the 20-mg. level the weighing error with a good microbalance becomes less significant. LITERATURE CITED
(1) Bastizii, R., ANAL. CHEW 21, 972 (1949); 23,580 (1951); 25,259 (1953).
(2) Bastian, R., Weberling, R., Palilla, F., Ibid., 22, 160 (1950). (3) Cogbill, E. C., Yoe, J. H., Ibid., 29, 1251 (1957). (4) Cogbill, E. C., Yoe, J. H., Anal. Chzm. Acta 12, 455 (1955).
( 5 ) Eberle. A. R., Lerner, M. W., ANAL. ~I
CHEM.32, 146 (1960).
(6) Eberle, A. R., Lerner, M. W., U. S.
Atomic Energy Comm., New Brunswick Laboratory, unpublished data, 1964.
( 7 ) Eberle, A. R., Pinto, L. J., Lerner, M. W., ANAL.CHEM.36,1282 (1964). (8) Hiskey, C. F., Ibid., 21, 1440 (1949). (9) Hiskey, C. F., Rabinowitz, J., Young, I. G., Ibid., 22, 1464 (1950). (10) Lauer, K. F., Le Duigou, Y., 2. Anal. Chem. 184, 4 (1961). (11) Le Duigou, Y., Lauer, K. F., “Pre-
cise Determination of Boron,” Meeting on High Precision Chemical Analvsis. European-American Nuclear Data corn: mittee, Brussels, Jan. 18-22, 1965. (12) O’Laughlin, J. W., Banks, (2; V., “Encyclopedia of Spectroscopy, G.
L. Clark, ed., p. 19, Reinhold, New York, 1960. (13) Platzer, R., Simenauer, A.. “Stand. ardization of Aqueous ‘Solutions of Boric Acid,” Meeting on High Precision Chemical Analvsis. EuroDeanAmerican rl’uclear Data ’ Comkittee, Brussels, Jan. 18-22, 1965. (14) Shalgosky, H. I., Ibid., “Precise Determination of Boron.” (15) Susano, C. D., Menis, O., Talbott.’ C. K., ANAL. CHEY. 28, 1072 (1956). (16) Young, I. G., Hiskey, C. F., Ibid., 23, 506 (1951).
RECEIVEDfor review July 12, 1965. Accepted September 3, 1965.
Mixed Carrier Gases for Quantitative Chromatog ra phy: A N e w Approach for Linearizing T her ma I Conductivity Based on Kinetic Theory SIR: On the basis of fundamental considerations a method has been developed to linearize the response of the thermal conductivity detector (TCD). The technique involves the use of eluents consisting of binary mixtures of gases judiciously selected to match certain significant molecular properties of the specific unknown (the sample) which is determined. If convective effects (2) are minimized by appropriate experimental design, the T C D operates on the principle that caloric energy is transferred from a heated sensor-e.g., a thermistor-to a cold wall by molecular collisions in the ambient gas. Under conditions of invariant heat input, the temperature of the sensor (as well as its resistance and the unbalance potential of the Wheatstone bridge in which it is customarily wired) is a function of the thermal conductivity coefficient, K,, effective in the gaseous milieu. Thus the physics of the T C D is essentially simple. However, its usefulness in quantitative analysis has been handicapped by empirical calibration procedures and implicit reliance on tenuous approximations, including the following generalizations. Experimental elution integrals and/or peak heights represent proportional measures of the amount of sample actually eluted in a given peak. Thermal conductivities are governed by a linear mixing rule of the type K , = XiKi XzKz (1)
+
1572
ANALYTICAL CHEMISTRY
where X denotes mole fractions and the subscripts 1 , 2 identify the sample and the carrier gas, respectively. I n recent years, significant advances in kinetic gas theory (5-7) have yielded improved mathematical descriptions of the thermal conductivity of gas mixtures. These treatments are in terms of complex collision integrals not readily applicable to situations prevailing in gas chromatography. Based on Xaxwellian intermolecular potentials, Mason ( 5 ) has provided a satisfactory rationalization for warranting the use of the Wassiljewa equation:
as a suitable approximation. I n Equation 2 the dimensionless coefficient A12 represents the ratio of the efficiencies with which molecules 2 and molecules 1, respectively, impede transport of heat by molecules 1, and Azl has the converse significance (3). Comparison of Equations 1 and 2 reveals that Aiz = A21 = 1 (3) is a sufficient condition for the linear mixing rule to prevail. I n previous work in these laboratories M. H. Barsky (1) has inferred theoretically and demonstrated experimentally that the requirement expressed in Equation 3 can be met in gas mixtures in which sample and carrier gas are matched with respect to the following molecular properties :
mass, collision diameter (distance of closest approach), and intermolecular force constants. Large disparity of molecular mass and collision diameter between molecules in a mixture causes (in conformity with theoretical expectations) a negative deviation from the thermal conductivity predicted by Equation 1, while differences in intermolecular forces yield a positive deviation. By proper selection of the components f and 2 it is possible to balance these effects and maintain effectively the linear Relationship 1. With the dual purpose of testing experimentally these interesting theoretical predictions and applying them concomitantly to a significant analytical problem, carrier gases were tailor made in the present investigation for the quantitative determination of carbon dioxide. The selection of this particular unknown was based on its general importance in organic elementary analysis, where gas chromatography may well displace classical methods in the future (4, 10, 2 1 ) . Helium was mixed with either argon or nitrogen and used t o elute carbon dioxide. The resulting ternary gas mixtures exhibited the expected behavior of pseudo-binary systems consisting of carrier (helium plus argon, or helium plus nitrogen) and sample (COz). A theoretically significant and experimentally accessible caloric transport parameter, AQ, defined in Equations 4 and 5 , was used successfully as a genuine linear measure of the amount of sample eluted in relevant gas chromatographic peaks :
AQ
- Qo
=
Q
Qo
= a&
=
aR, (T,- Tw) 0' - ti) (4)
(T.- T w )
(ti
- ti)
(5)
The symbols in Equations 4 and 5 have the following significance: t is the time in seconds; subscripts i and f denote emergence and tail of elution peak; 5" is the temperature in degrees Kelvin; subscripts s and w identify sensor and cold wall (T,is directly measurable; T,is calculable (8)from the bridge unbalance potential); Qo and Q are heat (calories) transferred during time (t/ - t i ) by the carrier gas alone (in the absence of sample) and by the carrier plus sample, respectively; a is a cell constant expressed in cm., dependent only on the design of the T C D ;
I?,
=
-
I
l &' - t i l j Kmdt
t'
is the mean thermal conductivity coefficient in cal./cm. sec. deg., averaged over the elution peak. A comparison of Equations l, 4,and 5 reveals that whenever the linear mixing rule applies, AQ = const. n
(6) ~
where n = XI (tf - t i ) F is the number of moles of COZ in the sample and F denotes the flow rate in moles/sec. Reliance on AQ in lieu of peak integrals
I
I
I l 1 1 1 1 1
20 50 SAMPLE SIZE Figure 1.
,"
I
I
I
I
I
I I l l
100 200 500 , micromolesof CO,
Elution of carbon dioxide with mixed carrier gases
Solid lines (left ordlnate): AQ/n plotted vs. sample size; dotted line (right ordinate)! g / n plotted VI. rornple slze. Zero slope corresponds to Ideal linearity. AQ = caloric transport parameter deflned in Equatlon 4; = mean unbolonce potential proportional to peak area (Equation 7). Carrier gas cornpositions are given In mole per cent
E
[where E denotes the transient bridge unbalance potential (8, 9) 1, obviates limitations inherent in the assumption
dT,
-
