Performance Characteristics of MembraneCovered Polarographic Gas Detectors Daniel P. Lucero Electro-Analytical Transducer Corp., Fullerton, Gal$ Membrane-covered polarographic gas detectors operate in the diffusion-limited condition. Their performance characteristics are established by the mass transport properties of the gas molecules in the membrane material. The membrane can be represented by a network of mass capacitance nodes interconnected by diffusion-resistance elements. A mathematical model of the network comprised of a simple system of equations and a defining set of boundary conditions describes the detector performance characteristics. The mathematical model can also be utilized to establish the detector performance limits of signal level, rate of response, temperature response, and pressure response. In addition, the model can be employed to design detectors and evaluate their performance and to design experiments and aid in the interpretation of experimental results.
MEMBRANE-COVERED polarographic gas detectors are electrochemical devices which measure the partial pressure of a particular molecular species present in gas mixtures and/or dissolved in liquids ( I ) . The detectors are comprised of the basic elements of an electrolytic cell and are contained within a sealed cartridge. The electroactive elements of the detector are exposed to the gas molecules of the external environment through a semipermeable membrane. The gas molecules of the external environment migrate through the semipermeable membrane and the electrolyte of the cell, to an electroactive surface that provides selective catalytic reaction sites. The electroactive gas molecules react and electrons are transferred at the surface. The rate of reaction or the rate of electron transfer at the surface establishes the detector signal level. The signal level is an electrical current measured through a load external to the cell and is directly proportional to the concentration of electroactive molecular species in the external environment. A unique combination of the working electrode surface material, electrode potential, supporting electrolyte, and the reference electrode can ensure that the working electrode reaction sites be active only to a single molecular species. Thus, a high degree of molecular specificity is attained by the detector. Operation and characteristics of the detector are described by a mathematical model which is written for the mass transport processes that transpire. Variations in design and environmental conditions are reflected in changes to the boundary conditions of the system of equations representing the model. OPERATION OF THE DETECTOR
The performance characteristics of membrane-covered polarographic gas detectors are established by the mass transport and electrochemical processes occurring in the system. The description of these processes adequately depicts the operation of the detector. The electroactive surface of the detector is separated from the external environment by a thin layer of liquid electrolyte (1) L. C . Clark, J . Applied Physiol., 6, 189 (1953).
and a semipermeable membrane (Figure 1). This illustration shows only the geometrical relationships of the detector elements; it does not show their dimensional characteristics. The transport of gas molecules from the external environment to the electroactive surface occurs by the following sequential processes : (1) gas molecules dissolve in the membrane at the rnembrane-gas interface; (2) gas molecules migrate through the bulk of the membrane to the membrane-electrolyte interface ; (3) gas molecules in the membrane come out of solution at the membrane-electrolyte interface ; (4) gas molecules dissolve in the electrolyte at the membrane-electrolyte interface ; (5) gas molecules migrate through the bulk of the electrolyte layer to the electrolyte-electrode surface interface ; (6) gas molecules in the electrolyte come out of solution at the electrolyte-electrode surface interface; (7) gas molecules or atoms are absorbed at catalytic sites on the electrode surface; (8) a chemical reaction involving the gas molecules and electron transfer occurs at the electrode surface ; (9) all the products of the reaction with the exception of the electrons migrate away from the surface into the bulk of the electrolyte. The rate of the electrochemical reaction at the electrode surface is a function of several kinetic parameters that may be collectively adjusted such that the reaction consumes the electroactive molecular species at a greater rate than it can migrate through the membrane and the electrolyte layer. The attainment of this relationship between the mass transport and electrochemical processes confines the detector to the diffusion-limited mode of operation (2). Operation of the detector in this condition and its performance characteristics are completely established by the mass transport mechanisms outlined in steps 1 through 6 above. DIFFUSION-LIMITED MODE OF OPERATION
The diffusion-limited mode of operation can be described by employing the analytical representation of the mass transport processes of the membrane and electrolyte layer to simulate the detector. The electrode configuration of Figure 1 depicts a system that is essentially one dimensional in its diffusion characteristics-Le., the transport of molecules from the external environment can be analytically described with reasonable accuracy by considering only those molecules that are diffusing through the system directly over the electroactive surface and perpendicular to it. Thus, it is assumed that the lateral dimensions of the membrane and electrolyte layer are infinite relative to their thickness dimensions. The general equation expressing the concentration of any particular molecule of the one-dimensional system is (3): (2) S. Glasstone, “Introduction to Electrochemistry,” 1st ed., Van Nostrand, New York, 1956, pp 444-8. (3) R. M. Barrer, “Diffusion in and through Solids,” Cambridge University Press, London, 1951, chap. I. VOL. 40, NO. 4, APRIL 1968
707
1[#] J?
m
KEHBWLYE REGION
(e,l) -------
7:
e =
Figure 2. Binodal membrane/electrolyte lattice model of membrane-covered polarographic gas detector
Figure 1. Molecular diffusion path of membrane-covered polarographic gas detector
where concentration of gas molecules dissolved within the electrode system at any point, moles/cc D = diffusion coefficient of gas in the mass conducting medium, cm2/sec t = time, sec X = distance along the thickness dimension of the membrane and electrolyte layer, cm C
=
Equation 1 is the basis of the subsequent analytical development and modifications are made to its primary form only to accommodate simplifying assumptions. These assumptions alleviate some of the unwieldy manipulations that are attendant to the analytical solution of the equation with discontinuities in the mass transport properties of the electrode system. Discontinuities are introduced by the nonhomogenous characteristics of the detector-Le., the abrupt changes in the material composition such as at the membrane-electrolyte interface. New boundary conditions are established at the interface and are defined by changes in the diffusion and solubility coefficients of the gas molecules in the media. A pure analytical solution of Equation 1 with more than one set of boundary conditions is a cumbersome numerical problem. However, an analytical solution is obtained when the diffusion coefficient is assumed to be constant. The equation is expressed as an expansion of the error function that relates the gas concentration to the exponential of distance and time (4). A solution that possesses greater utility and which will result in defining important parametric groups is obtained by (4) A. Sommerfeld, -partial Differential ~~~~~i~~~ in physics,” Academic Press, New York, 1949, pp 55-62.
708
e
ANALYTICAL CHEMISTRY
considering the mass conducting media a network or lattice of point mass capacitances interconnected by diffusion resistance. The nodal representation of the diffusion process can equal that of the pure analytical solution by employing an infinite number of nodes or infinite lattice density. However, the degree of accuracy necessary in describing the performance characteristics of the detector will actually determine the lattice density required for the analytical model. The equation expressing conservation of mass a t a node within the membrane or electrolyte layer is:
where A
=
Cz
=
C1
=
t = V = (Ax) =
diffusion cross-sectional area (apparent area of the electroactive surface), cm2 concentration of the gas in the medium at interface 2 , moles/cc concentration of the gas in the medium at interface 1, moles/cc time, sec volume of material contained by node, cm3 the straight line distance the molecule travels in diffusing through the nodal volume of material, cm
The equation for a single node characterizes the diffusionlimited mode of operation of the detector. Therefore, a single node may be employed in its analytical representation for the purposes of design and evaluation and as dictated by the level of accuracy required. The total system of equations representing the detector is comprised of one equation for each volume of mass conducting medium and equations relating the concentration, diffusing gas partial pressure, and solubility of the gas molecules in each medium at their interfaces. The analytical model is illustrated by the binodal system shown in Figure 2 ; one
node represents the membrane and one node represents the electrolyte. Thus, two equations identical to Equation 2 can be written with the appropriate subscripts to identify the node:
where
e
subscript denoting the electrolyte subscripts denoting electrolyte interfaces shown by Figure 2 m = subscript denoting the membrane 1, m; 2, m = subscripts denoting membrane interfaces shown by Figure 2 1, e ; 2, e
= =
Equations 3a and 3b can easily be solved for the gas concentration in spite of changes in the diffusion coefficient from D, to De. However, a further complicating factor to be resolved is introduced by a change in solubility coefficient or the discontinuous change in the concentration of the gas at the membrane-electrolyte interface-i.e., another equation must be introduced to relate the two nodal equations at the membrane-electrolyte interface. As the gas dissolved in the membrane comes out of solution it reverts to its gaseous state prior to dissolving in the electrolyte. Therefore, the partial pressure of the gas at the interface is identical over the membrane and electrolyte surface. This condition is utilized to relate the nodal equations. The concentrations of the gas in the membrane and the electrolyte at the membrane-electrolyte interface are related to the partial pressure of the gas at the interface by their respective solubility coefficients as expressed by Henry’s law for dilute solutions of gases in liquids (5):
Table I. Transport Properties and Characteristics of the Membrane and Electrolyte Layer of a Polarographic Oxygen Detector Parameter Membrane Electrolyte layer Thickness, cm 2.54 x 10-3 2.54 x 10-4 Diffusioncoefficient, cm2/sec 3 X 1.8 X 10-5 Solubility coefficient, moles/ cc-mm Hg 3.3 X 1.66 x 10-9 Time constant, sec 2.15 0.00359 Gas flux, moles/sec-cm*-mm 3.89 X 10-I2 1 . 1 8 x 10-10 Hg Time constant ratio, membrane/electrolyte 600 Gas flux ratio, membrane/ electrolyte 0.033
The Detector Analytical Model. The detector analytical model is the system of equations and the characterizing
set of boundary conditions that describe the operation of the mass capacitance and diffusion-resistance lattice in a series arrangement. The lattice density comprising the system is determined by the accuracy requirements imposed on the performance calculations, the mass transport properties of the conducting media, and the geometrical and dimensional characteristics of the detector elements. The minimum lattice density that is required to represent adequately the analytical model can be established by comparing the calculated value of detector signal level and time constant utilizing different lattice densities. The calculations are repeated with an increasing lattice density for each computational cycle until the difference in the signal level and time constant of succeeding cycles is within the accuracy limits desired. Sample calculations were conducted to illustrate the relative effects of the membrane and electrolyte layer on the signal level and time constant of an oxygen detector. The mass transport properties and dimensions of the membrane and electrolyte layer that were utilized in the calculation are shown in Table I. This calculation shows that the electrolyte layer is saturated with dissolved gas molecules at a greater rate than is the membrane, by a factor of 600. Therefore, the rate of response of the detector is almost entirely determined by the membrane. The mass transport resistance of the membrane is greater than the resistance of the electrolyte layer by a factor of 30. Therefore, the steady state flux of the electroactive species through the detector is almost entirely determined by the membrane. The calculation demonstrates that the node representing the electrolyte layer may be deleted from the analytical model without compromising its utility or accuracy. The number of nodes required to represent the membrane in the analytical model can be determined by comparing the difference in the time constants of detector models that contain a successively greater number of nodes, as described above. Identical results can be obtained by a mutually coupled finite difference operator as a representation of the general diffusion equation and of computing the dynamic error of the operator (6). This is equivalent to computing the time constant error. The per cent difference in the time constant and signal level of a one-node and a two-node lattice is less than 4% for a membrane with a thickness of 2.54 X cm and a diffusion coefficient of 3 x 10-6 cm*/sec.
(5) S. Glasstone, “Elements of Physical Chemistry,” 1st ed., Van Nostrand, New York, 1954, pp 350-2.
(6) M. C. Gilliland, Annales de 1’Association Internationale Pour le Calcul Analogique, No. 2, April 1962, pp 78-82.
where Ke
=
K,
=
pt
=
solubility coefficient of the gas in the electrolyte, moles/cc-mm Hg solubility coefficient of the gas in the membrane material, moles/cc-mm Hg partial pressure of the gas at the membrane-electrolyte layer interface, mm Hg
Division of Equation 4a by 4b establishes the relationship between the gas concentrations in the membrane and the electrolyte at the interface and the relationship of the membrane nodal equations and the electrolyte nodal equations. The system of equations can be made entirely representative of the detector in the diffusion-limited condition by introduction of a boundary condition that characterizes this mode of operation. The sole defining operational characteristic is the nature of the reaction at the electrode surface which consumes the electroactive species at a greater rate than it can diffuse into the detector and to the electrode surface. Therefore, the definitive boundary condition is given when the concentration of the molecules at the electrolyte-electrode interface is zero: C2,e = 0
VOL. 40, NO. 4, APRIL 1968
709
Therefore, the analytical model of the detectors can consist of a single membrane node. Steady State Performance Characteristics. The steady state signal level of a diffusion-limited detector is directly proportional to the flux of the electroactive molecular species entering the detector. The flux is described by the equation below: im =
[DA/(Ax>lm[ C l , m
- C~,ml
(5)
where j m = steady state flux of electroactive molecular species, moles/sec. The concentration of the gas dissolved in the membrane material is related to the partial pressure of the gas in the external environment by Henry's law:
brane when a step change in the gas concentration in the external environment occurs is described by the equation below:
c2=
C,@t>Oo,C2=C m @ t > 0
Equation 10 is reduced to more simple terms by expressing the volume term as the product of the membrane area and its thickness A ( A x ) :
The solution of Equation 11 is: C, = [l
- e-='], CY
= D/(Ax)*
(12)
is the reciprocal of the system time constant. The detector time constant is ( A X ) * / D . The time required for the detector to attain 99% of its final response is given by: CY
where Cm = concentration of gas in the membrane material, moles/cc Km = solubility coefficient of the gas in the membrane material, moles/cm2-mm Hg p = partial pressure of gas, mm Hg Substitution of Equation 6 into Equation 5 yields:
Time for 99% response = 5 ( A ) 2 / ~ D Equation 12 delineates the detector rate of response characteristics when it is confined to the diffusion-limited mode of operation. Its characteristics are completely ascribed to the mass transport properties and dimensional and geometricaI characteristics of the membrane.
(7) since P , = DmKm P,
=
permeability coefficient of the gas in the membrane material, moles/sec-cm2-mm Hg/cm.
The deletion of the nodal equation representing the electrolyte layer mathematically confines the gas pressure of the electroactive molecular species in the layer to zero at all times. Thus, the constraint imposed by this boundary condition reduces the flux equation to:
TEMPERATURE RESPONSE CHARACTERISTICS
The temperature response characteristics of a diffusionlimited detector are proportional to the temperature response of the flux of the electroactive species. An examination of the equation describing the flux shows that the only parameter that varies with temperature is the permeability coefficient of the membrane material. Dimensional changes of the membrane are assumed to be absent. The permeability coefficient varies with temperature as shown by (7): Pm = (pm)oe-E'RT
where where p a = partial pressure of the electroactive molecular species in the external environment of the detector, mm Hg. Equation 8 can be modified to yield the detector signal level in terms of the rate of electron transfer at the electrode surface or an electric current:
i
=
nFpa[PA/(Ax)]m
(9)
where F
=
Faraday constant 96,500 coulombs/equivalent
i = detector signal level, A
n
of equivalents in each mode of electroactive species, equivalents/mole,
= number
E
= permeability
Pm
= =
(P,&
R T
=
=
activation energy (point-to-point migration energy potential barrier of nonequilibrium thermodynamics), cal/mole permeability coefficient, moles/sec-cm2-mm Hg/cm reference permeability coefficient (permeability coefficient when temperature response data are extrapolated to infinite temperature), moles/seccm2-mm Hg/cm universal gas constant, 1.986 cal/mole-"K absolute temperature of the membrane, "K
The substitution of this expression into Equation 9 yields the temperature response characteristics of the detector: i
Equation 9 characterizes the detector steady state signal level when it is confined to the diffusion-limitedmode of operation. Its characteristics are completely ascribed to the mass transport properties and dimensional and geometrical characteristics of the membrane. Rate of Response Characteristics. The rate of response characteristics of a diffusion-limited detector is established by the rate at which the membrane is saturated with the gas of the electroactive species. The rate of saturation or rate of change of gas concentration within the bulk of the mem710
ANALYTICAL CHEMISTRY
=
[nFAp,(Pm)~/(Ax)m]e-E'RT
(1 3)
PRESSURE RESPONSE CHARACTERISTICS
The pressure response characteristics of the detector will be delineated by changes in the mass transport properties of the membrane material with gas pressure. The pressure characteristics of the material result from deviations of Henry's (7) B. J. Zwolinski, H. Eyring, and C . E. Reese, J. Phys. Co//oid Chem., 53, 1426-51 (1949).
law for dilute solutions of gases in liquids. It has been assumed throughout this development that the gas dissolves in the membrane material in its original molecular form and does not experience changes in its structure which would alter the equilibrium relationship defined by Henry’s law. Henry’s law is sufficiently accurate for gas partial pressures ranging from 0 to 1.5 atmospheres for most polymers. Thus, under normal circumstances of gas detection where the electroactive molecular species is the diluent gas, the pressure response characteristics will be completely defined by the flux and/or signal level equations presented earlier. The concentration of a gas dissolved in a polymer at relatively high pressures may be expressed by: C = Kpn
(14)
where
C = concentration of dissolved gas in the material, moles/cc K = solubility coefficient, moles/cc-mm Hgn n = exponent determined experimentally (n = 1 for Henry’s law) p = gas partial pressure, mm Hg The pressure response characteristics at these relatively high pressures can be defined by substitution of Equation 14 into Equation 13 : i = [nFA(P,)o/(Ax)m]pa”e-E’RT RECEIVED for review October 13, 1967. Accepted January 24. 1968.
Complete Identification of Chromatographic Effluents Using Interrupted Elution and Pyrolysis-Gas Chromatography John Q. Walker and Clarence J. Wolf McDonnell Co., St. Louis, Mo. 63166 One of the major problems remaining in gas chromatography is concerned with the identification of eluted compounds. An identification system which utilizes pyrolysis-gas chromatography is described. The instrument performs the same basic function as does a gas chromatograph-mass spectrometer combination. Organic compounds are separated in a temperature programed gas chromatograph (the separation GC). The effluent from this chromatograph passes into a high temperature pyrolysis unit in which the complex compounds are degraded into simpler ones. The products so formed are analyzed by means of a second gas chromatographic unit (the analysis GC). The technique of interrupted elution chromatography is adapted to the separation GC so that all the unknowns can be pyrolyzed and identified in a single GC analysis. The chromatograms resulting from the pyrolysis of several classes of compounds including hydrocarbons, methyl esters of fatty acids, and alcohols are presented and discussed.
ONEOF THE MAJOR problems remaining in gas chromatography today is concerned with the positive identification of eluted compounds. Identification of unknowns can be made by comparison with known calibration samples, by making characteristic derivatives, or by the determination of unique physical properties such as infrared or mass spectra. The combination of a gas chromatograph (GC) with a mass spectrometer or infrared spectrometer forms a versatile analytical instrument. The most desirable arrangement consists of a G C connected directly to the spectrometer through a molecular separator. However, spectrometers are not accessible to all who require their use, and even if available they require a specialist in order to interpret the data. The relatively simple and inexpensive technique of pyrolysis-gas chromatography (PGC) yields a unique signature characteristic of the sample pyrolyzed. In this method the material being analyzed is thermally fragmented in a controlled manner and the product distribution is recorded with
a GC. Recently, Levy has carefully and in great detail reviewed the entire field of PGC (I). Dhont (2,3) and Weurman ( 4 ) have successfully applied PGC to the analysis of chromatographic effluents. They trapped the eluted vapors at the outlet of the chromatograph and then pyrolyzed the compound. The resulting chromatogram was used t o identify the compound. Keulemans together with Perry ( 5 ) and Cramers (6) studied the techniques required for reproducible pyrolysis of organic vapors in order to obtain structural information about compounds eluted from a GC. Levy and Paul (7) described a two-unit G C system built into a single instrument. Their apparatus consisted of a conventional GC, a flow delay-line trap, pyrolyzer, and a second GC. A few selected compounds from the original mixture could be trapped and pyrolyzed. They showed that the pyrolysis of normal alkanes, olefins, alcohols, mercaptans, and saturated and unsaturated methyl esters of fatty acids was reproducible and analogous t o mass spectra. Sternberg et al. (8)also described a two-unit G C which contained a high pressure electrical discharge between the chromatographs. As a compound emerges from the first GC, it is fragmented in the discharge and the stable products identified and recorded in the second GC. This gives rise to fragmentation spectra which are analogous to mass spectra. (1) R. L. Levy, Chromatog. Reu., 8,48 (1966). (2) J. H. Dhont, Nature, 206, 882 (1963). (3) J. H. Dhont, Analyst, 89, 71 (1964). (4) C. Weurman, Chem. Weekblud, 59,489 (1963). ( 5 ) A. I. M. Keulemans and S. G. Perry, “Gas Chromatography,” M. Van Swaay, Ed., Butterworth, Washington, D. C., 1962, pp 356. (6) C. A. J. G. Cramers and A. I. M. Keulemans, J . Gus Chromarog., 5, 58 (1967). (7) E. J. Levy and D. G. Paul, Ibid., 5, 136 (1967). (8) J. C. Sternberg, I. H. Krull, and G. D. Friedel, ANAL.CHEM., 38, 1639 (1966). VOL 40, NO. 4, APRIL 1968
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