Time-Resolved Laser Induced Degradation of Polystyrene Stanley G. Coloffl and Nicholas E. Vanderborgh2 Department of Chemistry, University of New Mexico, Albuquerque, N. M. 87106
A new method is explored for high temperature pyrolysis and rapid analysis of organic polymers. The experimental technique combined a low power continuous wave CO2 laser for sample pyrolysis and a time-of-flight mass spectrometer for rapid analysis of degradation products. The laser beam, controlled by a solenoid actuated shutter, provided pulses from 0.01 to 1 second, whereas operation with the open shutter permitted continuous heating of the sample. Accurate timing of the pulse duration was accomplished with a fast response photo-diode used to trigger the oscillograph recorder timing galvanometer. The analog output system of the time-of-flight instrument permits the time-resolved observation of selected degradation products during the lifetime of the thermal event. Laser degradation studies of polystyrene show a temperature dependency on the formation of degradation products with increased fragmentation at increasingly higher temperature.
Degradation methods are routinely used for the characterization of materials which exhibit a low vapor pressure a t usual temperatures (1, 2 ) . In particular, polymeric materials are analyzed by pyrolysis-gas chromatography, thermal-gravimetric, and differential-thermal analyses. These well characterized techniques use a variety of methods to heat the sample. Currently, there is much activity in finding heating techniques that permit rapid and reproducible thermal rise times since there is ample evidence that fragmentation patterns depend upon the heating rate ( 3 ) . Most recently, pulsed lasers have been used to degrade samples (4-6). This technique has proved especially convenient since energy deposition times can be in the order of tenths of milliseconds, or shorter. Thus, the entire thermal excursion, rise and fall times, can be of shorter duration than the most promising competitor, the Curie point apparatus ( 1 ) . Mass spectrometry offers the most definitive technique for monitoring the degradation process. Pulsed laser heating, within the high vacuum chamber of a mass spectrometer, has been used on a variety of materials. Rather extensive work has been reported by Lincoln, Vastola, and Knox (6-12). Work has been done with both normal and Present address, U.S. Environmental Protection Agency, Nat i m a l Environmental Research Center, Research Triangle Park, N.C. 27711 2 To whom correspondence should be addressed. Levy, Chromatogr. Rev., 8, 48 (1966). Madorsky, “Thermal Degradation of Organic Polymers,” Interscience Publishers, New York, N . Y . , 1964, p 4. ( 3 ) F. Farre-Rius and G . Guiochon,Anal. Chem., 40, 996 (1968). ( 4 ) W. T . Ristau and N. E. Vanderborgh, Anal. Chem.. 43, 702 (1971). (5) R . Levy, C. J. Wolf, and J. Oro, J. Chromafogr, Sci., 8, 524 (1970). ( 6 ) K . A . Lincoln, High Temperature Technology. Proceedings of the Third International Symposium, Asilomar, Calif., 1967, p 323. ( 7 ) K . A . Lincoln, Ana/. Chem., 37, 541 (1965). (8) F. J. Vastola. Amer. Chem. SOC., Div. Fuel Chem., 11 ( 4 ) , 229
(1 ) R . L. ( 2 ) S. L.
(1967). (9) F. J. Vastola, R. V . Mumma, from.. 3, 101 (1970).
and A . J. Pirone. Org. Mass Spec-
$-switched lasers at energies of 694.3 nm (ruby) and 1060 nm (Nd-glass) with pulse energies ranging upwards from 0.1 J/pulse. Much of the emphasis of this work has been to determine the composition of the materials ejected from the surface of the materials under study and to determine if ionic fragments result directly from the pyrolysis event. The use of pulsed lasers of short duration, pulse width in the range of 1 millisecond or less, precludes the possibility of time resolving the thermal event-ie., the determination of the order of appearance of the specific molecular fragments. Such information would be valuable to construct a mechanistic description of the degradation process. Two possible techniques exist for such time resolution. Using repetitive pulses on the same sample, mass spectra can be determined a t various times after the initiation of the pulse; this would be an utilization of the technique “boxcar integration.” Unfortunately, laser energy deposition is not highly reproducible but still this technique shows some promise especially to characterize surface temperatures. Alternately, a mass spectral sampling technique might be designed that is rapid compared with the pulse width of the lasing energy and multiple spectra determined during that pulse. The sole mass spectral technique that offers promise here is that of time-of-flight mass spectrometry since this procedure permits the determination of discrete spectra in the millisecond time range. Even so, TOF techniques offer little opportunity to monitor time resolved fragmentation patterns resulting from pulsed lasers since their pulse widths are in the same time range and no more than a few discrete spectra can be obtained during a single thermal event. Clearly, a longer energy deposition time is necessary. In this paper, such a study is reported. A low power COz continuous wave laser was used to give pulse widths of, approximately, 100-millisecond duration. Fragmentation patterns were then determined during the thermal rise and fall times.
EXPERIMENTAL The laser T O F mass spectrometer instrumentation used for this investigation is shown in Figure 1. A low power continuous wave COz laser, described below, was led through a focusing lens and KC1 window into the ion source region of a time-of-flight mass spectrometer. A mechanical electrical shutter mechanism was placed in the laser beam. When the shutter was closed, the beam was reflected into a beam dump. Also passing onto the shutter was the output from a He-Ne alignment laser. When the shutter was opened, the COz radiation heated the sample contained within the mass spectrometer and the alignment laser illuminated a fast response photodiode which gave an output signal which was used to time the pyrolysis event. Using the full output power of the COz laser, rapid degradation resulted; it was necessary, a t times, to attenuate this energy. This was done by masks fabricated by drilling precise 1-, 2-, and 3-mm holes in a sheet of asbestos; the attenuators were placed directly upon the KC1 window. Time-of-flight measurements were made with a Bendix Model 12-101 T O F mass spectrometer. This instrument was equipped with a Model S-14-107 ion source and an M-105-G-6 ion multi(10) €3. E. Knox and V . S. Ban, J. Chem. Phys., 5 2 , 2 4 3 (1970). (11) 6.E. KnoxandV. S. Ban, J. Chem. Phys., 52, 248 (1970)
ANALYTICAL CHEMISTRY, VOL. 45, NO. 8 , JULY 1973
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PHOTO DIODE
Table I. Degradation Fragments from Continuous Wave Laser Degradation of Polystyrene Using Ionization COz CW LASER Energy of 100 eV I Relative Relative m/ea Intensity mlea Intensity He-Ne LASER 27 10.5 77 15.1
? TURNING MIQROR
SHUTTER
=
MIRROR
n
I
TOF-MASS SPECTROMETER
39
50 51 52 "Only peaks
ION MULTIPLlER SAMPLE SUPPORT
shown.
OSCILLOSCOPE
Figure 1. Schematic of the laser time-of-flight mass spectrometric instrumentation KC1 WIND014
,
ACCELERATING
ADJUSTABLE S A W E
\ ASSEMBLY BOTTOM HEADER:
SAMPLE SUPPORT
Figure 2. Mass spectrometer ion source assembly showing top window, gate valve, and sample support apparatus (shown in
lowered position) plier. The analog output system provided electrometers for the continuous monitoring of up to four separate m / e channels. Mass spectra were recorded on either a Honeywell Model 906C Visocorder (optical galvanometer) or a Tektronix Model 545A oscilloscope. The TOF spectrometer was modified to accept a sample holder. A gate valve assembly and vacuum tight sample support assembly are shown in Figure 2. This device permitted introduction of solid samples into the mass spectrometer without breaking the high vacuum. A stainless steel cylinder welded to the bottom blank-off header aligned the sample support assembly which consisted of a threaded stainless steel rod terminated in a steel sample cup welded to the rod. Sample introduction was accomplished by lowering the threaded rod, closing the gate valve, removing the bottom header, placing the sample in the cup, and then reversing the process. Indexing marks on the threaded rod permitted reproducible positioning of the sample directly beneath the ion-source assembly of the mass spectrometer. The COz laser was of conventional design ( 1 2 ) .A gaseous mixture of helium, nitrogen, and carbon dioxide with flow rates of 10, 7, and 5 ft3/hour, respectively, was used at an operating pressure of 20 Torr. The laser cavity was pumped by a Welch Duo-Seal Model 1397 vacuum pump and was powered by a 20 kV supply, Del Electronics Corporation Model PSC 20-150. The laser power was characterized by a Coherent Radiation Laboratory Model 201 Thermopile. Maximum continuous power was 10 watts: typical beam diameters were 5 mm. This system showed long term stability of better than f O . l watt and could be utilized for output powers in the range of 1.5 to 10.0 watts. The photodiode (1N2175) was connected in series with a 1.5-V battery and a 2-Mohm resistor. The timing circuit monitored the voltage drop across the current limiting resistor. Residual air present in the mass spectrometer offered convenient mass markers at m l e of 14 ( N + or N z 2 + ) , 16 (0+or 0 2 2 + ) , (12) C K N Patel. S o A r n e r , 219, 22, August 1968
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ANALYTICAL CHEMISTRY, VOL. 45, NO. 8, JULY 1973
13.0 13.0 23.2 10.5
78 91 103 104
24.4 29.0 37.2
100.0
larger than 10% of the parent peak, m/e = 104, are
18 (HzO+), 28 (Nz+), 32 (Oz+) and 40 ( A + ) as well as residual mercury, from the diffusion pumps, a t m / e of 196, 198, 199, 200, 201, 202, and 204. In addition to these markers, sulfurhexafluoride and perfluorokerosene were used to calibrate the mass scale. Three separate m / e channels were monitored during the lifetime of the laser event. The fourth recording channel was utilized to monitor the output from the photodiode circuit. Initially, a sample was placed in the mass spectrometer and illuminated under continuous conditions (shutter open) with attenuated power. The gate marker of each of the. monitoring electrometers was set using the oscilloscope display to position the gate directly on the desired mass peak. Mass peaks were identified from a previously recorded spectrum. Gate widths of 50 nanoseconds were utilized to ensure no interference occurred from adjacent peaks. Precise centering of the gate marker was effected by adjusting each separate galvanometer for maximum signal during the continuous radiation. Once the instrument was calibrated for specific fragment monitoring, the laser shutter was closed and a new sample was positioned. The camera shutter was set for the desired pulse duration and remotely triggered. Recorder chart speed was typically 2.5 in./sec. The degradations reported here was effected with ionization energies of 100 eV and electrometer sensitivities of 100 nA. Polystyrene samples were supplied by the Cadillac Plastic and Chemical Company, Detroit, Mich., as CADCO polystyrene No. 7. Specifications for this high purity polymer indicated a molecular weight of 230,000 and a density of 1.046 grams/cm3.
RESULTS AND DISCUSSION Polystyrene was chosen for these studies since degradation of this compound has been investigated using several different thermal techniques (2, 13, 14). Table I shows mass spectral results obtained by continuous radiation in the calibration experiments. Similar results are found in conventional mass spectrometric analyses. The principal fragments are a t m/e of 39, 51, 78, 91, and the most intense peak, the monomer a t 104. The fragment a t m / e 91 is considered to be toluene or the resonance-stabilized tropylium ion C7+ ( 1 5 ) . Polystyrene monomer, the monomer derived benzene fragment, and the ethylene fragment were selected for time resolved degradation studies. Not shown in Table I are the mass fragments corresponding to the dimer ( m / e = 220) and trimer (312). These as well as peaks a t 279 and 296 appear as low intensity peaks in the degradation spectra. Staudinger et d . (14) suggested a mechanism of chain scissions to account for the dimer and trimer degradation products. Degradation must begin by breaking either a C-H bond or a C-C bond; evidence is that C-H bonds are of lower stability and, thus, initially, hydrogen should be abstracted from a tertiary carbon and be transferred to another tertiary site. The smaller fragmentation products, of lower molecular weight than styrene, are typical of those found in mass spectral analyses of styrene. These results, therefore, suggest that laser induced thermal degradation of the (13) S. R . Urzendowski, AFWLTR-69-46, Air Force Weapons Laboratory, Kirtland AFB, New Mexico, 1969. (14) H. Staudinger, H. M . Brunner, and K. Frey, Ber. Deuf. Chem. Ges. 73,62, 241 (1929). (15) F . W . McLafferty, "Interpretation of Mass Spectra," W . A . Benjamin, Inc., New Y o r k . N . Y . . 1966, p 9 2 .
Table II. Rate Data Derived from Time Resolved Laser Pyrolysis (pyrogram shown in Figure 3) of Polystyrene Time (after pulse inittation, sec)
Peak intensity. /
Rate, d//dt
k,(sec-')
I
3.29 SECOND PULSE
1
TIHE
Temperature,a"C
Monomer fragment 0.09 0.1 1 0.12 0.13 0.14
6 22 35 48 56
780 1560 1300 1300 650
130 71 37 27
12
857 845 832 825 809
493 533 640 800 457
99 48 28 26 12
852 837 846 825 81 0
Benzene fragment 0.11 0.12 0.14
23 31 0.16 37 Estimated accuracv f20 "C.
0.15
a
5
11
sample surface pumps predominately styrene into the ionsource region together with a small portion of styrene dimer and trimer. These then undergo additional fragmentation in the ion source. Ions have been observed as a direct consequence of laser deposition into a sample (9, 16). These experiments used a pulsed laser source and positioned the sample within the drift tube of a T O F mass spectrometer. A mass spectrogram occurring with the ion-source switched off can result only from direct laser induced ionization, or from heterolytic bond scission. Experimental conditions in the present study were somewhat different. The sample was placed 3 cm below the ion source and ions produced directly from the thermal event could be detected but only a t the highest laser power and the highest sensitivity. This is due to the physical geometry of this instrumental design. Most of the ions should be trapped on the metallic surfaces (at ground potential) within the ion source before reaching the accelerating grids. Thus, the greatest majorit y of degraded sample would be deposited and not detected. However, these studies, highest laser power and no ionization voltage, did definitely show both styrene ( m l e of 104) and benzene ( m l e of 78) and thus give direct evidence that thermal degradation of styrene is affected a t the sample. Since these species were detectable, they, of necessity, were positicely charged ionic fragments. The monomer (styrene), the six carbon and the two carbon fragments were selected for time resolved studies; m l e channels 104, 78, and 27 were monitored. For convenience, these fragments are denoted as the styrene, benzene, and ethylene fragments, c8, cg, and C Z , respectively. Figures 3 and 4 show typical time-resolved results. Here time is plotted from left to right and the photodiode response is shown in the upper trace. Shortly after initiation of the laser pulse (0.29 second total duration) the styrene fragment appears; this peaks and decreases during the duration of the pulse. Next, the benzene fragment goes through a maximum and also decreases during the pulse lifetime. Similarly, the ethylene fragment reaches a maximum that decreases just prior to the end of the thermal event. Upon completion of the laser pulse, the peaks again appear for each of the monitored fragments in opposite order. These results were obtained with the ion source a t 100 eV. The different appearance times of each peak indicate the primary degradation process occurs on the sample sur(16) B E Knox, ' Molecularity and Charge States of Laser Vaporized Materials," Symposium on Thermoktnettc Effects in Pulsed Energy Deposition Albuquerque, New Mexico, October 1970
Figure 3. Time resolved mass spectra
of polystyrene
Ethylene, benzene, and styrene fragments monitored as a function of time. Upper trace shows 0.29-second laser pulse duration; laser power, 7.9 watts; total energy deposited = 2.3 joules
TIM PJLSE
Figure 4. Time resolved
mass spectra of polystyrene
Ethylene, benzene, and styrene fragments monitored during the lifetime of a 0.19-second pulse. Laser power (CW), 7.9 watts; energy deposited = 1.5 joules. Upper trace shows photodiode output
n
TIME
0.05 SECOND PLiLSE
Figure 5. Time
resolved mass spectra of polystyrene
Trace showing the intensity of the benzene and styrene fragments during a short duration pulse. Laser power (CW), 4.5 watts; total energy deposited = 0.22 joules. Upper trace shows photodiode output
face. As the sample surface is heated, the temperature rapidly rises to a point where primarily c8, m / e = 104, is evolved. As the thermal event continues, the surface temperature reaches a point where the Cg fragment begins to decompose and its intensity declines with a corresponding increase in the c6, m / e = 27, fragments. Next, c6 reaches peak intensity and with continued heating decomposes giving rise to a C Z peak which also declines upon decomposition to smaller fragments. Observations of the CI fragment a t m l e = 14 in separate experiments indicate an increase in this fragment corresponding to the decline in Cp; ANALYTICAL CHEMISTRY, VOL. 45, NO. 8, J U L Y 1973
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TIME
0 63 SECOND PULI[
fi
Figure 6. Time resolved mass spectra of polystyrene Trace showing the intensity of the ethylene, benzene, and styrene fragments during an experiment where a hole was burned through the polymer sample. Laser power (CW), 7.9 watts: total energy deposited during pulse = 5.0 joules. Upper trace (0.63-second duration) shows output from photodiode
however, the intensity of the C1 fragment was not large enough to totally account for the decline in Cz. This implies that there may be several one- and two-carbon fragments with m l e < 27 a t this point in the heating cycle. Upon completion of the laser pulse, the sample cools and first passes through the temperature region where the Cz fragment predominates, and again the intensity of this fragment rises to a broad maximum. In a similar manner, the continuing decrease in temperature gives a broad peak for the benzene component followed by a broad peak for the monomer. In the cooling phase, the peaks are broader because the rate of temperature change is much slower than during heating. Thus, a thorough picture of three principal degradation products of this material is obtained. This behavior is noted only when intense thermal fluxes are utilized. Figure 5 shows time resolved results at lower power levels. Here the intensity profile of benzene closely follows that of styrene and suggests that the benzene fragment totally results from electron bombardment (within the ion source) of the styrene that is ejected from the surface. This conclusion is consistent with the assumption that these lower power levels will not result in temperatures high enough to effect extensive degradation. The highly focused beam a t times burned a hole directly through the sample. This was evident by examination of the sample after irradiation. Figure 6 shows a typical time resolved spectra. With the onset of the laser pulse, a hole is quickly burned through the sample after which the laser beam strikes the stainless steel sample cup and reflects back into the sample. The plateau exists until the beam has deposited sufficient energy to again raise the sample temperature to effect decomposition. The CS and C6 fragments give rise to a binodal trace similar to that seen in Figures 3 and 4. However, the CZ fragment trace shows only one discernible maximum. The lack of a second maximum and the more gradual decline of C Z in this case may indicate the maximum temperature obtained on the sample was only slightly in excess of the temperature required for peak production of Cz, thus a slower decomposition rate. Even though the pulse duration was several times longer, the geometry of the hole in the sample may have precluded the beam from effectively heating the sample. These results are obtained under a continuous thermal flux which results in ever-increasing surface pyrolysis. This is different from most polymer degradation where isothermal results are obtained. The kinetics of polysty1510
ANALYTICAL CHEMISTRY, VOL. 45, N O . 8, JULY 1973
rene degradation are known to be first order and follow the functional form dI/dt = kP where I is the amount of polymer degraded, ie., the weight loss; t, is the time; k , the specific rate constant; and P, the amount of polymer present. During a pulsed thermal event, the polymer undergoes surface heating; thermal gradients exist through the sample from ambient temperature a t the holder to the higher temperature at the sample surface and across the beam diameter exposed on the sample. A strict kinetic analysis of these nonisothermal conditions is not practical; however, it is still tempting to explore the possibility of estimating surface temperatures using kinetic parameters. The traces in Figures 3, 4, 5, and 6 are a representation of the ion current of a particular component as a function of time. The slope of a time resolved trace at a particular time, is clearly the variable dlldt. By evaluating the polymer degradation rate a t various times during the thermal event, it should be possible to estimate the surface temperature by determining the temperature which corresponds to the measured rate dlldt. This analysis requires the knowledge of the specific rate parameters which have been evaluated. Solving the Arrhenius equation for temperature yields:
Ti = E / R (In A - In k i )
(2)
where Ti is temperature of the surface emanating product at the rate ki, E is the activation energy, and A is the Arrhenius factor. Urzendowski (13) using thermogravimetric analysis in vacuum a t 370-460 "C, reported the activation energy and the Arrhenius factor for polystyrene degradation to be 68.3 X lo3 caljmole and 2.3 x l O I 9 sec-I, respectively. There is no reason to conclude that these parameters are temperature independent, and they must be considered as but approximate values for other temperatures than those in the range of 370-460 "C. Surface temperature estimations were performed for several of the polystyrene degradations. Consideration of the estimated temperatures (see below) and experimental pressures (10-6 Torr) leads to the conclusion that fragment velocities are rapid enough that TOF data closely follows surface events; that is, under these experimental conditions, the transit time (from sample to ionizing region) is shorter than the sampling time of the instrument. These temperature estimates are shown in Table 11. Data indicate surface temperatures in the vicinity of 825 "C and that the temperature decreases during the lifetime of the heating cycle. The accuracy of these temperature estimates cannot be better than A20 "C; however, they do suggest a cooling trend. Conventional polymer degradations measure the total decrease in weight of the polymeric sample; this usually is the variable dlldt. Here we define separately the rate of production of two fragments, monomer and benzene. Thus, this decreasing temperature may well be an artifact which results from decreased production rates, due to increasing surface fragmentation, of these during the latter part of the heating cycle. Alternatively, these decreasing temperatures might accurately reflect the uptake of heat due to the endothermic polymer degradation. It is not possible to distinguish between these two possibilities. However, this analysis does give temperature estimates that are sensible and fit in with the proposed degradation model.
ACKNOWLEDGMENT The use of equipment and facilities of the Laser Division and Scientific Support Branch of the Air Force Weapons Laboratory, Kirtland Air Force Base, was instrumental in completing these studies. We thank Wayne Wasson and Willy Kunzler for their technical support and acknowledge A. Guenther and Keith Gilbert for their support and
encouragement of this project. The timely discussions with Kenneth Lincoln, NASA, Ames Research Center were greatly appreciated. Received for review September 25, 1972. Accepted January 29, 1973. This work was taken from the M.S. Thesis of S.G.C., The University of New Mexico, June 1972.
Estimating Precision for the Method of Standard Additions lngvar L. Larsen,l Norbert A. Hartmann,2and Jerome J. Wagner' Oregon State university, Corvallis, Ore. 97331
An estimate of the uncertainty term expected in the method of standard additions using linear regression analysis is presented. The method agrees favorably with the standard deviation for values which are not corrected for a blank as well as with the population standard error of difference for corrected samples. Analysis for zinc in an environmental sample yielded a concentration range within the expected value.
is determined by extrapolating a line to the abscissa. (If the line is not linear, often a transformation can be performed such as the conversion of light absorption to absorbance.) The best line which can be fitted to the data minimizes the sum of the squares of the vertical distance between data points and the constructed line, referred to as the line of least squares ( 4 ) . The least squares line can be described by the equation
? = M X + I The precision for an experimental process is usually determined through replication. When applied to calibration line data, however, sample replication may not only be insufficient but even misleading ( I ) . In addition, when applied to the method of standard additions, replication becomes tedious as well as time consuming. An estimated error term based upon linear least squares regression applied to the method of standard additions is presented as an approximation to the experimental error. The method of standard additions is often used to determine the concentration of an element in an interfering matrix (2). Caution, however, should be exercised when using this technique to ensure that the initial calibration of the instrument be accomplished with standards contained in as similar a matrix as the sample, both chemically and physically, in order to estimate the appropriate "blank" correction. In this way correction for light scattering due to the matrix can be approximated. Rains (3) discusses some aspects of elimination or control of interferences in atomic absorption spectrophotometry. In the method of standard additions, a small known concentration of the desired element is increasingly added to several samples of the unknown test solution, and the resulting solutions as well as an untreated one are then analyzed. The response readings from the particular instrument are then plotted linearly against the added concentrations and the amount of unknown element present School of Oceanography.
* Department of Statistics. (1) F. J. Linnig and John Mandel, Anal. Chem., 36 (13), 25A (1964). (2) 0. Menis and T. C. Rains, in "Analytical Flame Spectroscopy," R. Mavrodineanu. Ed., Philips Technical Library, Eindhoven, The Netherlands, 1970, Chap. 2, pp 47-77. (3) T. C. Rains, "Atomic Absorption Spectroscopy." ASTM STP 443, American Society for Testing and Materials, Philadelphia, Pa., 1969, pp 19-36.
where is the value predicted (absorbance) by the equation for a given value of X (concentration), M is the slope of the line, and I is the intercept of the line with the ordinate axis. Using data obtained from an analytical determination of treated and untreated samples, the slope of the least squares line can be calculated as follows: Slope, M = n Z X Y
- ZXZY
n Z X 2 - (ZX)*
(2)
where X refers to the concentration of the standard solution, Y is the linear instrument response reading, and n is the total number of readings made. Equation 1 is referred to as the regression of Y on X , X being the independent variable and Y the dependent variable. The X values are assumed to be known without error, whereas the Y values are randomly distributed about some mean Y value (5, 6). The intercept of the line with the ordinate axis is calculated as follows: Intercept, I =
7 - Mil
(3)
where Y is the arithmetic mean of the total Y readings, X is the arithmetic mean of the total X values, and A4 is the slope of the line. An estimate of the variability of the data points about the least squares line is the standard error of regression and is calculated as follows: ( 4 ) C. L. Grant, Statistics helps evaluate analytical methods. "Developments in Applied Spectroscopy," Vol. 6, Proc. of the 18th annual mid-American spectroscopy symposium, Chicago, 15-1 8 May 1967, W. K . Baer, A. J. Perkins, and E. L. Grove, Ed., Plenum Press, New York, N. Y., 1968, pp 115-26. (5) John Mandel and Frederic J. Linnig, Anal. Chem. 29, 743-749 (1957). (6) George W. Snedecor and William C. Cochran, "Statistical Methods," 6th ed., The Iowa State University Press, Ames, Iowa, 1968, 591 PP.
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