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Deriving Properties of Low-Volatile Substances from Isothermal Evaporation Curves

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Deriving Properties of Low-Volatile Substances from Isothermal Evaporation Curves J. Non-Equilib. Thermodyn. 2016; 41 (1):3–11 Research Article Ricardas V. Ralys*, Alexander A. Uspenskiy and Alexander A. Slobodov Deriving properties of low-volatile substances from isothermal evaporation curves DOI: 10.1515/jnet-2015-0030 Received May 31, 2015; revised October 8, 2015; accepted October 19, 2015 Abstract: Mass ux occurring when a substance evaporates from an open surface is proportional to its satu- rated vapor pressure at a given temperature. The proportionality coecient that relates this ux to the vapor pressure shows how far a system is from equilibrium and is called the accommodation coecient. Under vac- uum, when a system deviates from equilibrium to the greatest extent possible, the accommodation coecient equals unity. Under nite pressure, however, the accommodation coecient is no longer equal to unity, and in fact, it is much less than unity. In this article, we consider the isothermal evaporation or sublimation of low- volatile individual substances under conditions of thermogravimetric analysis, when the external pressure of the purging gas is equal to the atmospheric pressure and the purging gas rate varies. When properly treated, the dependence of sample mass over time provides us with various information on the properties of the exam- ined compound, such as saturated vapor pressure, diusion coecient, and density of the condensed (liquid or solid) phase at the temperature of experiment. We propose here the model describing the accommodation coecient as a function of both substance properties and experimental conditions. This model gives the - nal expression for evaporation rate, and thus for mass dependence over time, with approximation parameters resulting in the properties being sought. Keywords: Saturated vapor pressure, thermogravimetry, evaporation rate, diusion 1 Introduction Measurement of vapor pressures of low-volatile substances is not an easy task, and assessing this property usually involves both static and dynamic methods [1]. Static methods, such as manometry [2] or ebulliometry [3], require the sample that is being studied to reach its equilibrium state rst. These methods give reliable vapor pressure data, but they are highly demanding in terms of substance quantity, purity, labor, and time. Dynamic methods, on the contrary, do not need to equilibrate the condensed and vapor phases rst. In these methods, vapor pressure is derived from the measured mass ux, and basically, this group includes the tran- spiration, Knudsen [4], and Langmuir methods [5]. Of these, only the transpiration method uses carrier gas to vaporize sample, with the overall mass change at a given experimental time relating to its saturated vapor pressure via the ideal gas law [3, 6, 7]. The Knudsen and Langmuir methods explore mass change under evap- oration into vacuum, with the area of evaporation being the main dierence between them. In the Knudsen method, vapors of a substance escape from a small orice, causing either measurable mass reduction over time or some measurable force like in torsion eusion [8]. In contrast to the Knudsen model, in the Langmuir method, sample vaporizes from an open surface into vacuum. Dynamic methods have high reliability, but their drawback is the need to maintain a sophisticated experimental setup, which involves mass detection and high and tight vacuum. *Corresponding author: Ricardas V. Ralys: Department of Information Technologies in the Fuel and Energy Complex, ITMO University, 49 Kronverksky Pr., St. Petersburg 197101, Russia, e-mail: [email protected] Alexander A. Uspenskiy, Alexander A. Slobodov: Department of Information Technologies in the Fuel and Energy Complex, ITMO University, 49 Kronverksky Pr., St. Petersburg 197101, Russia, e-mail: [email protected], [email protected] 4 Ë R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances With the availability of highly accurate modern thermogravimetric analyzers allowing direct measure- ment of mass change with acceptable precision, the Langmuir method has been adapted to the conditions of such equipment. However, the basic assumption of this method is no longer valid. Under vacuum, the vaporized sample fully escapes the surface, and this is no longer observed when evaporating into a purging gas used in thermogravimetry. The equation that describes the evaporation rate in terms of saturated vapor pressure is known as the Langmuir equation: dm M = −αp S S { , dt V 2πRT ( ) dm S where dt is the evaporation rate (kg/s), α is the accommodation coecient, p is the saturated vapor pres- 2 sure (Pa), SV is the surface of phase boundary area (m ), M is the molar mass of( ) the substance (kg/mol), R is the universal gas constant (J/(mol K)), and T is the sample temperature (K). When vaporizing into vacuum, the accommodation coecient equals unity, i.e. no molecules return to the surface from which they escaped. However, under nite external pressure, most of the escaping molecules return, and this makes the accom- modation coecient much less than unity, and this parameter has 10 5 to 10 4 orders of magnitude (see, e.g., [9–14]). − − Such a large departure from the initial assumption of α = 1 requires that the value of the accommodation coecient be determined rst before deriving the vapor pressure from mass loss measurements. To do so, we need to carry out a series of experiments with calibration substances for which the saturated pressures at temperature of interest are available. This need imposes certain limitations: (1) calibration substances should be structurally similar to those being studied, and (2) experimental conditions should be identical when vaporizing the calibration substances and the samples being studied. Both conditions are not always satised due to evident lack of vapor pressure data for low-volatile compounds. To overcome these obstacles, attempts have been made to create models that explicitly describe the de- pendence of the accommodation coecient or even evaporation rate on substance properties and experimen- tal conditions [15–17]. These models consider evaporation in a thermogravimetric analyzer as a diusion- limited process. When applied to experimental data, the equations proposed in these works give good agree- ment with observed evaporation rates. However, these models do not explain the experimentally observed dependence of evaporation rates on purging gas rates [18–20]. To explain the observed evaporation phenomena, e.g. low accommodation coecient and its depen- dence on purge gas rate, we need to construct a model by taking into account both substance properties and experimental conditions. The model sought is supposed to give explicit expressions for saturated vapor pressure as well. 2 Initial assumptions and model construction Before deriving the equations of interest, consider under which assumptions they are valid. The following conditions should be met for the model to be valid: 1. Purge gas ux is parallel to the walls of the crucible with the sample. 2. No association or dissociation of the molecules occurs in the vapor phase. 3. Purging gas is neutral to the sample under investigation; thus, neither chemical interactions between them nor gas sorption into the sample takes place. 4. No chemical or phase transformations are observed in the sample during evaporation experiments. 5. The thermal resistivity in the sample is negligible; thus, the sample has the same temperature over its volume. This assumption is readily satised when working with thermogravimetry because substance quantities are small and the heat supply in the apparatus is sucient. 6. The hydrodynamic regimen is laminar; thus, there is stable diusion region over the sample surface. This assumption is also satised when working with thermogravimetry. R. V. Ralys, A. A. Uspenskiy and A. A. Slobodov, Properties of low-volatile substances Ë 5 7. Evaporation, on the one hand, is suciently fast in order to be detected at a reasonable experimental time, and on the other hand, is not very high in order to prevent the surface from undercooling. 8. The concentration of the vapor at the top of diusion layer is non-zero. 9. The amount of the substance diused through the stagnant gas layer is equal to the amount that the purge gas takes away. When the above validity criteria are fullled, the following equations describing the evaporation process are applicable: 1 j = αc S v̄, (2.1) T 4 D ( ) j = (c S − c ), (2.2) T x W V( ̇ ) jT = cW , (2.3) SV 2 where equation (2.1) describes the overall mass ux jT (kg/(m s)) in terms of the Langmuir equation, which involves the accommodation coecient α, saturated vapor concentration c S (kg/m3), and average molecules velocity v̄ (m/s); equation (2.2) describes the mass ux occurring in the diusion( ) layer in terms of the rst Fick’s law, which involves the diusion coecient D (m2/s), diusion path x (m), and concentration at the 3 layer top boundary cW (kg/m ); and equation (2.3) describes the mass ux resulting from convection of 2 diused vapor in the purge gas, with cross-sectional surface SV (m ) and volume rate of the purge gas V̇ (m3/s). The average molecular velocity is considered at equilibrium state of vapor, and thus its value is de- ned by Maxwell–Boltzmann distribution and equals v̄ = x8RT/(πM). The saturated vapor concentration equals c S = p S M/(RT). When( ) equating( ) the right-hand parts of (2.2) and (2.3), we easily obtain the expression for the concentra- tion at the top diusion layer boundary: DSV S cW = c (2.4) DSV + Vẋ ( ) Taking into account (2.3) and (2.4), we write for the mass ux the following expression: ̇ ̇ V DSV S DV S jT = c = c . (2.5) SV DSV + Vẋ DSV + Vẋ ( ) ( ) Comparing (2.5) with the Langmuir equation (2.1), we easily obtain the function that describes the accom- modation coecient: 1 DV̇ 4 DV̇ αc S v̄ = c S â⇒ α = .
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