Determination of alcohol partial molar volumes from single-droplet gravimetric and optical resonance data Glenn O. Rubel

The author is with the Chemical Research, Development, and Engineering Center, U. S. Army Armament, Muni­tions, and Chemical Command, Aberdeen Proving Ground, Maryland 21010-5423. Received 2 February 1990.

The molar volume of pentyl alcohol in dïbutylphthalate (DBP) is determined from weight changes in a DBP droplet during alcohol vapor absorption in conjunction with volu­metric changes that are determined from optical resonance spectra.

The classical method for determining the partial molar volume of a solute in a solvent involves the measurement of the volume or density of the solution for a known composi­tion. The accuracy of the method depends on the accuracy of the volume measurements, which can be difficult when either the solute or solvent are volatile. Furthermore impurities in the solute or solvent can interfere with the volume measurements, especially when the solute is spar­ingly soluble in the solvent. A new method for partial molar volume determination is presented that circumvents the problems that volatile solvents and impurities in the sol­vent introduce. This method is referred to as single-particle electrical levitation (SPEL).

SPEL technology utilizes static and oscillating electric fields to levitate single particles in gases. By exploiting the light-scattering properties of individual droplets, research­ers have used SPEL to measure the physicochemical proper­ties of droplets with improved accuracy.1 For example, the existence of morphological-dependent resonances in the light-scattering signature of single droplets has made it possible to evaluate the liquid index of refraction to a

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resolution of 5 × 10-5.2 These Mie resonances have also been used to measure accurately the size change of the growing solution droplets and ultimately the water accom­modation coefficient for the exchange of water between the gas and droplet phase.3 In that work the droplet growth rate was determined from the time-dependent resonance spec­tra by using a theoretical expression relating the resonance periodicity to the droplet size change. It is of interest to use this relationship between the periodicity of the morphologi­cal-dependent resonances and the droplet size change to determine the volumetric change of a single dibutylphtha-late (DBP) droplet during pentyl alcohol vapor absorption. By combining the volumetric data with gravimetric data that are obtained from weight-balancing electric fields, the partial molar volume of pentyl alcohol in DBP can be determined.

The experimental system consists of a single-particle electrodynamic cell that levitates the droplet in the scatter­ing volume of a He-Ne laser. The scattered light at 90° is collected by a lens that subtends an angle of 10°, and the time-dependent light intensity is detected by a photomulti-plier tube. The output of the photomultiplier tube is amplified 1000-fold by a current amplifier, and the signal is sent to a recorder. Simultaneously the levitation voltage, the voltage differential that is required to establish an electric force that balances the weight of the droplet, is recorded. The levitation voltage is proportional to the droplet weight, the moles of pentyl alcohol absorbed by the droplet are determined from this voltage differential. Be­cause the pentyl alcohol is absorbed into the droplet from the gas phase, the probability of impurity interference is reduced. Furthermore DBP droplet volume changes are determined once the pentyl alcohol has established equilib­rium between the gas and droplet phase. Therefore solute volatility does not affect the accuracy of the volume determi­nations.

It has been shown theoretically that in the limit where the droplet diameter is much greater than the radiation wavelength, the change in the droplet diameter between

successive Mie resonance peaks is given by

where m is the liquid index of refraction, λ is the laser wavelength, and d » λ. The relative droplet volume between successive resonance peaks can be expressed in terms of the droplet diameter change as

where ƒ and i are the final and initial volumes V and diameters d, respectively. In these experiments the initial droplet diameter di is determined from Eq. (2) by measur­ing the relative levitation voltage between successive reso­nance peaks for a purely evaporating DBP droplet. For a pure component droplet the relative levitation voltage is equal to the relative droplet volume, and with Eq. (1) the initial droplet diameter can be determined. In the present experiment the initial DBP droplet diameter is equal to 63.41 μm.

The droplet volume change ΔV between successive opti­cal resonance peaks can be written from Eq. (2) as

Because the levitation voltage is proportional to the droplet mass, the mole number of pentyl alcohol nal that is absorbed by the DBP droplet can be expressed in terms of the relative levitation voltage as

where LVi and LVƒ are the levitation voltages before and after pentyl alcohol vapor absorption, respectively, ρDBP is the density of DBP, and Mal is the molecular weight of pentyl alcohol. The volume change in the droplet caused by the absorption of alcohol vapor is expressed in terms of the species mole numbers nal and nDBP as

where v and υ° are the partial and molar volumes of the respective species. Because DBP is nonvolatile in these experiments nDBP is constant, and the concentration of pentyl alcohol in DBP is low (< 2% by weight) so that to a good approximation υDBP = υDBP°. Combining Eqs. (3), (4), and (5), we can express the partial molar volume of pentyl alcohol in DBP as

Equation (6) shows that the partial molar volume of the alcohol molecule is determined from knowledge of the droplet diameter change and the corresponding relative levitation voltage.

Figure 1 shows the resonance spectra of the DBP droplet during the absorption of pentyl alcohol vapor. Three sets of resonance spectra (A, B, C) are shown here. Thus, A1, B1, and C1 are successive resonance peaks, and the droplet diameter change Ad between these peaks is given by Eq. (1). Note that the period of the sets increases with time, which indicates that the alcohol vapor absorption rate decreases with time. The corresponding levitation voltage is also recorded, which shows an increasing value as the droplet mass increases during alcohol vapor absorption.

Fig. 1. Optical resonance and gravimetric data for a DBP droplet during the absorption of pentyl alcohol vapors.

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Table I. Partial Molar Volume of Peπtyl Alcohol in DBP as Determined from Levitation Voltages and Optical Resonance Data

Table I shows the pentyl alcohol partial molar volume as determined from Eq. (6) by using the levitation voltages at A1, B1, and C1, and the physical constants Mal = 88.15 g/mole and ρDBP = 1.047 g/cc. Also shown is the pentyl alcohol mole fraction that corresponds to the specific reso­nance peak. The pentyl alcohol partial molar volume is evaluated to be 95.784 cc/mole at 0.02 alcohol mole fraction and 96.884 cc/mole at 0.04 alcohol mole fraction. The molar volume of pentyl alcohol can be calculated from its density ρal = 0.8144 g/cc and its molecular weight Mal and is determined to be 108.24 cc/mole.4 Thus the partial molar volume of pentyl alcohol in DBP is ~ 11% less than the molar volume of pentyl alcohol in the neat state, which indicates the volume contraction of the alcohol molecule during absorption in DBP.

Density measurements of solutions of pentyl alcohol and DBP were measured at 25°C for concentrations of 0.10, 0.08, 0.06, 0.04, and 0.02 alcohol mole fraction. From these measurements the apparent partial molar volume of pentyl alcohol DBP was determined to be 98.345 and 99.785 cc/mole for alcohol mole fractions of 0.02 and 0.04, respec­tively. These independent measurements of the partial molar volume of pentyl alcohol in DBP are in good agree­ment with the single-particle analysis data and demon­strate the validity of SPEL for partial volume determina­tions.

The alcohol absorption kinetics can be analyzed by evaluating the droplet growth rate from the data in Fig. 1. The droplet diameter is determined at the resonance peaks A1, B1, and C1, and the corresponding time is noted. The droplet growth rate ä is calculated from the difference ratio Δd/Δt. Using this method, we determined the droplet

growth rate d to be 4.073 × 10-8 cm/s between resonances A1 and B1 and 2.405 × 10-8 cm/s between resonances B1 and CI. According to gas phase diffusion-controlled trans­port theory, the droplet growth rate d is

where P∞ is the ambient alcohol partial pressure and Pd is the alcohol partial pressure over the droplet surface. Ini­tially the alcohol partial pressure over the droplet surface is small, and it can be neglected in Eq. (7). Using the calculated partial molar volume of pentyl alcohol and measured values for the gas phase diffusion coefficient Dg and the ambient alcohol partial pressure, we calculated the droplet growth rate from Eq. (7) to be 3.912 × 10-8 cm/s. This is in good agreement with the experimental values for the droplet growth rate and shows that pentyl alcohol vapor absorption is gas phase diffusion controlled.

The partial molar volume of pentyl alcohol in DBP is evaluated for alcohol mole fractions of 0.02 and 0.04 by measuring optical resonance data and gravimetric data for a single DBP droplet-absorbing pentyl alcohol vapor. Mea­surements show that the pentyl alcohol partial molar volume in DBP is less than that of the alcohol in its neat state, which indicates alcohol molecule contraction in the DBP solution. It is also shown that the alcohol partial molar volume increases with the alcohol mole fraction.

References 1. E. J. Davis, "Transport phenomena with single aerosol

particles," Aerosol Sci. Technol. 2, 121-144 (1983). 2. P. Chylek, V. Ramaswamy, A. Ashkin, and J. M. Dziedic,

"Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data," Appl. Opt. 22, 2302-2312 (1983).

3. K. H. Fung, I. N. Tang, and H. R. Munkelwitz, "Study of condensational growth of water droplets by Mie resonance spectroscopy," Appl. Opt. 26, 1282-1287 (1987).

4. R. C. Weast, ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, Ohio, 1981), p. C-478.

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