parameters to be controlled for reproducibility. The theory can be used to optimize the extraction conditions with a minimum number of experiments and to correct for variations in extraction conditions, without the need to repeat calibration tests under the new conditions. For example, SPME analysis of outdoor air may be done at ambient temperatures that can vary significantly. A relationship that predicts the effect of temperature on the amount of analyte extracted allows calibration without the need for extensive experimentation. Extraction conditions that affect Kfs include temperature, inorganic salt concentration, pH and organic solvent content of the water.

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