Anal. Chem. 1984, 56, 1481-1487
Seven separate GC/MS determinations were made with unlabeled casbene (Table 111), with each run representing 30-40 s. An average value of 20.53% was obtained for the ratio of (M 1)/M for unlabeled casbene us. a theoretical value of 22.48%. Table IV contains the results of five runs using deuterium labeled casbene from the enzyme preparation. After subtracting the previously determined I3C contribution of unlabeled material (20.53%) from the M + 1ion intensity of the labeled material, we found an average value of 49.86% incorporation. This value is only slightly higher than previous results obtained by using the oscillographic technique (49.1%). A different enzymatic preparation of labeled casbene gave an incorporation of 50.60 f 0.85%. Within experimental error, these two results are identical and are well within the 1-5% error generally associated with this type of isotope measurement (25).
+
LITERATURE CITED Kennett, B. H. Anal. Chem. 1987, 3 9 , 1506. Blros, F. J. Anal. Chem. 1970, 5 2 , 5357. Peele, 0. L.; Brent, D. A. Biomed. Mass Spectrom. 1978, 5 , 180. Hass, J. R.; Friesen, M. D.; Harvan, D. J.; Parker, C. E. Anal. Chem. 1978, 5 0 , 1474. Van Ness, G. F.; Solch, J. 0.; Taylor, M. L.; Tiernan, T. 0. Chemosphere 1980, 9 , 553. Lenmann, W. D.; Schulten, H. R. Angew. Chem., Int. Ed. Engl. 1877. 16, 184. Ligon, W. V., Jr. Abstracts, 27th Annual Conference on Mass Spectrometry and Allied Topics, Seattle, WA, June 3-8, 1979; p 481. Ligon, W. V., Jr. Abstracts, 28th Annual Conference on Mass Spectrometry and Allied Topics, New York, May 25-30, 1980; p 490. Snelllng, C. R., Jr.; Cook, J. C., Jr.; Milberg, R. M.: Rlnehart, K. L., Jr. Abstracts, 29th Annual Conference on Mass Spectrometry and Aliled
1481
Topics, Minneapolis, MN, May 24-29, 1981; p 602. (10) Savitzky, A.; Goiay, M. J. E. Anal. Chem. 1964, 36, 4527. (11) Snelling, C. R., Jr; Milberg, R. M.; Cook, J. C., Jr.; Rinehart, K. L., Jr. Abstracts, 28th Annual Conference on Mass Spectrometry and Aiiied Topics, New York, May 25-30, 1980; p 484. (12) Ligon, W. V., Jr. Int. J. Mass Spectrom. Ion Phys. 1982, 41, 213. (13) Mueller, P.; Rudin, D. 0. Nature (London) 1968, 217, 713. (14) Roy, G. J. Membr. Biol. 1975, 2 4 , 71. (15) Symposium on Membrane Channels Fed. R o c . , f e d . Am. Soc. Exp. Biol. 1978, 37, 2626. (16) Rinehart, K. L., Jr.; Pandey, R. C.; Moore, M. L.; Tarbox, S. R.; Snelling, C. R.; Cook. J. C.. Jr.; Milberg, R. H. I n “Peptides: Structures and Biological Functlon 1979”; Gross, E.; Meienhofer, J., Eds.; Plerce Chemical Co.: Rockford, I L 1979; pp 59-71. (17) Rlnehart, K. L., Jr.; Gaudioso. L. A.; Moore, M. L.; Pandey. R. C.; Cook, J. C., Jr.; Barber, M.; Sedgwick, R. D.; Bordoli, R. S.; Tyler, A. N.; Green, B. N. J. Am. Chem. SOC.1981, 103, 6517. (18) Rlnehart, K. L., Jr.; Cook, J. C., Jr.; Meng, H.; Olson, K. L.; Pandey, R. C. Nature (London) 1977, 269, 832. (19) Beynon, J. H. “Mass Spectrometry and Its Applications to Organic Chemistry”; Elsevier: New York, 1960; pp 294-302. (20) Caprioli, R. M. “Biomedical Applications of Mass Spectrometry”; Waller, G. R., Ed.; Wliey-Interscience: New York, 1972; pp 735-776. (21) Caprloll, R. M.; Bler, D. M. “Blomedlcal Appiicatlons of Mass Spectrometry, First Supplementary Volume”; Wailer, G. R., Dermer, 0. C., Eds.; Wiley-Interscience: New York, 1980; pp 895-925. (22) Reference 19, pp 201-203. (23) Millard, B. J. “Quantltatlve Mass Spectrometry 1979”; Heyden: London 1979; pp 62-66. (24) Robinson, D. R.; West, C. A. Blochemistry 1970, 9, 70. (25) Reference 19, p 100.
RECEIVED for review December 28,1983. Accepted March 26, 1984, This work was supported in part by a grant from the National Institute of General Medical Sciences (GM27029). This report is taken in part from the Ph.D. thesis of C. R. Snelling, Jr., University of Illinois, Urbana, 1981.
Supercritical Fluids as Spectroscopic Solvents for Thermooptical Absorption Measurements R. A. Leach and J. M. Harris* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
Dense gases near the liquid-vapor critical point are shown to possess advantageous properties for use as solvents In thermoopticai absorption measurements. The sensltivlties of thermal lens and photothermal deflection spectroscopies have been studied in carbon dioxide near its crlticai point as a function of temperature and density. A 100-fold sensitivity Increase, relative to measurements in carbon tetrachloride, has been achieved for both thermoopticai methods tested. The theory of critical point sensitivity enhancement is developed by using P-Y-T data on C02 from the literature. Experimental results obtained are found to closely follow theoretical predictions.
A new class of sensitive spectrophotometric techniques has been developed in which the heat produced by nonradiative decay of excited species acts to modify the optical properties of the sample (I). These thermooptical absorption techniques can be classified by their spatial temperature distribution and corresponding refractive index change, which determines the particular method of detecting the heat produced in the sample. For example, if the sample if uniformly heated by a large-area excitation beam, the change in optical path can be detected interferometrically (2). When a focused laser 0003-2700/84/0356-1481$01.50/0
beam is used for excitation, the resulting temperature gradient produces a lenslike optical element which can be measured by its effect on the divergence of the laser beam (3,4). A periodic temperature distribution can be generated by an excitation interference pattern and probed as a thermal transmission grating by diffraction of a laser beam into a detector (5,6). Finally, by creating a thermal gradient in a sample with optical excitation at an interface, the resulting thermal prism can be detected by its deflection of a laser beam (7). Unlike conventional transmission or reflection measurements, the sensitivity of these thermoopticalmethods depends on the power of the radiation used for excitation and the thermophysical properties of the sample. Solvents which exhibit a large change in refractive index with temperature, dn/dT, are advantageous since a given increase in temperature produces a large change in optical path. For continuous wave (CW) excitation, solvents of smaller thermal conductivity produce larger temperature gradients at steady state, where the rate of heating is balanced by the rate of thermal diffusion. Supercritical fluids or dense gases are a class of solvents which are generating considerable interest and applications in analytical chemistry. As mobile phases for chromatography, supercritical fluids provide greater chromatographicefficiency than normal liquids due to their larger diffusion coefficients, 0 1984 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
(b.)
Kr+
t
A
\
LI D - S h
HeNe Oven
Flgure 1. Optical diagrams of (a) thermal lens instrument and (b) photothermal deflection instrument: L, L1, and L2 are plano-convex lenses, BS in a beam splitter, Sh is an electronic shutter, S Is the sample, P is a pinhole, PD is a photodiode, and PSD is a posltionsensitive detector. while allowing separation of higher molecular weight compounds than gas chromatography due to their liquidlike densities (8-10). Supercritical fluids appear to be better suited to the mass spectrometer/chromatographinterface than liquid mobile phases (11). Furthermore, since many dense gases having favorable critical conditions (small critical pressure, critical temperature close to ambient) are simple molecules, they offer useful spectroscopic windows in the ultraviolet and infrared spectral regions. The recently accomplished interface between the supercritical fluid chromatograph and the Fourier transform infrared spectrometer (12) has demonstrated the suitability of supercritical COz as a solvent for infrared spectroscopic detection. In this work, the sensitivity of thermooptical absorption measurements, made in a solvent near its critical point, is studied. Since the coefficient of thermal expansion of a fluid diverges at its critical point, the resulting change in refractive index with temperature, dnldT, becomes unbounded. This results in enormous optical sensitivity to temperature gradients and gives rise to the phenomenon of critical poiht opalescence commonly demonstrated in lectures on states of matter (13). The analytical sensitivity of thermooptical absorbance measurements should therefore be enhanced when the solvent in which the sample is dissolved is near its critical point, as a subcritical liquid or supercritical fluid. The concept of critical point sensitivity enhancement is tested in this work with two thermooptical techniques, photothermal deflection and the thermal lens effect. The theory of the enhancement is developed by using the thermal lens effect as the primary example and then extended to other thermooptical methods. the predictions of the theory are verified experimentally. EXPERIMENTAL SECTION Thermal Lens Experiment. The ion laser based, thermal lens calorimeter used in this work, shown schematically in Figure la, differs somewhat from that described previously (14). A krypton ion laser (Coherent, Model 90-K) was operated at 568.2 nm under light regulation. A 50-cm focal length, plano-convex lens focused the beam to a waist where an electronic shutter (Uniblitz,Model 225L2A) blocked and unblocked the beam before the sample cell. The intensity of the beam center was measured at a distance of 12 m beyond the sample cell by using a 5.1 mm diameter silicon photodiode (Silicon Detector Corp., Model SD200-12-12-041)wired for photovoltaic response. Shutter control and data collection were accomplished by using a DEC LSI-11 based microcomputer system (Terak, Model 8510A) with a
multiple channel, ADC interface (Data Translation, Model DT1761). Typically 200 intensity points were sampled during the thermal lens traqsient at a rate of 100 Hz. At each experimental condition, four data sets consisting of 15 transients were ingathered. Only the initial, I&), and steady-state, tensities were used to determine the relative change in far-field spot size. These intensities were estimated by a modified (15) Savitsky quadratic fit (16) of the first and last 15-pointsegments of each data set. Samples were contained in a stainless steel high-pressure cell (17)fitted with three 6.4 mm thick sapphire windows resulting in an optical path of 6.5 cm and a volume of 25.5 mL. While this path length was too long for observing ideal thin thermal lens behavior, the cell was already available for these first experiments, having been constructed previously for other unrelated highpressure spectroscopy. During thermooptical experiments, the cell was placed in a thermostated oven constructed of 1.9 cm thick plywood and equipped with glass microscope slides for windows. The oven temperature was regulated to within *0.05 "C by using a 100-W tungsten lamp, a proportional control circuit built from a modified published design (18), and a fan for air circulation. Samples were prepared individually by pipetting a volume of Sudan red 7B (Pfaltz and Bauer) stock solution, 1.79 mM in acetone, into the open, high-pressurecell and allowing the acetone to evaporate. The cell was closed and connected to a high-pressure gas manifold and purged with COzgas (U.S. Welding, 99%). The cell was then cooled a few degrees below ambient and filled with liquid COz by slow distillation from the supply cylinder. This procedure prevented the transfer of any particulate contamination from the cylinder to the sample. The cell was then sealed off and removed from the manifold; the mass of COztransferred into the cell was measured with hO.1 g precision by using a top loading balance. Solvation of the dye by liquid COz was verified by obtaining the visible absorbance spectrum of the solution at room temperature with a Cary 17D spectrophotometer. The strong absorbance of this dye in the blue region of the visible spectrum allowed a spectrophotometriccheck on the dye concentrationwhile providing a weakly absorbing sample for thermooptical measurements at the laser wavelength. The fiied cell was placed in the oven, the temperature of which was initially regulated at the low end of the range of interest. After aligning the cell in the optical path, the cell was allowed 1 h or more to reach thermal equilibrium before data collection was begun. Following data collection, the oven temperature was increased and the procedure repeated. Photothermal Deflection Experiments. The krypton ion laser pumped photothermd deflection experimentwas constructed by modification of the above apparatus, as shown in Figure lb. The solid sample was mounted inside the three-window highpressure cell described above. The krypton ion laser beam, X = 568.2 nm, was aligned to strike the center of the sample normal to its surface. A larger power density was achieved by focusing the radiation to a waist located at the sample surface with a 50-cm focal length, plano-convex lens, L1. The power of the pumping radiation at the sample was reduced to a maximum of 1.4 mW by use of the front surface reflected beam from a wedged beam splitter. Chopping the pump laser beam with an electronic shutter caused the formation of a periodic thermal gradient at the sample surface which was probed with a He:Ne laser (Spectra Physics, Model 124B) operated at 632.8 nm. A 30-cm focal length lens, L2 focused the probe beam to a waist located at the center of the sample. The distance between the probe beam and the surface of the sample was varied by translating the sample cell normal to the propagation direction of the probe beam. The sample surface was typically located a distance of 1mm from the center of the probe beam producing a reasonably fast and sensitive response without severe distortion of the beam profile by diffraction. The refraction of the probe beam by the thermal gradient was monitored 1.3 m beyond the sample with a one-dimensional, position-sensitivedetector (Hamamatsu, S1545) having a 1 X 12 mm active area. A differential current follower provided an output voltage which responded linearly to the probe beam position on the long axis of the detector. Voltage changes due to the timedependent deflection of the probe beam were digitized at a 20-Hz
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8,JULY 1984
rate for a period of 5 s. Typically, four sets of five transients were gathered and averaged at each experimental condition. The initial and steady-state beam positions were determined by using 15point quadratic fits to the two ends of the voltage transients, as described for thermal lens data treatment. The sample used for photothermal deflection studies was a 0.13 mm X 1.5 cm X 0.5 cm piece of brass shim stock. The surface was cleaned and roughened with an aqueous slurry of carborundum powder and rinsed with deionized water and acetone in an ultrasonic bath to remove particulates and oils. A diffuse reflectance spectrum of the sample, acquired on a Cary 17D spectrophotometer, indicated a reflectance at the pumping laser wavelength of R = 0.89 relative to a MgO reference. The sample was fixed firmly at its edge to an aluminum support inside the high-pressure cell. The cell was sealed, purged, and loaded with a measured mass of carbon dioxide in the same manner as the thermal lens experiments. Theoretical Methods. Pressure-volume-temperature data for COz in the region of the critical point were compiled from several sources (19-22). These data, tabulated as isotherms, were transformed into a family of isobaric curves for this work. This transformation was achieved by first fitting each isotherm (pressure vs. density) to a fifth-order polynomial in reduced density, pr = p / p c , where pc is the critical density. The maximum deviation between the data and polynomial fits was 2%. The fitted isotherms allowed interpolation between data points so that density and temperature could be evaluated at constant pressure, producing a set of isobars (temperature vs. density), which were interpolated by using a fifth-order polynomial to achieve a constant spacing of points along the density axis. For conditions along the liquid-vapor coexistence line, the liquid density vs. temperature relationship was modeled according to the following empirical equation (21): p1
= 0.51413 - 1.479
X
10-3T
+ O.12613(Tc- T)0.357 (1)
where the temperature, T, is given in OC, the critical temperature ~. of COz,T, = 31.04 "C,and the liquid density, pI, is in g ~ m - For each isobar, the derivative of temperature with respect to density, dT/dp, was obtained by using a nine-point, cubic-polynomial digital filter (16). These derivative values were inverted and then converted to a derivative of refractive index with temperature, (dn/dT), through the Lorentz-Lorenz relation (23). Since experiments in this study were carried out under constant volume conditions, it was instructive to present the final theoretical predictions as a family of isochores, (dnldT) vs. temperature. Thermal conductivity data for COzin the vicinity of the critical point have also been reported in the literature (24). The original data in the form of eight isotherms, thermal conductivity vs. density, were interpolated by using a cubic spline at a series of densities. This allowed transformation of the data to a series of isochores, thermal conductivity vs. temperature. The data were interpolated on a log-log scale with a cubic spline to obtain values of thermal conductivity for the same conditions as the (dnldT) vs. temperature results, described above.
THEORY Thermal Lens Absorption Measurements. A thermal lens is formed in a sample by absorption of radiation from a laser beam having radially symmetric, Gaussian intensity distribution. Nonradiative relaxation of the excited molecules results in the greatest heating of the sample a t the center of the beam, producing a temperature distribution which maximizes a t the beam center. In liquids which expand upon heating, this temperature gradient reduces the optical path a t the beam center, acting as a diverging lens. Under conditions of continuous illumination, this thermal lens reaches a steady-state strength given by the reciprocal focal length (3):
2.303P(d n / dT)A (2) akw2 where P and w are the laser power and spot size at the sample, d n l d T is the change in refractive index with temperature, A is the sample absorbance, and k is the thermal conductivity. l/f(m) =
1483
0.8
0.7
0.3 0.2 22
24
26
28
30 32 3$ 36 Tempuakua, C
30
40
42
Flgure 2. Density vs. temperature diagram for C02 near the critical point. Maximum and minimum pressure isobars considered in this work are labeled, in atmospheres, along wlth the crkical isobar, P,. Adjacent isobars differ by 2.2 atm. The heavy line indicates the density and pressure values as the temperature of a constant-volume sample is raised, where the average density is 0.55 g ~ m - ~ .
The effect of the thermal lens on the beam is generally measured as a relative change in the beam-center intensity, hlb,/Ib,, in the far field. This relative change optimizes for weak absorbance measurements when the sample is located 31/2Z, beyond the beam waist where (25, 26)
Mbc/Ibc = 2.303EA + (2.303EA)2/2
(3)
where E is the sensitivity enhancement relative to a transmission measurement (26) E = -P(dn/dT)/l.SlXk (4) and the confocal distance is 2, = r w O 2 / X ,where wo is the spot size of the beam waist and X is the laser wavelength. The above relationship indicates that the sensitivity of a thermal lens measurement depends on the choice of sample matrix or solvent, the thermooptical properties of which determine the enhancement per unit laser power. Over the range of typical spectroscopic solvents, one observes significant differences in dn/dT and k (26) leading to large differences in thermal lens sensitivity. For example, taking as the laser wavelength X = 568.2 nm, the enhancement per unit laser power in milliwatts, E / P , for thermal lens measurements in carbon tetrachloride is predicted to be 5.2 mW-' while E / P for an aqueous sample is predicted to be 0.15 mW-l. Thermooptical Properties near the Critical Point, The anomalously rapid changes in density, compressibility, specific heat, and thermal conductivity which occur near the liquid-gas critical point of a material (27) should greatly influence the properties of the material relevant to thermooptical absorption measurements. Due to the rather mild critical conditions of COZ (T, = 31.04 "C, P, = 72.85 atm, and pc = 0.468 g ~ m - ~ ) , the critical point behavior of this material has been the subject of considerable investigation (19-22, 24) and is therefore chosen for this work. While the thermooptical properties of only COPwill be explicitly derived, expressing the thermodynamic parameters as reduced variables, pr = p/p,, P, = PIPc, and T, = TIT,, should allow the prediction of the thermooptical behavior of any fluid near its critical point, according to the principle of corresponding states (27). Since the change in refractive index with temperature used to predict the thermooptical sensitivity is related to the volume expansion coefficient, P-V-T data on COz compiled from the above sources have been presented in Figure 2 as a series of isobars showing density as a function of temperature near the critical point. The liquid-vapor coexistence line is also included, which describes the system below the critical tem-
1484
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
350
-
300
-
200
-
150
-
0.10
0.08
u ..\ c
0.06 h
B
\
0.04 I
0.02 22
0.00
24
28
28
30 32 36 Temperaturc,34T
38
40
42
Flgure 4. Theoretical thermal lens enhancement per milliwatt of laser
power for C02 near its critical point. Values are shown for liquid-vapor Coexistence line (left) and 12 isochores. AdJacent isochores differ by 0.025 g ~ m - ~ .
Flgure 3. Thermooptical properties of CO, near the critical point: (a)
(dnldT) vs. temperature for the coexistence line and 12 isochores; (b) thermal conductivity vs. temperature for the coexistence line and the same 12 isochores. I n both graphs, the density increment between adjacent isochores is 0.025 g ~ m - ~ .
perature. The slope of an isobar at a particular temperature may be evaluated to determine dp/dT, which is used to predict d n / d T by the Lorentz-Lorenz relation (23):
(n2- l ) / ( n 2 + 2) = L p
(5)
where L is a constant empirically determined for COZ to be L = 0.150 f 0.001 g-' cm3within the temperature and density range shown in Figure 2. By solving eq 5 for n and differentiating with respect to T, the change in refractive index with temperature can be expressed as d n / d T = (3L/2)(1
+ 2Lp)-lJ2(l - L ~ ) - ~ / ~ ( d p / d r )(6)
Experiments were carried out in this study by using a constant-volume sample cell, where the temperature ranged from below to above T, and the sample densities were above pc. The heavy line in Figure 2 marks the course of a typical experiment, where measurements below the critical temperature are being made in the denser, liquid phase. Moving along the liquid-vapor coexistence line, the liquid density decreases as the temperature is raised until it reaches the average density of solvent within the cell, at which point the sample becomes a single phase of constant density. Further increases in temperature result in no further change in density, thus defining an isochore of the system. Conforming to constant-volume experimental conditions, dn f d T was evaluated by using eq 6 and the slope of the isobars, dp f dT, in Figure 2 and plotted as a function of temperature for the coexistence line and several isochores in Figure 3a. As expected, the magnitude of dn f d T maximizes in the region of
the critical point and decreases at higher temperatures. The sharp border between the rising and falling portions of each curve corresponds to the transfer from liquid-vapor equilibrium to a single dense gas phase. In order to complete the prediction of thermal lens behavior near the critical point, values of dn/dT must be divided by the thermal conductivity to obtain the enhancement. Thermal conductivity data for COZ in absence of convection have been reported (24) and are presented in Figure 3b for the coexistence region and several isochores as above. Thermal conductivity, like dn/dT, diverges at the critical point (24) and is not a smooth function of temperature at the boundary between two-phase and single-phase systems. By substitution of the data in Figure 3a, b into eq 4 and use of a laser wavelength X = 568.2 nm, the enhancement of thermal lens measurements relative to transmission is predicted in Figure 4 for COP near its critical point. Along the coexistence line approaching the critical point, the ratio of dn/dT and k increases but remains finite. Along the isochores several degrees above the critical temperature the predicted enhancement is enormous due to the larger width in temperature of the dn/dT anomaly as shown in Figure 3. The enhancements predicted are several orders of magnitude larger than for normal liquids, even for densities and temperatures well away from critical conditions. Uncertainties in these results arise from several sources. Values for d n / d T above 0.1 are subject to numerical errors in calculating the large derivatives and to determinate errors from lack of fit to the original data. Uncertainties in the thermal conductivity in the vicinity of the critical point arise from a lack of adequate data for reduced temperatures T, 6 1.16 where K is rapidly changing. Consequently, near the critical density where E / P > 250 mW-l the results plotted in Figure 4 should be considered only a qualitative prediction. For values of E / P < 250 mW-l, relative uncertainties in the reported values are not greater than 10%. Photothermal Deflection Spectroscopy. The gain in sensitivity by use of fluids near their critical points as solvents for thermal lens measurements should equivalently apply to all thermooptical methods. As a second example, the use of COPas a medium for photothermal deflection measurements was also studied in this work. In this technique (7), optical energy absorbed at a sample surface forms a thermal gradient in a fluid medium (liquid or gas) in contact with the sample. Since the source of heating is localized at the sample surface, the temperature increase is greatest a t the sample-fluid interface and decreases with distance away from the sample as
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
thermal diffusion carries the heat into the bulk of the fluid. The temperature gradient formed modifies the refractive index of the fluid. The resulting refractive index gradient may be conveniently monitored by passing a probe laser beam through the gradient, parallel to the sample surface and measuring the deflection of the beam. The magnitude of the deflection angle in radians, 6,depends (7) on the length of the temperature gradient through which the probe beam passes, 1, the magnitude of that gradient, dT/dx, and variation of refractive index with temperature, dnldT, analogous to a thermal lens:
4 = Z(dn/dT)(dT/dx)/n
(7)
60
has been derived (28). When the excitation radiation is chopped sufficiently slowly that the thermal diffusion length is much longer than the probe beam spot size, then the temperature gradient simplifies to the form (dT/dx) = BP(1 - 10-A)/k
(8)
where P is the power of the excitation beam, A is the sample absorbance, k is thermal conductivity, and B is a collection of terms which includes the spot sizes of the two beams and the distance between the probe beam and the sample surface. If the optical geometry is fixed, B is a constant. The displacement of the beam, Ax, some distance 2 from the sample is proportional to the deflection angle Ax = 2 sin 4 24 (9)
=
for sufficiently large values of 2, 2 >> Z,, and small values of 6. Substituting the temperature gradient from eq 8 into eq 7, one predicts that the magnitude of the photothermal deflection, Ax, will depend on the ratio (dn/dT)/k, as does the change in beam-center intensity in thermal lens measurements. Several sensitive measurements of surface optical absorption have been demonstrated with photothermal deflection spectroscopy utilizing air as the contacting fluid medium. Recently, Fournier et al. (29) took advantage of the greater (dn/dT)/k afforded by a liquid surrounding medium and immersed a solid sample in CC14for a photothermal deflection measurement. Use of a fluid near its critical point as the contacting medium should produce additional, significant sensitivity gains as described above.
RESULTS Thermal lens measurements were made over a range of temperatures extending from below to above the critical point, for three average densities of COz. The AIh/lb,results were corrected for differences in sample concentration due to volume changes of the liquid phase along the liquid-vapor coexistence line. The absorbance of the sample at 505 nm was first measured spectrophotometrically at room temperature (T 22 "C) for each average density of COz and used to predict the much weaker absorbance at 568.2 nm from the ratio of molar absorptivities at the two wavelengths. Assuming the solute is significantly more soluble in the denser liquid phase, the absorbance of the liquid phase at a given temperature, T, along the coexistence line is given by AT
where
Vl,T
= A22Vl,22/Vl,T
(10)
is the volume of the liquid phase given by VI,T
Vcbav
-Pg)/h
- Pg)
(11)
where V, is the volume of the cell, pav, pg, and p1 are the average, gaseous, and liquid densities, respectively. The latter two values are predicted by formulas (21),as in eq 1. The sample absorbances, thus determined, were used with the AIbc/zbc results, eq 3, and the measured laser power to
I
0 0.54 0
500
* 0.68
0
f 0.74
0
+ t+
+ +
*
+
0 *tQo 25
*
*
20 I 20
o
* * 0
+ t
Q
+t$+
30!
A complete expression for the temperature gradient, dTldx,
1485
22
24
28
28
30
32
Ternperdure, 'C
34
36
J
38
Flgure 5. Measured thermal lens enhancement per milliwatt of laser power for C 0 2 near its critical point. Left axis shows the measured enhancements; right axis indicates values corrected for the sample path length used. Symbols correspond to different average densities of CO,, given in g ~ m - ~ .
determine E/P, the thermal lens enhancement per unit laser power plotted in Figure 5. For comparison, thermal lens measurements were performed with the sample cell on a carbon tetrachloride solution of Sudan red. The resulting enhancement per unit laser power for CC14was found to be EIP = 0.56 (f0.02) mW-'. The largest sensitivity observed in COz at a density of 0.54 g and a temperature of 29 "C was EIP = 53 (f2) mW-l. These results indicate nearly a 100-fold sensitivity improvement for C02 over CClk The value of EIP measured for CC14 is smaller than the 5.2-mW-l prediction by eq 4, which is an expected consequence of using a sample cell having a path length longer than the confocal distance (30). The relative loss of enhancement should, as a result, be constant since the same cell and focusing optics were used for all experiments. The ratio of predicted to measured EIP for CCll can, therefore, be used as a scaling factor for the observed COz response to obtain a thin sample sensitivity estimate as shown in Figure 5, which can be compared with the purely theoretical results of Figure 4. The comparison between these figures is favorable both in the shape and in magnitude of response. The sensitivity tracks a common liquid-vapor coexistence line until the breaks in the curves where the samples become a single phase. The closer the sample approaches critical conditions in density and temperature, the greater the observed sensitivity. Two sources of uncertainty influence the results reported in Figure 5. First, the volume correction applied to the data was based on the assumption that the total amount of solute was dissolved in the liquid phase. As the densities of the liquid and vapor phases approach one another as the temperature is increased (see Figure 2), the relative solubilities must become equal leading to the distribution of solute in both phases. Due to the steepness of the liquid-vapor coexistence line near the critical isobar in Figure 2, one would expect this phenomenon to influence only a small temperature range in the data. A second possible source of error is the variation in the shape of the absorption spectrum of the solute in response to changes in the density of the solvent (32). This error would not significantly affect the lower temperature portion of the data where the liquid density changes are small, which is reflected in the consistency of the three sets of data in this region. The thermal lens results in the isochore region could, however, include a contribution from changes in the sample absorptivity which has not been taken into account. To eliminate these possible sources of error which depend on the distribution and spectral behavior of the solute, analogous experiments using photothermal deflection were carried out. In this case, a solid sample could be used, the
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
3.0
1
A 0.52 0.56 0.61
-6007
0.65
-400
c B
- 300
fa 0
- zoo H Y
0.5
- loo
1
0.0 1 20
22
24
26
28 30 52 Temperature, 'C
34
58
$-
JO
58
Figure 6. Photothermal deflection results in COP near its crlticai point. The right has been scaled by using a measurement in CCI, to predict the thermal lens performance. Symbols correspond to different average densities of COPgiven in g ~ m - ~ .
location of which in the dense liquid phase could be verified visually. Furthermore, the optical absorption properties of the metal sample should not be sensitive to the temperature or density of its surroundings. The photothermal deflection results are plotted in Figure 6. The corresponding value of the photothermal deflection signal made with CC1, is 0.30 (*0.005), in the same arbitrary units as the left axis of Figure 6. For the lowest density of COz used, p = 0.52 g ~ m - a~ , greater than 100-fold increase in sensitivity relative to CCll was realized at a temperature of 31.5 OC. The ratio of photothermal deflection result for CC14 and its E / P value from eq 4 were used as a scaling factor to estimate the thermal lens sensitivity for C02 from the photothermal deflection data. These predictions, also shown in Figure 6, compare vary favorably with the theoretical results in Figure 4. Any uncertainty in the photothermal deflection data would be expected to arise from differences in optical alignment. Such errors would be apparent along the liquidvapor coexistence line where at different filling densities one would expect to observe the same enhancement in a liquid phase, the properties of which depend only on temperature. The variations from one filling density to the next in this region appear to be small, within the statistical fluctuations of the individual data points.
DISCUSSION For both thermal lens and photothermal deflection measurements, use of a solvent near its critical point produces significant increases in sensitivity relative to normal liquids. There are a t least two analytical situations where such capability could be immediately useful. For thermal lens detection of samples in small volumes where path length must be sacrificed because of the divergence of the focused beam (32),the resulting loss of concentration sensitivity could potentially be regained by using solvents near their critical point. This would be an important benefit for ultralow-volume, thermal lens detection in supercritical fluid chromatography, for example. A second possible benefit of critical point sensitivity enhancement would be a reduction in the laser power needed to achieve trace level detection. Lowering the laser power requirements from hundreds of milliwatts to a few milliwatts reduces the cost of the instrumentation and opens up new spectral regions in the UV and IR to thermooptical methods, where high-powered CW laser sources have not been available. The ultimate detection limits achievable near the critical point of a solvent will not only depend on sensitivity but also on noise sources present in these systems. Neglecting mi-
croscopic density fluctuations which lead to opalescence at the critical point, rendering this region inaccessible to thermooptical measurements, the largest source of noise appears to be residual, macroscopic density gradients present well above the critical density. Sources of these gradients might include inhomogeneous temperature control of the sample, Joule-Thompson cooling at very slow leaks in the highpressure system, and convection (33)when the sample absorbs a significant amount of laser power. Gravitationaleffeds have also been shown to produce large density gradients several degrees away from T,(34). The effect of these slowly varying, density gradients is to deflect the laser beam passing through the fluid, corrupting the optical alignment of the system. A sample-related, slow source of signal drift was observed in these experiments and found, as expected, to be proportional to thermooptical sensitivity which limited the maximum enhancement which could be explored. The severity of this drift was probably aggravated in this study by the nature of the data collection process, where the temperature of the cell was changed approximately each hour. In addition, the long path length of the cell increased the sensitivity to drift since the angle of beam deflection is proportional to interaction length as indicated by eq 7. The tedious nature of the batch-style sample processing and the resulting uncertainties in concentration due to carry-over and loss precluded a thorough assessment of detection limits and noise sources in this work. To overcome these limitations, research is being directed toward the development of a flow injection system to conveniently and reproducibly introduce samples into high-pressure solvents (35). Such a system will allow the use of a small-volumeflow cell which should improve the temperature stability of the sample compared to the much larger volume, static cell used here. These developments could turn the observed critical point sensitivity enhancement into improved thermoopticaldetection capability suitable for lower power laser excitation.
ACKNOWLEDGMENT We are indebted to Tomas Hirschfeld for his original suggestion to use fluids near their critical point in thermal lens measurements. The generous loan of a high-pressurecell by Edward M. Eyring is also acknowledged. Registry No. COz, 124-38-9. LITERATURE CITED (1) (2) (3) (4) (5) (8) (7) (8)
(9) (10) (11) (12) (13) (14) (15) (16)
(17) (18)
(19) (20) (21) (22) (23)
Hordvik, A. Appl. Opt. 1973, 76, 2827-2833. Woodruff, S. D.; Yeung, E. S. Anal. Chem. 1982, 5 4 , 1174-1178. Whinnery, J. R. Acc. Chem. Res. 1974, 7 , 225-231. Harris, J. M.; Dovlchl, N. J. Anal. Chem. 1980, 5 2 , 695A-706A. Pelletler, M. J.; Thorsheim, H. R.; Harris, J. M. Anal. Chem. 1982, 5 4 , 239-242. Pelletler, M. J.; Harris, J. M. Anal. Chem. 1983, 55, 1537-1543. Boccara, A. C.; Fournler, D.; Badoz, J. Appl. Phys. Lett. 1980, 3 6 , 130- 132. Klesper, E.; Corwin, A. H.; Turner, D. A. J . Org. Chem. 1962, 2 7 , 700-701. Glddings, J. C.; Manwaring, W. A.; Myers, M. N. Science (Washlngton, D . C . ) 1986, 154, 146-150. Novotny, M.; Springston, S. R.; Peaden, P. A,; Fjeldsted, J. C.; Lee, M. L. Anal. Chem. 1981, 5 3 , 407A-414A. Randall, L. 0.; Wahrhaftlg, A. L. Rev. Scl. Instrum. 1981, 5 2 , 1283-1295. Shafer, K. H.; Griffiths, P. R. Anal. Chem. 1983, 55, 1939-1942. Habgood, H. W. J. Chem. €doc. 1958, 3 3 , 557-558. Dovlchl, N. J.; Harris, J. M. Anal. Chem. 1981, 5 3 , 106-109. Carter, C. A,; Harris, J. M. Anal. Chem. 1983, 55, 1256-1261. Savltsky, A,; Golay, M. J. E. Anal. Chem. 1964, 3 6 , 1627-1639. Bullier, A,; Levl, 0.; Marsault-Herail, F.; Marsault, J. P. C. R . Hebd. Seances. Acad. Scl., Ser. B 1974, 279, 597. Sheng, A. C. N.; Granleri, G. J.; Yellin, J. "RCA Thyristor/Rectlfiers Databook"; RCA Corporation: Lancaster, PA, 1974; pp 445-472. Michels, A,; Mlchels, C. Proc. R . SOC. London, Ser. A 1935. 153, 201-214. Michels, A,; Mlchels, C.; Wouters, H. Proc. R . SOC.London, Ser. A 1935, 153, 214-224. Michels, A.; Blaisse, B., Mlchels, C. Proc. R. SOC. London, Ser. A 1937, 160, 358-375. Wentorf, R. H., Jr. J . Chem. Phys. 1956, 2 4 , 607-615. Michels, A,; Hamers, J. Physica 1937, 10, 995-1006.
Anal. Chem. 1984, 56, 1487-1492 (24) Micheis, A.; Sengers, J. V.; Van der Guiik, P. S. Physics (Amsterdam) 1962, 2 8 , 1216-1237. (25) Sheldon, S. J.; Knight, L. V.; Thorne, J. M. Appl. Opt. 1082, 2 1 , 1663.- - - 1664. --(26) Carter, C. A.; Harris, J. M. Appl. Opt. 1984. 2 3 , 476-481. (27) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. “Molecular Theory of Gases and Liquids”: Wiiey: New York, 1954; Chapters 4 and 5. (28) Jackson, W. B.; Amer, N. M.; Boccara, A. C.; Fournier, D. Appl. Opt. 1981, 2 0 , 1333-1344. (29) Fournier. D.; Boccara, A. C.; Badoz. J. Appl. Opt. 1082, 2 1 , 74-46. (30) Carter, C. A.; Harris, J. M. Appl. Spectrosc. 1983, 37, 166-172. (31) Isaacs, Neil S. “Liquid Phase High Pressure Chemistry”; Wiiey: New York, 1981; Chapter 6.
1487
(32) Carter, C. A.; Harris, J. M. Anal. Chem. 1084, 56, 922-925. (33) Buffet, C. E.; Morris, M. D. Appl. Spectrosc. 1983, 37, 455-458. (34) Moldover, M. R.; Sengers, J. V.; Gammon, R. W.; Hocken, R. J. Rev. Mod. Phys. 1979. 51, 79-99. (35) Harris, J. M.; Leach, R. A.; Hardcastie, F., presented in part at the Pittsburgh Conference and Exposition, Atlantic City, NJ, March 1984; Paper 595.
RECEIVED for review January 17,1984. Accepted March 23, 1984. This material is based upon work supported by the National Science Foundation under Grant CHE82-06898.
Statistical Uncertainties of End Points at Intersecting Straight Lines Lowell M. Schwartz* and Robert I. Gelb Department of Chemistry, University of Massachusetts, Boston, Massachusetts 02125
Procedures are described for calculating the statlstlcal uncertalntly of a tltratlon end polnt deflned by the lntersectlon of stralght llne segments. Thls type of tltratlon curve often occurs In analytlcal technlques such as conductometry, spectrophotometry, and amperometry. Equatlons are derlved for the standard error estlmate and the confldence Interval for the end polnt. Also dlscussed are statlstlcal uncertalntles of end polnt difference assays, i.e., assays calculated from the dlfference of two end points, both of whlch are deflned by lntersectlng stralght Ilnes. An lllustratlve example shows calculational details, lncludlng the effect of data points which devlate from the straight segments and the effect of Increasing noise level.
Segmented linear titration curves are often encountered in routine chemical analysis. For example, this type of curve is observed in Gran plots and in titrations monitored by conductometry, spectrophotometry, and amperometry employing dropping mercury or rotating platinum indicator electrodes. In these techniques, deviations from linearity are often observed directly at the end point. Other techniques, such as the so-called “dead-stop” or “biamperometric”titration, yield curves which feature linear segments contiguous to both sides of the end point but deviations occur elsewhere. We are not aware of a previously published discussion of the statistical uncertainty of end points obtained from linear segmented titration curves and so in this paper will develop procedures for assigning these uncertainties. These procedures are based on the statistical fluctuation in discrete titration data. Thus titration data must be available in digital form or, if the original titration curve is recorded in continuous (analog) form, discrete points whose values include a random sampling of the noise fluctuation must be selected. Titration curves of this nature consist of two or more branches each of which consists partly of a linear portion. We will denote the coordinates of two adjacent branches as yAvs. X A and Y B vs. xg. The end point defined by these two branches is the intersection of the two straight segments. In many conductometric, spectrophotometric, and amperometric titrations this intersection lies beyond the linear ranges of both, The end point of interest is the abscissa value of this inter0003-2700/84/0356-1487$01.50/0
section, which we will denote by X . Once the parameters of the two lines are found, it is a simple matter to calculate X algebraically. In order to facilitate the upcoming statistical analysis, we will write the two linear segments as
Y A = QA
bA(xA - ZA)
(la)
Y B = QB
+ ~ B ( X -B ZB)
Ob)
where (zA,gA)and ( z B , gB) are the centroids of the A and B linear segments, respectively. The intersection occurs when YA = y B and when X A = X B = X . Application of these conditions to eq l a and l b yields
x = -& - QA + bAZA - bBfB)/(bB - b
~ E) - A a / A b
(2)
for the end point. The notations Aa and Ab represent the differences in y intercepts and slopes of the two lines, respectively. One problem we pose in this paper is to estimate statistical uncertainties for X . Our aproach to this problem depends on the statistical natures of discrete observations of the y and x . If the relative statistical uncertainties of the x data are negligible compared to the y data, the method of least squares is appropriate and is the most common choice. It has been shown (1,2)that the method of least squares is also the correct method to use even when the x data have substantial errors provided that these data are experimental settings as contrasted with experimentalmeasurements. In order to illustrate this difference, consider a titration done by using an ordinary manual buret. If the incremental volumes x are recorded by reading the positions of the meniscus on the graduated scale, these volumes are subject to randomly varying reading errors. This means that if the titration could be replicated in such a way that the same volume increments are transferred, the recorded volumes x would differ from replicate to replicate because of the reading errors. Volume data obtained in this way are random variables. On the other hand, suppose the experimenter decides to transfer titrant in increments of 1 mL. After setting the initial meniscus position to zero or to some other integer milliliter level, he proceeds to transfer titrant by 1 mL at a time, stopping to record the response datum and the integer volume setting. He does not examine the level of meniscus after each transfer but rather is satisfied that 1 mL has been transferred as long as the meniscus is reasonably near an integer mark. There is no doubt that the 0 1984 American Chemical Society
