such integrators. A second advantage is that the area is registered by a digital counter to four or five digits rather than by an oscillating pen blip or by a radial dial as is the case for the ball-and-disk and the low-inertia motor integrators. The ball-and-disk type of integrator has the distinct advantage of presenting the integral information directly on the chart along with the chromatographic peak. The results for the ball-and-disk integrator are in agreement with those for the voltage-to-frequency integrator, but the average deviations for the areas obtained with it are approximately three times as great. The low values for the first two peaks obtained with the low-inertia motor prolxihly can be attributed to inadequate response speed for the fast-rising, early peaks of the gas chromatogram s h o \ w in Figure 1. Thus, the integrator probably has accounted for less than the total area for each of the first two peaks and as a result high values are determined for the areas of the last two peaks (where relative per cent composition is the analytical approach). The voltage-to-frequency integrator gives a counting rate of 100 counts per mv.-second, which is the same as the ball-and-disk integrator with a 1-mv. recorder a t full scale deflection. However. with a 10-mv. recorder, the ball-and-disk integrator gives only 10 counts per mv.-second, while the voltage-to-frequency integrator still
gives 100 counts per mv.-second because of its independence of the recorder. Thus, the latter integrator permits a less expensive 10-mv. recorder t o be used to monitor the gas chromatogram without sacrificing any sensitivity in terms of integration. This type of integrator also is easily adaptable to automatic datahandling systems and printers (4). The low-inertia d.c. motor has a much lower counting rate and thus cannot give areas to more than three significant figures. If a retransmitting slide-wire with 6 volts across it is used on a 1-mv. recorder, the counting rate is the equivalent of 16.67 counts per mv.-second (equivalent of full-scale deflection of recorder), which is one sixth of the best t o be realized with either the ball-anddisk integrator or the voltage-to-frequency integrator. Although the motor integrator has the lowest counting rate and probably the least accuracy, it is entirely satisfactory for many applications. Furthermore, i t requires a fraction of the investment necessary for most other types of integrators and it is readily adapted t o the recorders used b y most gas chromatographers.
ACKNOWLEDGMENT
The authors are grateful to the Bell and Howell Research Center for a grantin-aid in.support of this work, which in-
cluded a graduate fellowship t o one of the authors (J.K.B.). LITERATURE CITED
(1) Bell, S . W.,Chuinti, V., Electronics 35 ( l l ) , 64 (1962). ( 2 ) Dal Sogxre, S., Bennett, C. E;:
Harden, J. C., “Gas Chromatography, V. J. Coates, H. J. Noebels, I. S. Fagerson. eds.. D. 117. -4cademic Press, Sew York, 1958. (3) DeFord, I). D., Division of Analytical Chemistry, 133rd Meeting, ACS, San Francisco, Calif., hpril 1958. (4!, Infortronics Corp., Houston, Tex., CRS-1 Digital Chromatograph Integrator,’’ Bull. CRS-102. (5) JFnak, J., J . Chromatog. 3, 308 (1960). (6) heulemans, A . I. M., “Gas Chromatography,” p. 209, Reinhold, New York, 1957. ( 7 ) Lane, It. S., Cameron, D. B., Electronics 32 (9), 53 (1959). (8) Lingane, J. J., -4nal. Chim. Acta 18, 349 (1958). (9) Martin, ,J. W., Cox, J. R., Electronics 35 (12), 46 (1962).
(10) Foebels, H. J., Wall, R. F., Brenner, S . , ”Gas Chromatography,” Academic Press, S e n York, 1961. . (11) Orr, C. H., ANAL. CHEM.33, 158 (1961). (12) Pecsok, R. L., ed., “Principles and Practice of Gas Chromatography,” Wiley, Xew York, 1959. (13) Perrine, W. L., “Gas Chrornatography,” H. J. Noebels, R. F. Wall, N. Brenner, eds., p. 119, Academic Press, Yew York, 1961. (14) Simmons, hl. C., Second Annual Research Conference on Gas Chromatography, University of California, Los Angeles, Calif., January 1962. RECEIVEDfor review April 26, 1962. -4ccepted June 20, 1962.
CaIcuIation of Relative Molar Response Factors of Thermal Conductivity Detectors in Gas Chromatography E. G. HOFFMANN Max-Planck-lnstitut fur Kohlenforschung, ML’lheim (Ruhr), Germany
b In spite of the introduction of new detecting devices, the thermal conductivity cell is still the most commonly used detector in gas chromatography when GC is used in quantitative analysis. Relations have been derived from the kinetic theory of gases for calculation of sensitivity factors of thermal conductivity cells as detectors in gas chromatography. They have been applied to 18 substances in the gases Hz, He, and Nz as carriers. The calculations could b e verified in part by related measurements with thermistors as sensitive elements and an electronic integrating device. 1216
ANALYTICAL CHEMISTRY
N
INVESTIGATIONS have stated t h a t the signal strength coming from a sample in a thermal conductivity cell and related to equal molar concentration is considerably dependent on the nature of the substance. As Messner, Rosie, and Argabreight (28) pointed out, the signal strength within a homologous series increases nearly proportional with the molecular weight. I n 1958 we reported (13) this to be a consequence of the elementary kinetic theory of transportation phenomena in binary gas mixtures. We stated that the different influence of the substance vapors on the thermal conductivity of
UMEROUS
the carrier gas is mainly determined by the different cross sectional areas of the substance molecules by which the heat transport of the carrier gas is impeded. Later on Littlewood (26), using some derivations from the more rigorous Chapman-Enskog theory of transport properties, confirmed this result. I n the meantime we could extend our former results and apply them to other carrier gases (12). Earlier, some authors expressed the opinion (9-11,22,23, 29, 31) that the use of light carrier gases like H e or HPshould lead to signal strengths nearly independent of the nature of the materials
to be analyzed. They ovcrlooked the fact that the kinetic theory does not predict simple additivity in thermal conductivity of miatures. On the contrary, linear additivity can be expected only in some special cases. I n general, the thermal conductivity of a mixture is given by an expression, Equation 2 , essentially derived by Wassiljewa in 1904 (40). Only when both the coefficients Alz and Azl are unity, real linear additivity is given. I n general, the following relation holds.
(For A ~ I change , the subscripts.) For the calculation of the A’s, i t is better in practice to use the viscosity data from which the collision diameters are derived instead of the diameters directly, because the exact values of the collision diameters cited in the literature depend somewhat on the method of estrapolation applied to eliminate the temperature dependence. Using the relation between dynamic viscosities, ‘7, and collision diameters, Tu, we then obtain
Thermal Conductivity = Factor X Transported Energy X Velocity of Transportation X Mean Free Path Length between Collisions
Ai?
When t n o gases are mixed, the effect of the change of thermal conductivity is not limited solely to replacing molecules with high speed of energy transport by slower ones-if this were the case, one should expect additivity. The change of thermal conductivity also affects the indi\idual changes of the mean free path lengths by introduction of molecules with a larger cross section for collision. For the thermal conductivity of a mixture, this consideration leads to an expression of the general form: x?n
=
A1
A’i Ai
-
+
xp
A’.
-“
Az
instead of
+
AM = 1 1 x 1 X2Xz (la) The last formula expresses simple additivity. (AM, All and XI represent thermal conductivities of the mixture, the carrier gas, and the component in question, respectively; X represents mole fractions.) I n Equation 1, the ratios of the partial quantities of mean free path lengths (A’, A) of the components in the mixture and the pure state replace the mole fractions in a simple rule of mixtures, Equation l a . This corresponds to the formula of Wassiljewa (40): AM
A?
A1
=
1
x X+ 1 + 1 -x +T x ‘412
A21
(2)
The same type of relation has been derived earlier by Sutherland (38) and by Thiesen (39) for the viscosity of gas mixtures, but there is no proof that the corresponding A’s must be equal for the transport of energy and momentum. The results of the more rigorous theories also lead to this type of concentration dependence if allowance for some simplification is made (43). The appiication of a simplified theory of mean free path length leads to the following expression for the A’s in Equation 2 .
=
1 (Y
-
4
X
Here Tul, Tup represent effective collision diameters a t temperature T valid for collisions betneen molecules of the same sort 1 or 2; Tu12 = ( = ~ 1 where m u is the mean collision diameter which is independent of the influence of attracting forces obtained by estrapolating to infinite temperature with Sutherland’s formula M1,pare molecular weights of carrier gas and vapor, respectively Cl,z are Sutherland’s constants valid for collisions between carrier gas molecules and substance molecules, respectively C12 = C1C2is Sutherland’s constant valid for collisions of carrier gas molecules with those of substance vapor Some remarks on Equation 4 should be made. There are a great variety of formulas of this type (1, 24, 26, 38, 42, 43) which differ in the exponents of the mass dependent members. Values from -l/p to + 3 / 4 are recommended. All these formulas describe the concentration dependence of the viscosity or thermal conductivity of gas mixtures fairly well for a number of substance pairs. Obviously the reason is that there exists not only a certain pair but a group of pairs of the A’s (43) which permit approximating the experimental values of mixtures. The factor (AI2/ -U1)1/2 (under the square root) in the first term originates from expressing collision diameters by viscosities and refers only to the pure gases for which there is no difference in the mass dependence in an elementary or in a more rigorous approach of kinetic theory, Therefore, its exponent should not deviate from contrary to the propositions of Lindsay and Bromley (Zd),who used a n exponent of 3 / p . We could check this for Hzas a carrier where the mass differences are extreme. Only sensitivity factors (see below) which are calculated by employing ( M 2 / M 1l )/ 2 under the square root of the first term yield fair agreement with the experi-
+
ments; the eaponcnts
and
3/4
do not
fit the cxperiniental data as well. The exact value of the second term of Equation 4, which has the form [;(I is not important if M z is appreciably higher than AIl, which is usually the case for organic vapors in light carrier gases. I n this case, the term varies only slightly with changes of the molecular masses of the heavier vapors. .is will be shown below, the relative A12’s are significant and as in the ratio of two Alp’sthis term is practically canceled, the exact form of this term does not affect our calculations with light carrier gases. The proportionality factor (Y introduced in Equation 4 makes allowance for some refinements in the elementary kinetic theory. It originates from the fact that the persistence of velocities (6, 15) is not identical for collisions between molecules of the same sort and between different ones. It is close to unity (-0.94) and varies only by 1-2% for the substances tabulated below (Tables 1-11) if light carrier gases are used. As only the ratios of the A’s are of interest in our case, the calculations can be done by using CY = 1 without an essential loss of accuracy. I n our first publication ( I S ) , which contains some calculations for helium as a carrier, we used a wrong factor which was in the average, CY = 0.64. Therefore, the F-values derived in this way are correspondingly lower compared with Table I. Despite this fault, the sensitivity factors calculated from these lower F-values (and related to benzene as a standard) could be obtained in agreement with the experience (33). For the response of a thermal conductivity cell Schmauch and Dinerstein (34) derived a product of two expressions, one of which depends on the cell geometry and temperature gradient as well as the electric parameters of cell and bridge, and the other on the thermal conductivity of the carrier gas and the mixture. For our purpose, i t is more useful to split the expression of Schmauch and Dinerstein into four factors in the following way:
+ %)In,
Response = Cell Factor X Carrier Gas Factor ( = 4 G )X Molar Thermal Conductivity Change X Concentration ( X ) Only the first two factors which depend on the thermal conductivity cell and the carrier gas remain constant during a GLC experiment. They determine the sensitivity level of the equipment. Schmauch and Dinerstein mainly discuss in detail the experimental parameters nhich are included in these two factors. They also tried to calculate absolute signal strengths (expressed in electrical units) in a few cases from their experimental parameters, but the agreement with their experiVOL. 34, NO. 10, SEPTEMBER 1962
1217
Table 1. ( A = 0.05
Substance JI n-Butane 58 n-Pentane i2 n-Hexane 86 n-Heptane 100 n-Octane 114 n-Nonane 128 Cyclohexane 84 Benzene i8 Toluene 92 Methyl chloride 51 Methylene chloride 85 Chloroform 119 Carbon tetrachloride 154 Carbon disulfide 76 Methanol 32 Ethanol 46 Diethyl ether 74 Acetone 58 a Setting benzene = 100. Experimental values this work. See reference (88). d Using Equation 9 directly.
-4 12 2.49 2.92 3.18 3.53 4.13 4.72 2.97 2.65 3.04 1.82 2.24 2.56 2.85 2.14 1.69 2.08 2.71 2.33
Helium . . . 0.13) c
An 1.03 1.07 1.09
2.36 2.80 3.07 3.43 4.04 4.64 2.88 2.56 2.95 1 .73 2 19 2.52 2.82 2.08 1.54 1.96 2.58 2.23
1.11 1.22 1.31
1.15 1.09 1.18 1.os 1.24 1.43 1.55 1.17 0.90 0.99 1.09 0.95
mental values was poor. Apart from the fact that it is difficult to know all the experimental conditions with sufficient accuracy for absolute determinations, i t is probable that the Wassiljewa constants used by the authors are not adequate for the problem. For calculating the A's, they used a relation proposed b y Lindsay and Bromley (94) in which the exponents of the mass dependent factors of Equation 4 are approximated empirically to cover the whole range of concentration. We also tried the formula of Lindsay and Bromley for calculation of the Wassiljewa constants. B u t the relative response factors derived in this way do not agree as well with the experimental data, as was stated above.
1.85 2.23 2.48 2.78 3.37 3.92 2.44 2.16 2.46 1.50 2 01 2.37 2 65 1.88 1.24 1.58 2.05 1.86
94 111 120 134 156 1% 112 100 115 G9 85 97 108 81 64 79 102 88
Here we are interested in the changes of the response caused by the nature of the different substances separated during GLC analysis, that is (XI - AM)/ ( X h ) = F, the molar thermal conductivity change. -4s can easily be derived, this expression, F , should equal a value close to unity for light carrier gases and low mole fractions, X , only if additivity of thermal conductivity can be accepted. Otherwise me derive from Equation 2 in a first approsirnation
- Xxy)'xx = Fo + fX (X
