Characterization of coal by laser pyrolysis gas chromatography

(5) D. J. Bennett and E. H. Creaser, Anal. Biochem., 37, 191 (1970). (6) A. R. Tschida and H. Markowitz, Anal. Biochem., 26, 337 (1968). (7) H. Small, T. S. Stevens, and W. C. Bauman, Anal. Chem., 47, 1801 (1975). (8) K. C. Blanshard, H. F. Bradford, P. R . Dodd, and A. J. Thomas, Anal. Biochem., 67, 233 (1975). (9) D. L. Stalling, R . C. Tindle, and J. L. Johnson, J. Assoc. Off. Anal. Chem., 55, 32 (1972). (10) R . C. Tindle and D. L. Stalling, Anal. Chem., 44, 1768 (1972). (11) K. R. Griffitt and J. C. Craun, J. Assoc. Off. Anal. Chem., 57, 168 11974). (12) GPC IbOl-Autoprep, Analytical Biochemistry Laboratories Inc., Columbia, Mo. 65201. (13) D. L. Stalling, J. Johnson, and J. N. Huckins, "Automated Gel Permeation-Carbon Chromatographic Cleanup of Dioxins, PCBs, Pesticides, and industrial Chemicals", in "Environmental Quality and Safety, Supplement Vol. 111, Pesticides Lectures of the IUPAC Third International Congress of

Pesticide Chemistry" (Helsinki, July 1974), F. Coulston and F. Korte, Ed., G. Thieme Publ., Stuttgart, Germany, 1975, pp 12-18. (14) D. L. Stalling, Proc. Conf. Environ. Sensing Assessment, Las Vegas, Nev., Institute of Electrical and Electronic Engineers, Inc., USA Annals No. 75CH1004-1. New York, 1976, Section 7-5. (15) B. A. Karlhuber and D. 0. Eberle, Anal. Chem., 47, 1094 (1975). (16) H. A. McLeod, J. Chromatogr. Sci., 13, 302 (1975). (17) D. E. Ott, Residue Rev., 55, 1, 1975. (18) J. F. Morot-Gaudry, V. Fiala, J. C. Huet, and E. Jolivet, J. Chromafogr., 117, 279 (1976). (19) W. B. Crummett and R. H. Stehl,. Environ. Health Perspectives, 15, Sept., 1973.

RECEIVEDfor review October 8, 1976. Accepted December 2, 1976.

Characterization of Coal by Laser Pyrolysis Gas Chromatography R. L. Hanson,"' N. E. Vanderborgh,* and D. G. Brookins The University of New Mexico, Albuquerque, N.M. 87131

Applications of laser pyrolysis gas chromatography (LPGC) . to study the influence of compositlon of coals on the distribution of gaseous products are presented. LPGC provides a rapid method when used in conjunction wlth plasma stolchiometrlc analysis for determining the relatlve concentratlonsof carbon, hydrogen, and oxygen in coals. Pyrograms and correlations between experimental products and elemental compositlons are presented.

The method of free energy minimization was used in a computer program to calculate the gaseous product distribution for coal samples a t temperatures from 2000 to 3500 K a t a pressure of 2280 Torr (22).Correlations are presented of the experimental gaseous product compositions to the elemental composition of the coal. The comparison between the calculated product distributions a t 3300 K and the experimental results agree for Hz, CO, and CzHz to 5% by volume or better for the five coal samples. EXPERIMENTAL

Pyrolysis gas chromatography has been used to characterize coals of various rank (1-3). The highest pyrolysis temperature used in these studies was 1273 K, resulting in the evolution of large quantities of low molecular weight gases. Benzene and toluene are the major aromatic products ( I ) . Changes in the chromatograms with coal rank were apparent (I, 2). The volatility of the organic compounds in the coal was enhanced by the addition of water to the coal samples before pyrolysis ( I ). Coal pyrolysis in nitrogen a t temperatures of 623-773 K produced methane and ethane as major products with carbon monoxide, ethylene, and propane also formed ( 4 ) . An argon plasma torch reactor has also been used to study the rapid devolatilization of small coal particles ( 5 ) . The weight loss increased with both increased heating rate and increased temperature. Heating rates from 105-106 K/s and a final temperature of 2000 K produced a greater loss of volatile matter than is obtained by proximate analysis. A microwave discharge in argon was used to study the gasification reactions of coal (6). The major products were Hz and CO with the major hydrocarbon products being CH4, CzHz, and CzH4. Also formed in smaller quantities were HzO, COz, HCN, and higher molecular weight hydrocarbons up to CS. Rapid pyrolysis of coal has been achieved by having pulsed lasers irradiate the samples and using mass spectrometry for identification and quantitation (7-18). Laser pyrolysis gas chromatography (LPGC) has been used in studies of organic compounds and oil shales (19-21). Because of the rapidity of analysis, this technique was selected to use to characterize coal samples. Present address, Inhalation Toxicology Research Institute, Lovelace Biomedical and Environmental Research Institute, P. 0. Box 5890, Albuquerque, N.M. 87115. Present address, Los Alamos Scientific Laboratories, University of California, Los Alamos, N.M. 87544. 390

ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977

The coal samples were received as powders from Paul Weir Company, Chicago, Ill. Table I lists the composition of each sample. The samples were pelletized at 20 000 psi to form pellets of 1-2 mm thickness. The samples were sectioned and placed in quartz sample tubes of 6-mm 0.d. A Perkin-Elmer model 3920 gas chromatograph was used with a beta-induced luminescence detector (BILD) and the hydrogen flame ionization detector connected in series. The BILD detects components in the column effluent by measuring the quenching of the luminescence from trace levels of nitrogen in the helium carrier gas that is excited by beta's from a tritium foil. The design and tesponse of this detector has been described (23).The quartz sample tube containing the pelletized sample was mounted on the injection port of the gas chromatograph. Helium carrier gas purged the system and carried the pyrolysis products onto the analytical column. A pulsed ruby laser with an output of 2.6 joules in the normal-mode was used in these studies. The beam was focused on the sample surface in the quartz tubes. Only about 0.01 cm2of the sample was irradiated by the focused laser beam. RESULTS AND DISCUSSION Hydrogen sulfide and hydrogen cyanide have been reported as products from laser pyrolysis of coal (7, 11). Hydrogen sulfide and hydrogen cyanide have retention times similar to the retention times of water and acetylene, respectively, under the experimental conditions. The experimental peaks are reported as water and acetylene with recognition that hydrogen sulfide and hydrogen cyanide may be formed. Equilibrium product distribution calculations were made for five coal samples for the elements carbon, oxygen, hydrogen, nitrogen, and sulfur. An initial calculation at 2280 Torr and 2500 K was made for 33 compounds: CH4, CzH4, CzHz, CzH, CH3, CH20, HCO, CO, COz, Hz, H, OH, HzO, C2N2, HCN, CSz, HNCO, HNO, Nz, N, NO, NOz, NzO, NH, "2, "3, SN, SH, Sz, SO, SOz, H2S, and C(+ Nine species with insignificant concentrations were excluded from further consideration. They were OH, HNCO, HNO, N, NO2, N20,

Table I. Ultimate Analysis of Coal Samples (weight % ) Sample No. Location (Type)

1 Texas Lignite

2 New Mexico Subbituminous

3 Louisiana Lignite

4 N. Ill. No. 6

5 Central Ill. No. 6

Carbon, % Hydrogen, % Nitrogen, % Chlorine, % Sulfur, % Oxygen, % Ash, %

43.13 3.39 0.73 0.08 2.45 11.96 38.26

65.26 4.73 1.07 0.07 1.47 12.78 14.62

60.10 4.23 1.26 0.14 0.83 14.60 18.84

72.21 5.03 1.29 0.04 2.99 9.60 8.84

65.40

Coal 2

Table I.][. Calculated C(s)Mole Fractions Formed

lofl,-;

Sample

2000 K

3500 K

Coal No. 1 Coal No. 2 Coal No. 3 Coal No. 4 Coal No. 5

0.5304 0.5894 0.5694 0.6278 0.6245

0.3490 0.3943 0.3822 0.4251 0.4223

Coal 1

loo,

I

I

C-3.59

I

;

H-3.39 j

I

010.748

I

I

N-0.052

I

I

;

C=5.64

4.59

1.08 0.06 4.41 8.65 15.81

H=4.73

I

I

0=0.799

I

I

I

N=0.076

I

I

S=O.b46

I

;

,

S=O.O07

I

I

,

l O - 1

10-2.k

Temperature K

Figure 2. Calculated products from coal 2 vs. temperature

Temperature K


NH, Son, and "2. Product distributions for the five samples were also calculated again at 2280 Torr a t 100 K increments from 2000-3500 K. Table I1 lists the mole fractions for C(sl calculated for these of samples a t 2000 and 3500 K. The calculations for the the coals at 2000 K indicated that over half of the product should be solid carbon, and carbon deposits were formed on the sample tube in the experimental pyrolysis. Other investigators have shown that these residues had a lower H/C ratio than the original coal samples (16). Figures 1through 5 are plots of the calculated gaseous mole fractions of products for coal samples from 2000-3400 K. Hydrogen and carbon monoxide were the predominant gas-

eous species. The following stable products, CzH2, HCN, CS2, N2, S, H2S, H2, CO, and CH4, had calculated gaseous mole fractions above mole fractions in portions of this temperature range. The relative quantities of these species varied with the relative stoichiometric abundances of the respective elements. Mass spectrometric studies of laser pyrolysis of coal agree with these computer predictions in that H2, CO, C2H2, CH4, HCN, Nz, and H2S have been observed. Apparently, S and CSz were formed in such small quantities that they have not been detected. Table I11 contains the experimental distribution of CH4, C2H2, H2, and CO and the calculated distribution for these products a t 3300 K, assuming the C2H and H species quench to CzH2 and Hz,respectively. The experimental CH4 concentration was much higher than the value calculated a t 3300 K. The agreement between the experimental and calculated values for Hz, CzH2, and CO is good. This indicated that the assumption of free radical quenching worked well in predicting experimental product distributions for the predominant plasma quenching molecules. Karn e t al. (15)reported 31.5 mol % methane as a product of carbonizing coal at 1173 K. Thus, much of the methane detected in laser pyrolysis resulted from the lower temperature degradation of samples in the vicinity of the crater. ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977

391

Table 111. Ruby Laser Pyrolysis of Coal

Sample

Exnerimental comDosition mole percentages from BILD CH4 CzHz Hz CO

Coal 1 Coal 2 Coal 3 Coal 4 Coal 5

2.05 3.70 1.93 3.05 3.83

15.3 20.5 17.4 22.1 22.0

44.5 46.6 50.2 51.4 45.4

22.9 20.9 25.1 17.7 21.7

Calculated gaseous percentage composition at 3300 K CH4 CzHz CzH Hz + H

+

1.36 X 1.42 X 1.30 X lo-" 1.46 X lo-$ 1.45 X

15.6 20.3

46.08 46.69 47.73 47.19 47.04

18.2

23.7 23.4

CO

26.35 21.50 25.90 16.20 15.77

Table IV. Coal Product Distributions in Mole Percent of Gaseous Products Sample

Hz

CO

CHI

HzO

CzHz

Coal 1 Coal 1 Coal 2 Coal 3 Coal 4 Coal 4 Coal 5

43.7 45.3 4.6 50.2 54.3 48.6 45.4

22.7 23.0 20.9 25.1 16.6

0.92 4.18 3.70 1.93 2.56 3.53 3.83

13.4 12.5 8.28 5.41 6.63 4.71 7.03

15.6 15.0 20.5 17.4 19.9 24.3 22.0

Co:il 4

C=h.02

18.8

21.7

11-5.03

0=0.600 N-0.092

S=0.093

HCO CZNZ

Temperature K

Flgure 3. Calculated products from coal 3 vs. temperature

Table IV lists the mole percentage gaseous product distributions from ruby laser pyrolysis of the coal samples with a 2.64-joule pulse in helium carrier gas as determined from the BILD response. The two sets of results for coals 1and 4 indicate the reproducibility obtainable with these samples. There was a broad peak in each pyrogram. Knox and Vastola (7) reported detection of hydrogen sulfide by irradiating coal in the source of a time of flight mass spectrometer with a 100millijoule ruby pulse. This broad peak was thought to be HzO although HzS has a similar retention time and is included in Table IV. The HCN retention time was similar to the retention time for CzH2. If HCN was pesent, it would appear with the CzHz peak in the BILD response. Hydrogen cyanide has been reported as a product from laser pyrolysis of coal (IO). The mole percentage of HCN produced varied from 0.7-1.5% by laser irradiation with five pulses of 1.5joules upon different rank coal samples in sealed tubes ( 11 ). Thus there could be a small quantity of HCN in the peak labeled CzH2. The first product considered was CO. The experimental and theoretical CO percentage compositions are given in Table 111. The calculated values are for equilibrium a t 3300 K using the sample stoichiometries from Table I. As can be seen, the theoretical and experimental values are in agreement, indicating that CO production is primarily due to plasma 392


Temperature K


quenching. The slight scatter in the values may result from the fact that oxygen was determined by difference and these values were used for the calculations. Hydrogen products also agreed with the theoretical concentrations of H and Hz with the assumption that 2H quenches to Hz.The experimental Hz concentrations are shown in Figure 6; plotted vs. the theoretical concentration of Hz from Hz and 2H quenching to give Hz as the gases cool. Figure 7 shows the plots of the experimental CO/H2 ratio and the CO/Hz ratio calculated at 3300 K (assuming that H

Coal 5 0'5.45 1 0 0 , ; ; ; ;

8-4.59

;

;

0-0.541

N=0.078 5-0.138

;

;

;

I

;

,

quenches to H2) vs. the O/H ratio of the coal samples. The least squares fit of the experimental values gave a line with a slope of 0.495 and a standard error of 0.0301. The calculated values gave a least squares line with a slope of 0.419 and a standard error of only 0.0026. The good agreement of the ex0.2:

0.21

-0€I

0.1

Temperature K


0.1 I

I

1

0.4

0.5

016

HZ Figure 7. Experimental CO/H2 ratio and calculatedCO/H2 ratio at 3300 K vs. O/H ratio of coals Ruby laser pyrolysis of coals with 2.6 joules per pulse. Calculated HPincludes H assumed to quench to HP. ( 0 )for experimental values. (X) for calculated values. Line A is the least squares line for the experimental results with a slope of 0.495, an intercept of -0.0534, and a standard error of 0.0301. Line B is the least squares line for the calculated results with a slope of 0.419, an intercept of -0.0231, and a standard error of 0.0026

---

Five Points

- Seven

.-4-I0

40

-445

Points

I

30

55

Percent Experimental H 2

Flgure 6. Experimental H2 volume percentage vs. H2 f centage a s hydrogen calculated at 3300 K

H volume per-

Ruby laser pyrolysis of coal samples with a pulse of 2.6 joules. Least squares line for seven points is solid. It has a slope of 0.119, an intercept of 41.16, and a standard error of 0.402. Least squares line for five points is broken. It has a slope of 0.271, an intercept of 34.07, and a standard error of 0.143

Figure 8. Neodymium laser pyrogram of coal No. 4 Experimental conditions: 1.8-joule pulse. Porapak Q column with helium flow rate of 30 ml/min. Temperature program of 2 min at 323 K, 8 K/rnin to 473 K, isothermal at 473 for remainder of pyrogram. X indicates when the laser is fired with time increasing to the right. Upper trace from the BlLD with lower trace from the FID. Peak A has a retention time corresponding to the retention time for benzene under these conditions ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977

393

LEGEND: Experimental Conditions: Carbosieve B column a t 373 K 30 ml/min. of Helium X is where l a s e r i f f i r e d ; time jncreases t o the l e f t .

-

Peak I d e n t i f i c a t i o n : A Hydrogen B = Carbon Monoxide C = Methane D * Water and/or Hydrogen S u l f i d e E = Acetylene

E

X

Figure 9. BILD pyrogram from ruby laser pyrolysis of coal No. 1

perimental values with the calculated line indicates that the Hz and CO relative compositions are dependent upon the relative stoichiometries of oxygen and hydrogen in the samples. The similarity of the experimental plot to the calculated plot suggests that the hydrogen and carbon monoxide produced in laser pyrolysis of coal can be used to determine the relative oxygen to hydrogen content of the coal samples. Acetylene was the third major product from laser pyrolysis of coal. The experimental acetylene concentrations and the calculated acetylene concentrations at 3300 K (assuming that C2H quenched to form acetylene) are given in Table 111. The calculated and experimental values are in good agreement. The acetylene produced was dependentupon the carbon and hydrogen content of the samples. An increase in acetylene production occurred with increases in the carbon and hydrogen contents of the coals. There is also an increase in the relative acetylene produced with increasing C/H ratio of the coal samples. These results indicate that the distribution of the plasma quenching products (H2, CO, and CzH2) correlate well to the hydrogen, oxygen, and carbon stoichiometry of the coal samples. The experimental results support the calculated equilibrium distribution approach. Acetylene was a plasma quenching product resulting from two species, one of which was the C2H precursor. Methane was formed mostly as a thermal blow-off product from less energetic degradation and devolatilization of coal. Karn et al. (18) reported formation of 52 volume percent methane from coal heated to 723 K while about 2 to 4 volume percent methane was produced in these laser pyrolysis experiments. Ruby laser pyrolysis of the San Juan coal sample with carbon, hydrogen, and sulfur analysis (C = 58.6%, H = 4.99%, S = 1.04%) produced the following volume percentages of the major product gases: 66.5% hydrogen, 19.2% carbon monoxide, and 12.9% acetylene. The acetylene yield corresponded to a C/H ratio of 1.058, which compared well with the C/H ratio of 0.978 obtained from the carbon and hydrogen composition. The experimental CO/H2 ratio gave a corresponding O/H ratio of 0.093, which would be 7.2% oxygen using the hydrogen value for comparison. Figure 8 is the pyrogram from neodymium laser pyrolysis of coal sample No. 4. The upper trace is from the BILD and lower trace from the FID. The second peak in the FID trace (the Cz hydrocarbons) had a relative peak area of 77% of the 394


total products detected. The first peak in the BILD trace is as large as the second peak because it contained hydrogen and carbon monoxide along with the methane that is detected by the FID. The third peak in the BILD trace is from carbon dioxide which the FID did not detect. The large broad peak after the fourth peak in the BILD trace is water and possibly other components that are not detected by the FID. The peak labeled A has a retention time similar to that of benzene under these experimental conditions. The BILD pyrogram shows a baseline shift with temperature programing. Since the low molecular weight gases are the predominant product of laser pyrolysis, they were chosen for the characterization studies. Figure 9 is the pyrogram of the BILD response to the products from ruby laser pyrolysis of coal No. 1. Hydrogen, carbon monoxide, and acetylene are the predominant products in this pyrogram. The concentrations of hydrogen and acetylene yield a plasma quenching temperature estimated a t 3300 K. Ristau (23)estimated the plasma quenching temperature for inorganic systems at about 3500 K. LPGC of coals has been shown to be useful for characterizing the relative carbon, hydrogen, and oxygen content of coals. The nitrogen and sulfur content could possibly also be correlated with the gaseous products, HCN and HzS, when these are separated and detected.

ACKNOWLEDGMENT The authors acknowledge the critical review of T. R. Henderson, s. H. Weissman, J. J. Miglio, and R. Clark, the technical illustrating of E. E. Goff, the technical editing of N. J. Barnett, and the typing of V. C. Gonzales.

LITERATURE CITED (1) J. Romovacek and J. Kubat, Anal. Chem., 40, 11 19 (1968). (2) R. P. Suggate, N.Z. J. Sci., 15, 601 (1972). (3) "Analysis of Fossil Fuels by Pyrolysis GLC", Chromatofacts, Fisher Scientific Co., Pittsburgh, Pa. (4) H. Luther, G. Bergmann, H. D. Engelmann, and J. Zajonitz, Chem. lng. Tech., 41(13), 743 (1969). (5) G. M. Kimber and M. D. Gray, Combust. Flame., 11, 360 (1967). (6) Y. C. Fu, Chem. lnd., 876 (July 31, 1971). (7) 9.E. Knox and F. J. Vastola, LaserFocus, 3(1), 15 (1967). (8) W. K. Joy, W. R. Ladner, and E. Pritchard, Nature (London), 217, 640 (1968). (9) W. K. Joy, W. R . Ladner, and E. Pritchard, Fuel, 49, 26 (1970). (10) A. G. Sharkey, Jr., J. L. Schultz, and R. A. Friedel, Nature (London), 202, 988 (1964).

F. S. Karn, R. A. Friedel, and A. G. Sharkey, Jr., Carbon, 5 , 25 (1967). J. L. Shultz and A. G. Sharkey, Jr., Carbon, 5 , 57 (1967). F. S. Karn and A. G. Sharkey, Jr., Fuel, 47, 193 (1968). F. S. Karn and J. M. Singer, Fuel, 47, 235 (1968). F. S. Karn, R . A. Friedel, and A. G. Sharkey, Jr., Fuel, 48, 297 (1969). F. S. Karn, R. A. Friedel, and A. G. Sharkey, Jr., Fuel, 5 1 , 113 (1972). A. G.Sharkey, Jr., A. F. Logar, and F. S. Karn, Fuel, 48, 95 (1969). F. S. Karn, R. A. Friedel, and A. G. Sharkey, Jr., Chem. lnd., No. 7 , 239 (1970). R. L. Hanson, N. E. Vanderborgh, and D. G. Brookins, Anal. Chem., 47,335 (1975). N . E. Vanderborgh, R. L. Hanson, and C. Brower, Anal. Chem., 47, 2277 (1975). J. P . Biscar, J. Chromatogr., 56, 348 (1971).

(22) R. L. Hanson, “Laser Pyrolysis Gas Chromatography and Plasma Stoichiometric Analysis”, Ph.D. dissertation, The University of New Mexico, August 1975. (23) W. T. Ristau, “Analytical Aspects of Laser-Induced Degradation of Organic and Inorganic Compounds”, Ph.D. dissertation, The University of New Mexico, June 1971.

RECEIVEDfor review September 23,1976. Accepted December 6, 1976. This work is taken from the Ph.D. dissertation of R.L.H., The University of New Mexico, 1975, who received support from the Coal Gasification Project a t The University of New Mexico.

Characterizing Skewed Chromatographic Band Broadening W. W. Yau E. 1. du Pont de Nemours and Company, Central Research and Development Department, Wilmington, Del. 19898

A new method of extracting column band broadening parameters from a skewed and noisy chromatographicpeak is derived from an exponentially modified Gaussian-peak model. The method Is more accurate than the James-Martin method which uses a simple Gaussian-peak model and Is more precise, Le., less susceptible to baseline noise, than the common statistical moments approach. Characteristicproperties of the skewed peak model are derived. These are then used to eliminate the reliance of the peak variance and skewness calculations on the 2d and 3d moments. The improvedcalculations depend only on the more stable, less noise influenced, zeroth and 1st moment calculations. The computatlonal algorithm of the method is slmpler and more straightforward than the curve fitting approaches.

A precise characterization of column band broadening in terms of peak shape parameters (variance and skewness) is needed for quantitative interpretation of chromatographic data. The information is used to evaluate and optimize column resolution and efficiency. Functionally, a chromatographic peak is simply a time distribution of the chromatographic height h ( t )at any retention time, t . The statistical moments of the peak are mathematically defined as: the zeroth order, the first order,

M O= MI =

Sornt

h ( t )dt

(1)

h ( t )dtlMo

(2)

and the higher order central moments,

a,,= im ( t - M I ) % @ )dtlMo

and

(8)

In principle, chromatographic peaks can be characterized by the moment calculations. However, in practice the precision of these calculations is poor. The baseline noise in actual, experimental chromatograms can greatly affect the precision of the higher order moments calculated by Equation 3. The commonly used James-Martin Method (1) uses the property of an assumed Gaussian peak model to calculate the peak variance. The method is based on Equation 9, 1 variance = - (area/peak height)2 2*

(9)

The variance calculated by Equation 9 uses the peak area, or the zeroth order moment, and is therefore much more precise (less influenced by noise) than that calculated as the 2d moment using Equation 3. While the precisidn of the area-height method is good, the accuracy of the method is poor. First, the method does not provide for calculation of peak skewness and, second, for highly skewed peaks, the method tends to grossly underestimate the true peak variance. Thus to improve the characterization of chromatographic peaks, a more general peak shape model which permits describing skewed peaks is needed. The exponentially modified Gaussian model (2) has been chosen in this work for this purpose. The model is justified both theoretically and experimentally since it is known that the chromatographic “Gaussian” peaks are modified exponentially by extra column effects and nonequilibrium mass transfer processes ( 3 , 4 ) .The model contains only four peak shape parameters, just one more than the simple Gaussian model. The peak contour of this model is described by the following convolute integral:

(3)

where n = 2 , 3 , 4 , . . . . The statistical moments of a peak are related to the peak shape parameters as defined by Equations 4-8. peak area = Mo

(4)

average retention time = M I

(5)

peak variance = M2

(6)

peak skew =

peak excess = M4/M22 - 3

(7)

where the four essential peak shape parameters are: A , the area, 7,the time constant of the exponential modifier, tR, the peak retention time, and u the standard deviation of the Gaussian constituent. The quantity t’ in the equation is a dummy variable of integration. In Figure 1 the shape of the convoluted peak is compared to its Gaussian constituent. As shown in the figure, the peak maximum of the convoluted peak falls on the contour of the Gaussian constituent. This property of this peak shape model can be expressed mathematically as the following: ANALYTICAL CHEMISTRY, VOL. 49, NO. 3, MARCH 1977

395

Characterization of coal by laser pyrolysis gas chromatography

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