In the Laboratory
Determination of Heat Capacity of Liquids with Time-Resolved Thermal Lens Calorimetry A More Accurate Procedure Kurt Seidman* and Amy Payne Department of Chemistry, Randolph-Macon Woman’s College, Lynchburg VA 24503
When a laser beam is passed through a liquid solvent that contains an absorbing species, a temperature gradient forms along the beam path. The gradient results from the nonradiative relaxation of the absorbing species’ excited state. As this thermal energy is transferred to the solvent, a density gradient is established that is very similar to the intensity profile of the laser beam. This causes the formation of a “thermal lens” that produces a significant increase in the divergence of the laser beam. This phenomenon was first described by Gordon et al. in 1965 (1). Since then, a number of applications of thermal lensing have been developed. These applications include trace analysis (2, 3), determination of pK (4), and small volume detection (5, 6 ). More recently, Salcido et al. have described an experiment for determining the heat capacities of liquids with thermal lens calorimetry (7). This experiment, which the authors recommend for a physical chemistry course, has the virtue of being a relatively simple and inexpensive introduction to lasers and oscilloscopes. It has the disadvantage of being inaccurate. A 63% error was reported for the heat capacity of methanol. Errors of this magnitude clearly detract from the experiment. We wish to report a modified procedure for determining heat capacities with thermal lens calorimetry that dramatically reduces the amount of error.
If a lens is placed between the laser and the diaphragm and a sample is positioned between the lens and the diaphragm, the signal undergoes a remarkable change. As the thermal lens forms and the divergence of the laser beam increases, the square wave suffers an exponential decay similar to that illustrated in Figure 2. Salcido et al. (7) point out that the decay is very sensitive to the positioning of the sample. The sample should be positioned a short distance beyond the focal length of the lens. However, the authors do not mention that the decay is also very sensitive to the size of the aperture in the diaphragm. We will say more about this shortly. The time constant, tc , is the 1/e lifetime of the exponential decay, and it can be determined from the parameters I BC and ∆IBC that are defined in Figure 2. The height of the signal after it has decayed to 1/e of its initial height is
I C = I BC – ∆ I BC 1 – 1e
(1)
and the time constant is the time, measured in milliseconds, required for the signal to decay to a height of IC. The time Diaphragm
Lens
Photodiode
Laser
Theory
Chopper
*Corresponding author.
Sample
Oscilloscope
Figure 1. The thermal lens experiment.
Voltage from Photodiode
Gaussian laser beams are ideal for producing a thermal lens. Besides being highly directional and monochromatic, the intensity of the beam is not uniform. The beam is most intense at its center, and the intensity decreases as we move radially outward from its center. A description of the characteristics of Gaussian laser beam propagation is given by Harris and Dovichi (5). This intensity gradient produces a temperature gradient when the beam enters the sample and excites the absorbing species. The temperature of the sample is highest where the beam is most intense. As the sample warms, it expands and its density decreases. The greatest decrease in density occurs in the region where the beam is most intense —along the center of the beam’s path. The density of the sample increases as we move radially outward from the center of the beam’s path. This density gradient behaves like a lens that enhances the divergence of the laser beam. A schematic diagram of the components used to produce the thermal lens effect is provided in Figure 1. When the beam from a He/Ne laser is chopped and passed through a diaphragm with an adjustable aperture to a photodiode that is connected to an oscilloscope, a square wave signal is obtained.
∆IBC IBC
Time (ms)
Figure 2. Idealized exponential decay resulting from the thermal lens effect.
JChemEd.chem.wisc.edu • Vol. 75 No. 7 July 1998 • Journal of Chemical Education
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constant is used to calculate the heat capacity of the sample. The appropriate formula is
Cp =
4t c κ ωc2 ρ
(2)
where κ is the thermal conductivity of the sample, ρ is the density of the sample, and ω c is the confocal beam waist. The lens focuses the laser beam to a minimum waist, defined to be ω 0. This beam waist is
ω 0 = πλωf
i
(3)
where λ is the wavelength of the radiation emitted by the laser, f is the focal length of the lens, and ωi is the laser beam waist before it is focused. The confocal beam waist is at a point one confocal length beyond the minimum beam waist, the point where the maximum thermal lens effect is obtained (5). At the confocal length the beam is diverging and the _ beam waist is √ 2 larger than the minimum beam waist. Obtaining the “correct” waveform is critical. Small variations in either the lens-to-sample distance or the aperture in the diaphragm cause significant changes in the time constant. Salcido et al. state that “With stable optical mounts and a little patience, the correct waveform can be achieved” (7 ). The waveform must be optimized for “shape and maximum depletion”. Unfortunately, these terms are subjective and can be the source of significant error in the experiment. These authors also point out that determination of the laser beam waist before focusing is another potential source of error. This is a ubiquitous problem in thermal lens experiments, and at least one article has been devoted to solving it (8). The magnitudes of the errors resulting from these problems can be greatly reduced by calibrating the method with a set of standards. During our preliminary studies we discovered that the setting of the diaphragm’s aperture has a marked influence on the waveform. In fact, if the same lens-to-sample distance is used for a set of samples, the aperture size must be adjusted for each sample to obtain the correct waveform. We were most interested in discovering a way to objectively set the aperture. We noticed that there was a relationship between the correct aperture diameter and the diameter of the thermal bloom on the diaphragm. In general, a larger aperture is required for samples that have larger thermal blooms. Furthermore, if some precautions are taken during sample preparation, the relationship between the bloom diameter and the diameter of the aperture is linear. If we first prepare a graph of bloom diameter versus the correct aperture diameter from a set of standards, we can use that graph to obtain the correct aperture setting from the measurement of the bloom diameter of an unknown sample. The procedure is described below.
diode detector (UDT model P-10D) was mounted behind the diaphragm and connected to an oscilloscope (Tektronix TDS-350). The laser was turned on and the laser, diaphragm, and photodiode were adjusted so that the beam was centered on the diaphragm and diode. Next, a lens with a 120-mm focal length was mounted about 30 cm from the laser head and adjusted so that the laser beam remained centered on the diaphragm and the detector. A set of solutions was prepared from common solvents and the dye indophenol blue (Aldrich). All solutions were prepared from stock solutions in which the dye concentration was ~2 × 10᎑4 M. The dye serves as the absorbing species that is required to produce heating along the path of the laser beam. The solvents included HPLC-grade acetone and methanol (Aldrich), chloroform and carbon tetrachloride (Fisher, Spectranalyzed), benzene and toluene (Fisher, A.C.S. Certified), and absolute ethanol (AAPER). The concentration of dye in each solution was adjusted so that all solutions had the same absorbance at 633 nm, namely, the wavelength of the laser light. Absorbance was determined with a Hewlett-Packard 8452A diode array spectrophotometer using cuvettes with a 10mm path length. Five of the samples (acetone, methanol, ethanol, benzene, and carbon tetrachloride) were chosen to serve as standards. These solvents provide a range of heat capacities, thermal conductivities, and densities (see Table 1). Three of the solvents are polar and two are nonpolar. Two parameters, lens-to-sample distance and diaphragm aperture, must be adjusted to obtain the correct waveform. We decided to adjust the lens-to-sample distance for one of the standards and to use that distance for all other samples. The adjustment of the diaphragm’s aperture was used to obtain the final waveform for each sample. We used acetone to adjust the lens-to-sample distance, although this choice is arbitrary. A 2-mm cuvette (Whatman) was filled with the sample and placed on the optical rail between the lens and the diaphragm. Sample placement was adjusted to ensure that the laser beam remained centered on the aperture of the diaphragm. The chopper was turned on and set to a frequency of 25 Hz. The lens-to-sample distance was varied by moving the sample along the rail until a proper waveform (one similar to that illustrated in Fig. 2) was observed on the oscilloscope. The rail carrier was then locked into position, and this distance was used for all other samples. The aperture of the diaphragm was adjusted until a waveform was obtained with an associated time constant (tc ) that yielded the correct heat capacity for the solvent when it was substituted into eq 2. This calculation requires a knowledge
Experimental Procedure The experimental design is illustrated in Figure 1. All components were mounted on a 1.5-m optical rail (MellesGriot). The optical rail was placed on an optical table to reduce vibration, but any stable surface can serve as a base. A 10mW He/Ne laser (Uniphase 1125) was mounted on one end of the rail. The laser was leveled and aligned with the rail. A diaphragm with an adjustable iris-type aperture (Edmund Scientific) was mounted at the other end of the rail. A photo898
Journal of Chemical Education • Vol. 75 No. 7 July 1998 • JChemEd.chem.wisc.edu
In the Laboratory
of the confocal beam waist ωc. This parameter was calculated from the laser beam waist before focusing (ωi) as described above. The laser beam waist before focusing was crudely measured with a millimeter-scale ruler. The confocal beam waist obtained from this measurement was 57 µm, a very reasonable value. The knowledge of an exact spot size is not necessary, since a calibration line will be derived from standards of known heat capacity. Finally, the diameters of the aperture and the thermal bloom were measured with a millimeter-scale ruler. All other standards were treated as described above. Each standard was placed in the holder and adjusted until the laser beam was centered on the aperture of the diaphragm. The position of the sample holder on the rail was not readjusted! The diaphragm of the aperture was adjusted until a time constant that yielded the correct heat capacity of the solvent was obtained. The diameters of the aperture and the thermal bloom were measured. A graph of the aperture’s diameter versus the bloom diameter was constructed. An equation for the resulting line was constructed from a least squares analysis of the data. The heat capacities of two “unknowns”, toluene and chloroform,
were determined. Each unknown was placed in the sample holder and its bloom diameter was measured. The aperture diameter was calculated from the equation for the line obtained from the standards. The diameter of the aperture was then adjusted and the time constant and heat capacity were determined. Results Solutions of standards and unknowns were prepared at three absorbances: 1.80, 1.46, and 0.86. There were small variations in absorbance of the solutions within each set. The absorbances of all the solutions are presented in Table 2. The scatter of absorbance data is greater for the first trial than for the second and third trials. We will say more about this shortly. Each graph of bloom diameter versus diaphragm aperture was linear. The crucial parameters from these graphs are presented in Table 3. Two graphs (trials 1 and 2) were constructed from four data points; the graph for trial three was constructed from five data points. The concentration of indophenol in methanol was too dilute at absorbances of 0.86 and 1.46 to produce reliable waveforms. The results obtained for the “unknowns” from each trial are presented in Table 4. All aperture diameters reported in this table were calculated from the corresponding bloom diameter with the equations for the lines reported in Table 3. One might reasonably expect that the bloom diameters should exhibit a steady decrease as the concentration of dye in the solvent decreases. This would, in fact, be true if we had maintained the same distance between the lens, sample, and diaphragm from one trial to the next. Although we maintained a constant distance during any one trial, we did not maintain a constant distance from trial to trial. The results show considerable improvement over others reported using this technique. The largest relative error was 3.8%, and the average relative error was 2.8%. The results obtained from trial 1 are as good as those from the other trials, despite the fact that the absorbances in trial 1 exhibit greater scatter. This is reassuring, because sample preparation consumes a large fraction of the time spent in these studies. A small error in the absorbances seems to be of little consequence. The greatest source of error in this experiment is the measurement of the diameter of the aperture and the bloom. Bloom diameter is a particularly difficult determination because the edge of the bloom is not sharply defined. For the sake of consistency, it is best that the same person make these measurements for all samples in a trial. This experiment requires more time to perform than the experiment described by Salcido et al. (7 ). A significant amount of this time can be saved by preparing the solutions before the laboratory. The additional effort that must be expended when performing this experiment is more than compensated by the considerable improvement in the final results. Acknowledgment We would like to express our gratitude to the National Science Foundation for an ILI Grant (DUE-9450899) with which much of the instrumentation described in this paper was purchased.
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Literature Cited 1. Gordon, J. P.; Leite, R. C. C.; Moore, R. S.; Porto, S. P. S.; Whinnery, J. R. J. Appl. Phys. 1965, 36, 3–8. 2. Dovichi, N. J.; Harris, J. M. Anal. Chem. 1979, 51, 728–731. 3. Alfheim, J. A.; Langford, C. H. Anal. Chem. 1985, 57, 861–864. 4. Erskine, S. R.; Bobbitt, D. R. J. Chem. Educ. 1989, 66, 354–357.
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5. Harris, J. M.; Dovichi, N. J. Anal. Chem. 1980, 52, 695A–706A. 6. Carter, C. A.; Harris, J. M. Anal. Chem. 1984, 56, 922–925. 7. Salcido, J. E.; Pilgrim, J. S.; Duncan, M. A. Physical Chemistry: Developing a Dynamic Curriculum; American Chemical Society: Washington, DC, 1993; pp 232–241. 8. Jansen, K. L.; Harris, J. M. Anal. Chem. 1985, 57, 1698–1703.
Journal of Chemical Education • Vol. 75 No. 7 July 1998 • JChemEd.chem.wisc.edu
