Estimating evaporation rates and contaminant air concentrations due to small spills of non-ideal aqueous organic solvent mixtures in a controlled environment

Abstract Although small spills of non-ideal organic solvent mixtures are ubiquitous undesirable events in occupational settings, the potential risk of exposure associated with such scenarios remains insufficiently investigated. This study aimed to examine the impact of non-ideality on evaporation rates and contaminant air concentrations resulting from small spills of organic solvent mixtures. Evaporation rate constants alphas (α) were experimentally measured for five pure solvents using a gravimetric approach during solvent evaporation tests designed to simulate small spills of solvents. Two equations were used for estimating contaminants’ evaporation rates from aqueous mixtures assuming either ideal or non-ideal behavior based on the pure-chemical alpha values. A spill model also known as the well-mixed room model with exponentially decreasing emission rate was used to predict air concentrations during various spill scenarios based on the two sets of estimated evaporation rates. Model predictive performance was evaluated by comparing the estimates against real-time concentrations measured for the same scenarios. Evaluations for 12 binary non-ideal aqueous mixtures found that the estimated evaporation rates accounting for the correction by the activity coefficients of the solvents (median = 0.0318 min−1) were higher than the evaporation rates estimated without the correction factor (median = 0.00632 min−1). Model estimates using the corrected evaporation rates reasonably agreed with the measured values, with a median predicted peak concentrations-to-measured peak concentrations ratio of 0.92 (0.81 to 1.32) and a median difference between the predicted and the measured peak times of −5 min. By contrast, when the non-corrected evaporation rates were used, the median predicted peak concentrations-to-measured peak concentrations ratio was 0.31 (0.08 to 0.75) and the median difference between the predicted and the measured peak times was +33 min. Results from this study demonstrate the importance of considering the non-ideality effect for accurately estimating evaporation rates and contaminant air concentrations generated by solvent mixtures. Moreover, this study is a step further in improving knowledge of modeling exposures related to small spills of organic solvent mixtures.


Introduction
Organic solvent mixtures are widely used in industrial settings. During their application, undesirable events such as accidental leaks or small spills of solvents can occur, generating possibly unacceptable or high exposure levels to airborne contaminants, which can threaten the workers' health and safety. Small spillages of volatile liquid chemicals are important in occupational hygiene as they are representative of exposure scenarios in which contaminant mass emissions are exponentially decreasing during the evaporation process, with peak concentrations occurring shortly after the beginning of the emission (Keil and Nicas 2003;Jayjock et al. 2011a;Hewett and Ganser 2017;Abattan et al. 2022).
Moreover, exposure scenarios involving chemical mixtures require meticulous attention since multicomponent volatile liquids are known to behave differently in terms of vapor generation, especially when the components of the mixtures are structurally different or are strongly diluted (Popendorf 2006;Debia et al. 2009;Popendorf 2019). This phenomenon, referred to as non-ideality, is commonly observed with water-based mixtures Popendorf 2019) which are increasingly used as substitutes for conventional solvent mixtures due to workplace and environmental legal restrictions (Wolf et al. 1991;Lavoue et al. 2003;Kikuchi et al. 2011). In general, deviation from ideality occurs in a positive direction, causing the chemical compounds of such mixtures to become more volatile than what is expected assuming ideal conditions (Popendorf 2006(Popendorf , 2019. Exposure assessment in small spills of chemical mixture scenarios using air sampling methods can be time and resource-consuming and/or particularly challenging, such as a lack of valid sampling protocol for new or complex mixtures (Bishop et al. 1982) or transient and rapid occurrence of peak concentrations that could be missed before the measurements are carried out (Jayjock et al. 2011a;Keil and Miller 2020). Mathematical modeling is often the only available approach for estimating vapor concentrations from these particular exposure scenarios. Several evaporation models have been developed for estimating airborne contaminant levels from solvent mixtures Smith 2001;Nicas 2003;Nicas et al. 2006;Guo et al. 2008;Nicas and Neuhaus 2008;Okamoto et al. 2010;Robbins et al. 2012;Earnest and Corsi 2013;Hofstetter et al. 2013;Maier et al. 2015;Nicas 2016;LeBlanc et al. 2018;Arnold et al. 2020;Tischer and Roitzsch 2022). Some of them addressed non-ideality Smith 2001;Okamoto et al. 2010;Hofstetter et al. 2013;LeBlanc et al. 2018) and increasing/decreasing emission rates in combination with non-ideality (Tischer and Roitzsch 2022); but only the present study used exponentially decreasing emission for the released contaminants in the context of non-ideality.
The current study expands on previous research by examining the impact of the non-ideality of solvent mixtures on vapor generation during small spills of non-ideal organic solvent mixtures. The aims of this study were specifically to: (i) estimate contaminants evaporation rates in situations of small spills of organic solvent mixtures, correcting or not for non-ideality; (ii) predict the resulting contaminants air concentrations using a small spill model (i.e., the well-mixed room model with exponentially decreasing emission rate); and (iii) evaluate the model's predictive performance by comparing the estimates made with or without correction for non-ideality against measured concentrations.

Overview of the study
Evaporation rate constants alphas (a) were determined for five pure solvents using a gravimetric approach, during solvent evaporation tests designed to simulate small spills of solvents. Two equations were used for estimating contaminants' evaporation rates from aqueous mixtures assuming either ideal or non-ideal behavior based on the pure-chemical alpha values. A spill model also known as the well-mixed room model with exponentially decreasing emission rate was used to predict air concentrations during various spill scenarios based on the two sets of estimated values. Model predictive accuracy, assuming or not ideal behavior, was evaluated by comparing the estimates against real-time air monitoring data obtained for the same scenarios. All tests were performed under controlled environmental conditions in a university teaching laboratory.

Theoretical background
Emission rates from small spills of solvents Small spills of volatile liquids are representative of exposure scenarios in which emission rates are shown to be exponentially decreasing over time due to the evaporative cooling of the spilled liquids, the decrease in the surface areas of the spills during the evaporation process, and the decrease in the masses of contaminants as evaporation occurs (Keil and Nicas 2003;Abattan et al. 2022). In such exposure scenarios, emission rates (G t ) can be calculated using the following equation (Keil and Nicas 2003): where G t is the instantaneous or time-varying mass emission rate (mg/min), a the evaporation rate parameter/constant (min À1 ), M 0 is the initial solvent mass spilled (mg), and t the elapsed time (min) since the spill occurred. The evaporation rate constant (a) is often determined experimentally using either the gravimetric measurement of the solvent mass loss over time or the curve of the gas phase concentration of the solvent measured over time within a well-mixed test chamber (Keil and Nicas 2003;Jayjock et al. 2011bJayjock et al. , 2011aHewett and Ganser 2017).

Estimation of emission rates from solvent mixtures
In general, the volatility of a liquid chemical depends on its saturated vapor pressure (Ishidao et al. 2016). The vapor pressure of the chemical corresponds to the pressure that the liquid exerts as it moves to the vapor phase (tendency of molecules and atoms to escape from the liquid chemical) (Keil 2009). The higher the pressure, the faster the liquid evaporates. For solvent mixtures, the equilibrium vapor pressures of the chemicals that constitute the solution are different from those of pure compounds. Therefore, estimating the partial vapor pressures of the constituents of the mixture is of great interest for predicting emission rates (Reinke 2009). Three approaches are available for estimating partial vapor pressures for solvent mixtures: Raoult's Law, Raoult's Law adjusted by the activity coefficient, and Henry's Law (Popendorf 2006(Popendorf , 2019. Raoult's Law applies to ideal mixtures, which are composed of structurally similar chemical components, with presence of weak molecular interactions (e.g., nonpolar-nonpolar organic solvents). This law also applies when there is no electrolyte in the mixture (Popendorf 2006(Popendorf , 2019. Raoult's Law indicates that the partial vapor pressure of a chemical i of the mixture (P vapor, i ) is a function of its molar fraction (X i ) and the vapor pressure of the pure form of the chemical i (P vapor ) (Popendorf 2006;Debia et al. 2009;Popendorf 2019), as shown in Equation (2): where P vapor, i ¼ the vapor pressure of each component within a mixture; Xi ¼ the molar fraction of each component of the mixture; and P vapor ¼ the vapor pressure of each component as a pure chemical. Contrariwise, liquid mixtures composed of chemical compounds that are structurally different generally tend to deviate from ideality as the chemicals involved exhibit strong molecular interactions due to their activity coefficients (e.g., isopropyl alcohol diluted in water). This phenomenon (termed nonideality), required Raoult's Law to be corrected or adjusted by the activity coefficients (c i ) of the components of the mixture as shown in Equation (3): where P vapor, i ¼ the vapor pressure of each component within a mixture; (c i ) ¼ the activity coefficient of each component of the mixture; X i ¼ the molar fraction of each component of the mixture; and P vapor ¼ the vapor pressure of each component as a pure chemical.
Note that for an ideal solution, the activity coefficient equals one (Raoult's Law). The more the activity coefficient deviates from one, the more the system moves away from ideality.
The values of (c) can be obtained from published tables (Grain 1990;Popendorf 2006Popendorf , 2019 or can be calculated using the UNIFAC (Universal Quasichemical Functional Group Activity Coefficient) model (Fredenslund et al. 1975).
Estimating evaporation rates for the exposure scenarios evaluated in this study require accounting for two factors: (i) the exponentially decreasing emission pattern that characterizes small spills and (ii) the nonideality of the liquid mixtures which takes into account the activity coefficients when indicated. To do so, we used the information provided by Equation (1), relating the time-varying contaminant mass emission rate (G t ) to the evaporation rate constant alpha (a), which itself linearly depends on the parameter "vapor pressure" (P vapor ) as shown in previously published studies (Keil and Nicas 2003;Abattan et al. 2022). As alpha (a) is related to both the vapor pressure and the emission rate G t , we replaced vapor pressure (P vapor ) with alpha in Equations (2) (Raoult's Law) and (3) (Raoult's Law adjusted by the activity coefficients) to approximate evaporation rates for the chemicals involved in the mixtures used in this study.
Thus, two equations were proposed for estimating evaporation rates for the exposure scenarios evaluated in this study: the first, assuming ideal conditions as presented in Equation (4), and the second assuming non-ideal behavior for the tested chemicals, as shown in Equation (5): where a i ¼ the estimated evaporation rate constant alpha of component i within a mixture; X i ¼ the molar fraction of component i within the mixture, c i ¼ the activity coefficient of the component i within the mixture; and a ¼ the measured evaporation rate constant alpha of the component i as a pure chemical.

Experimental design
A four-part study was designed to evaluate the impact of the non-ideality of liquid mixtures on contaminant evaporation rates as well as on contaminant air concentrations due to small spills of non-ideal organic solvent mixtures. Figure 1 presents the major steps of the study's experimental process.

Estimating contaminant evaporation rates from small spills of non-ideal solvent mixtures
Determining the evaporation rate constant alpha (a) for five organic solvents Solvents used. The solvents of interest were acetone (ACE), butan-2-one (MEK), methanol (MeOH), propan-2-ol (IPA), and toluene (TOL). These solvents were selected because they cover a wide range of vapor pressures and they are often found in commercial mixtures used in workplaces such as the International Thinner (mixture of butan-2-one and toluene), the Lacquer Thinner (mixture of toluene, methanol, acetone and butan-2-one), and a Global Tech Company aqueous mixture (propan-2-ol and water). Table 1 shows the physical-chemical properties of the solvents. All the values were taken from CNESST chemical substances toxicological repository (CNESST 2022), as several TLV V R -TWA (Threshold Limit Values -Time Weighted Average) were taken from the ACGIH booklet (ACGIH V R 2021). Note that the five solvents used in the study were laboratory grades: ACS grade or optima grade.
Gravimetric measurement of evaporation rate constant alpha (a). Evaporation rate constants alphas (a) were measured for the five organic solvents, during solvent evaporation tests, designed to represent various small spill exposure scenarios, using the gravimetric approach proposed by Keil and Nicas (2003). Evaporation tests were carried out under a controlled environment with air temperature ranging from 20 to 22 C, atmospheric pressure ranging from 735 to 765 mmHg, relative humidity ranging from 24 to 34%, and measured air speeds across the liquid pool approximately around 0.025 m/sec. A petri dish of 10 cm in diameter was used as the solvent's container. 0.5 mL of the solvent of interest was transferred in the petri-dish that was placed on top of an analytical balance which was connected to a computer that recorded the contaminant mass evaporation until seventy-five percent (75%) of the initial contaminant mass had evaporated. Graphs of mass decay curves were plotted according to the time (t) and the natural logarithm (ln) of the remaining mass to the initial mass ratios using Equation (6): where M t is the remaining solvent mass at time t (mg); M 0 is the initial solvent mass spilled (mg); a the evaporation rate constant (min À1 ); and t is the elapsed time (min) since the spill. Alpha (a) corresponds to the absolute value of the slope of the natural logarithm of the mass decay curve covering the loss of the first 75% of mass.
A Sartorius CPA423S analytical balance was used for the gravimetric measurement of the solvent's mass losses over time. A VelociCalc Plus (TSI, Inc., Shoreview, MN) was used to quantify the environmental conditions (i.e., ambient temperature, atmospheric pressure, relative humidity, and airspeed above the emission source).
Evaporation rates for the chemicals released from the mixtures. Equations (4) and (5) were used to estimate the contaminant evaporation rates for the exposure scenarios evaluated. For simplicity, the estimated alpha (a i ) using Equation (4) will be referred to as the non-corrected alpha while the one obtained using Equation (5) will be referred to as the corrected alpha.

Modeling contaminant air concentrations from the solvent mixtures
Vapor concentrations of the chemical compounds composing the mixtures were estimated using the AIHA IHMOD2.0 well-mixed room model with an exponentially decreasing emission rate (Drolet and Armstrong 2018) also referred to as the Small Spill Model (Arnold et al. 2020). The construct of the well-mixed box model is presented in Figure S1 in the supplemental materials. The model input parameters are: the initial contaminant mass (M 0 ), the evaporation rate constant alpha (a), the general ventilation rate (Q), and the volume of the room (V), as shown in Figure S2 in the supplemental materials. Time-varying vapor concentrations provided by the model can be calculated using Equation (7): where C(t) is the time-varying concentration (mg/m 3 ), a is the evaporation rate constant (min À1 ), M 0 is the Non-ideal initial solvent mass spilled (mg), V is the room volume (m 3 ), Q is the general ventilation rate of the room (m 3 /min), C 0 is the concentration at t ¼ 0, and t is the elapsed time (min) since the spill.
To represent the well-mixed room, an experimental test chamber of 0.085 m 3 in volume (0.42 m Â 0.44 m Â 0.46 m) was constructed using antistatic polycarbonate. A petri dish containing 1 to 10.5 mL of solvent mixtures (Table 4) was placed into the chamber, serving as the emission source (Figure 2A).
For the modeling process, evaporation rate constants alphas (a) estimated using Equation (4) (the non-corrected alphas) and Equation (5) (the corrected alphas) were used as emission rates in the model for each test. To estimate the corrected alphas, activity coefficients (c i ) for each component of the mixtures were calculated using the UNIFAC calculator proposed by Choy and Reible (Choy and Reible 1996), as shown in Figure S3 in the supplemental materials. The room volume (V) was the volume of the experimental chamber (0.085 m 3 ), the general ventilation rate (Q) was set to 2.5 L/min (0.0025 m 3 /min) ( Figure  2B), the values for the initial contaminant masses (M 0 ) were provided by the solvents evaporation tests described in the section dedicated to the gravimetric measurement of the evaporation rate constants, the initial concentration (C 0 ) in the box was always set to zero, and the maximum simulation time was 400 minutes. The values for the model input parameters for all the exposure scenarios evaluated are summarized in Table S1 in the supplemental materials.
For each test, two model estimates were provided: the non-corrected predicted concentrations (using the non-corrected alphas), and the corrected predicted concentrations (using the corrected alphas).

Real-time contaminant air concentrations in the experimental chamber
To measure the solvent concentrations in the experimental chamber, we used a gas chromatography (GC) technique equipped with a thermoconductivity detector (TCD). The GC system was composed of a gas chromatograph (GC) (Variant CP2003-P), a carrier gas (mobile phase, Helium), an injector, a 10-m column (CP-Sil-5 containing a 100% dimethylpolysiloxane stationary phase), a detector (TCD), and a recorder (Personal Computer with "stars" software preinstalled). The GC was connected to a sampling pump that removed the solvent vapors from the box via a Teflon tubing (Saint-Gobain Chemfluor 367 Scientific Grade Natural Tubing) at a rate of 0.04 L/min. The inlet of the Teflon tubing was located 20 cm above the petri dish (i.e., emission source) ( Figure 2B). The removed air (i.e., carrying the solvent vapors) was directly exhausted into the GC system via the sampling pump, and thereafter was processed for the separation, identification, and quantification of the components of the solvent mixtures, depending on the retention characteristics of the chemical compounds to be analyzed.
Before the measurements, the GC was calibrated for all the tested solvents, for several concentrations (250, 500, 750, 1,000, and 1,500 ppm) using Tedlar calibration bags. Figure S4 in the supplemental materials shows the calibration curve obtained for toluene.
Analytical methods were also developed for each solvent and the binary mixtures to optimize the detection parameters. The common chromatographic conditions under which all the tests were carried out were: injection time (255 msec), sampling time (10 sec), and sensitivity (high).

Model performance evaluations and data analysis
Model estimates were compared against the measured concentrations using both graphical and numerical approaches. Before the evaluations were carried out, the IHMOD predicted concentrations initially estimated in mg/m 3 were converted to ppm (parts per million) to harmonize with the GC concentrations. The model performance evaluation approach consisted of evaluating how close the time-varying predicted concentration profiles were to the measured concentration profiles in terms of the values for the peak concentrations (ppm) and the time (min) to reach the peak concentrations.
Both parameters were determined for each scenario and for each concentration profile (non-corrected predicted, corrected predicted and GC measured). Ratios between the predicted and the measured values were then calculated for the peak concentrations. Differences between the predicted and the measured peak times were also calculated for evaluations.
All the analyses were performed using the software program R (version 4.1.2, R Core Team, Vienna).

Estimating contaminant evaporation rates from small spills of non-ideal solvent mixtures
Determining the evaporation rate constant alpha (a) for the five organic solvents Figure 3 shows an example of the evaporation data plot for methanol using a spill volume of 0.5 mL in a petri dish of 10 cm in diameter.
The values for alpha for each solvent evaporation test are presented in Table 3.
Estimating evaporation rates for the solvents released from the mixtures Evaporation rate constants alphas (a) for the chemicals used in the mixtures were estimated based on the experimental alphas (Table 3), the molar fractions, and the activity coefficients of the constituents of the mixtures (according to Equations (4) and (5)). The values for the molar fractions, activity coefficients, and estimated alphas for each exposure scenario are summarized in Table 4.
The activity coefficients varied from 1 to 20. The activity coefficient equaled one for chemicals used in the ideal mixture (toluene and butan-2-one) while being greater than one for chemicals involved in the non-ideal mixtures. As molar fractions of the components of the mixtures decreased, the activity coefficients increased. The largest activity coefficients were observed for the binary non-ideal aqueous mixtures of butan-2-one and propan-2-ol at 1% of molar fractions (activity coefficients being 20 and 19.9, respectively).
For the binary ideal mixture of toluene (46%) and butan-2-one (54%), the estimated corrected alphas were the same as the estimated non-corrected alphas (0.0229 min À1 for toluene and 0.0860 min À1 for butan-2-one), as for each substance, the activity coefficient was equal to one.
For the 12 binary non-ideal aqueous mixtures, the estimated corrected alphas ranged from 0.00457 min À1 to 0.214 min À1 , with an average value of 0.0646 min À1 and a median value of 0.0318 min À1 , while the estimated non-corrected alphas varied from 0.00044 min À1 to 0.0429 min À1 , with an average value of 0.0111 min À1 and a median value of 0.00632 min À1 .

Modeling contaminant air concentrations and model performance evaluations
The evolution of vapor concentrations in the test chamber predicted by the IHMOD well-mixed room model combined with the exponentially decreasing emission rate using both the non-corrected and the corrected alphas are presented as follows.
For the binary ideal mixture Figure 4 presents the concentration profiles of each solvent involved in the ideal mixture of toluene (46%) and butan-2-ol (54%). The IHMOD model predicted the measured concentrations well in terms of the values for the peak concentrations and the timing for the peak concentrations. However, the predicted peak concentrations were slightly lower for both chemical compounds compared to measured values (the lowest values being observed for the toluene evaluations).
The peak concentrations and the peak times measured for butan-2-one were 1,122 ppm (19 min) while being 583 ppm (40 min) for toluene. The peak concentrations and peak times estimated by the IHMOD model for butan-2-one were 901 ppm (20 min) while being 432 ppm (40 min) for toluene. The predicted peak concentrations-to-measured peak concentrations ratios were 0.80 for butan-2-one and 0.74 for toluene. Differences between the predicted and the measured peak time values were þ1 min for butan-2-one and zero min for toluene (Tables 5 and 6).
For the 12 binary non-ideal aqueous mixtures Figure 5 presents the measured concentrations as well as the concentration profiles estimated by the IHMOD model considering the corrected evaporation rates (i.e., evaporation rates corrected by the activity coefficients of the solvents) and the non-corrected evaporation rates, for all the solvents, and for all the  The values for the molar fractions, the activity coefficients, and the estimated alphas were calculated only for the targeted chemical compounds.
evaluations. The concentrations modeled using the corrected evaporation rates were closer to the experimental measurements than those obtained using the non-corrected evaporation rates in all evaluations. The median measured peak concentrations were 493.5 ppm (248 to 764), 805.5 ppm (602 to 1076), and 979 ppm (811 to 1,209), respectively, for the 1%, 5%, and 10% mixtures. The median peak concentrations estimated by the modeling accounting for the correction factor were 430.5 ppm (234 to 891), 747 ppm (543 to 1,311), and 909 ppm (697 to 1,394), respectively, for the 1%, 5%, and 10% mixtures. The median peak concentrations estimated by the modeling without accounting for the correction factor were 99 ppm (26 to 219), 338.5 ppm (105 to 619), and 513.5 ppm (194 to 846), respectively, for the 1%, 5%, and 10% mixtures. The predicted peak concentrations-to-measured peak concentrations ratios obtained using the corrected evaporation rates ranged from 0.81 to 1.32 (Table 5), with an average value of 0.98 and a median value of 0.92, while those obtained using the non-corrected evaporation rates varied from 0.08 to 0.75 (Table 5), with an average value of 0.40 and a median value of 0.31. The median measured peak times were 46 min (35 to 88), 31.5 min (15 to 61), and 27 min (18 to 40), respectively, for the 1%, 5%, and 10% mixtures. The   Table 6. Model performance evaluations using the differences between the predicted and the measured peak times.   peak times estimated by the modeling taking into account the correction factor were 46 min (28 to 76), 32 min (12 to 44), and 24 min (12 to 32), respectively, for the 1%, 5%, and 10% mixtures. The median peak times estimated by the modeling without taking into account the correction factor were 100.5 min (76 to 160), 58 min (40 to 96), and 44 min (28 to 80), respectively, for the 1%, 5%, and 10% mixtures. The differences between the predicted and the measured peak time values obtained using the corrected evaporation rates ranged from À28 min to þ41 min (Table  6), with an average value of À4.8 min and a median value of À5 min, while those obtained using the noncorrected evaporation rates varied from þ10 min to þ72 min (Table 6), with an average value of þ34.92 min and a median value of þ33 min.

Estimating contaminant evaporation rates from small spills of non-ideal solvent mixtures
For the ideal solvents mixture of toluene (46%) and butan-2-one (54%), there was no difference between the estimated corrected and non-corrected alphas, as for each substance, the activity coefficient was set equal to one.
For the 12 binary non-ideal aqueous mixtures, the estimated corrected alphas were higher than the estimated non-corrected alphas in all evaluations (with the median values of 0.0318 min À1 versus 0.00632 min À1 ), which suggests that wrongfully assuming ideal behavior for non-ideal liquid mixtures can lead to significant errors in the estimation of evaporation rates. These results are in line with those obtained by Nielsen and Olsen, who used a theoretical evaporation model to predict evaporation rates for eight binary and strongly non-ideal aqueous mixtures in which the molar fractions of the dilute chemicals ranged from 8.5 Â 10 À5 to 5.1 Â 10 À3 . They found that predicted evaporation rates using non-ideal calculations were significantly higher in all cases (median ¼ 6,650 Â 10 À4 mmol m À2 s À1 ) compared to the evaporation rates predicted using ideal calculations (median ¼ 7 Â 10 À4 mmol m À2 s À1 ). Moreover, when comparisons were made against experimentally measured evaporation rates, results showed a satisfactory agreement between the predicted evaporation rates using non-ideal calculations and the measured values (median predicted-to-measured evaporation rate ratio ¼ 1.14 (0.84 to 1.57). Calculations that did not take into account the non-ideality were found to be meaningless as the resulting predicted evaporation rates drastically underestimated the measured values in most cases (median predicted-to-measured evaporation rate ratio ¼ 0.002 (0.0002 to 0.08)) . These results emphasize the importance of accounting for non-ideality in the process of estimating evaporation rates from solvent mixtures.

Modeling contaminant air concentrations and model performance evaluations
Solvent vapor concentrations in the test chamber were estimated by applying the IHMOD small spill model (well-mixed room model combined with the exponentially decreasing emission rate) using both the corrected and the non-corrected alphas.
For the ideal solvent mixture of toluene (46%) and butan-2-one (54%), the modeled concentrations were in excellent agreement with the measured concentrations, with model estimates within a factor of 0.74 to 0.80 the measured values. Differences between the modeled and the measured peak times were zero min and þ1 min for the toluene and butan-2-one evaluations, respectively. These observations highlighted the ability of the equation adapted from Raoult's Law to reasonably estimate evaporation rate constants alphas for ideal mixtures.
For the 12 binary non-ideal aqueous mixtures, the modeled concentrations using the corrected evaporation rates closely matched the measured concentrations in all evaluations, with the predicted peak concentrations-to-measured peak concentrations ratios ranging from 0.81 to 1.32, and a median value of 0.92. By contrast, when the non-corrected evaporation rates were used, the predicted peak concentrations-tomeasured peak concentrations ratios varied from 0.08 to 0.75, with a median value of 0.31. Moreover, the median difference between the modeled and the measured peak times was À5 min (with a maximum value of þ41 min) when evaporation rates were corrected versus a median value of þ33 min (with a maximum value of þ72 min) when evaporation rates were not corrected. Thus, ignoring the correction by the activity coefficients for non-ideal systems can result in considerably underestimating the peak levels and misjudging the timing of the peaks. This could be especially important to consider when workers would be assigned to the clean-up and/or emergency response activities in a supposedly "safe time period" during which peak concentrations would actually occur. These results underline the effectiveness of the equations proposed in this study for estimating evaporation rates in exposure scenarios involving small spills of non-ideal organic solvent mixtures. Finally, the UNIFAC model has been proven useful for estimating the activity coefficients (c) of the solvents involved in the liquid mixtures evaluated in this study.
To our knowledge, the current study is the first of its kind to have considered the non-ideality of chemical substances in the estimation of an exponentially decreasing evaporation rate and the modeling of vapor concentrations resulting from small spills of organic non-ideal solvent mixtures using a physical-chemical model (notably the IHMOD spill model also known as the well-mixed room model with exponentially decreasing emission rate).
It is also among the rare studies that have used the gas chromatography technique with direct reading of the emitted solvent vapors to measure the real-time concentrations generated by the solvents evaluated. In reviewed scientific literature, studies either used multigas detection monitors with different types of detectors (photoionization detectors (PID), flame ionization detector (FID), photoacoustic infrared detectors) for solvent vapors measurement Reinke and Brosseau 1997;Guo et al. 2008;Earnest and Corsi 2013;Keil 2015;Keil and Zhao 2017;Arnold et al. 2017aArnold et al. , 2017b, portable Fourier Transform Infrared (FTIR) spectrophotometers (Arnold et al. 2020), or gas chromatography systems in which solvent vapors were secondarily injected after being manually collected using materials such as syringes and septum-sealed glass flasks (Bishop et al. 1982;Keil and Nicas 2003). The direct reading procedure may have the advantage of reducing, to some extent, the measurement errors due to sampling and analysis methods (for example, errors associated with air sampling using charcoal tubes, syringes, etc.).
There are also a few limitations to this study. First, the evaporation tests were not carried out in replicates. However, a previously published study, conducted in identical experimental and environmental conditions as this study, showed little variability in alpha values when alphas were measured in triplicates (the average coefficient of variation for alphas was equal to 5.4%) (Abattan et al. 2022). Therefore, the absence of replicate measurements for alpha in this study is expected to have had only minor impacts on the results.
Second, evaluations were carried out only for binary non-ideal aqueous mixtures. Although results from this study are innovative, they should be regarded as preliminary, pending future studies which may cover a variety of more complex mixtures (including for example ternary non-ideal aqueous mixtures) with a broader range of concentration dilutions (i.e., molar fractions).
Finally, this research was conducted in a specific laboratory setting and greater variations are expected in the real world as the workplace environmental conditions may significantly differ from the ones observed in the laboratory. However, in any case, the consideration of non-ideality would bring the estimate closer to reality.

Conclusion
Two equations derived from Raoult's Law were proposed to estimate evaporation rate constants alphas (a) resulting from various small spills of organic solvent mixtures. For the 12 binary non-ideal aqueous mixtures evaluated, the estimated alphas accounting for the correction by the activity coefficients of the solvents were higher than the alphas estimated without the correction.
Time-varying solvent concentrations measured in a small-scale experimental chamber were close to those estimated by the IHMOD small spill model (wellmixed room model combined with the exponentially decreasing emission rate) using corrected alphas. When the non-corrected alphas were used, peak concentrations were underestimated and peak times were overestimated.
Occupational health professionals should be aware of the non-ideality effect, as failing to do so can influence both the magnitude and the kinetics of exposures by significantly underestimating contaminant airborne concentrations and causing the professionals to misjudge the time required to reach peak concentrations. The consideration of non-ideality in the estimates of evaporation rate constants alphas constitutes an improvement in the accuracy of the exposure models used to estimate occupational exposures to solvent mixtures.

Funding
This project was financially supported by the Institut de recherche Robert-Sauv e en sant e et en s ecurit e du travail (IRSST #2012-0044). S.F.A received a scholarship from the Institut de recherche Robert-Sauv e en sant e et en s ecurit e du travail.

Data availability statement
The data supporting the findings of this study are available within the article, its online supplemental materials and in the Open Science Framework (OSF) data repository accessible at: https://osf.io/tdfmj/?view_only¼fabda7562f29476d86 ab7ef1d4cb7f3b.