Comparing Antoine parameter sources for accurate vapor pressure prediction across a range of temperatures

Abstract

Determining the vapor pressure of a substance at the relevant process temperature is a key component in conducting an exposure assessment to ascertain worker exposure. However, vapor pressure data at various temperatures relevant to the work environment is not readily available for many chemicals. The Antoine equation is a mathematical expression that relates temperature and vapor pressure. The objective of this analysis was to compare Antoine parameter data from 3 independent data sources; Hansen, Yaws, and Custom data and identify the source that generates the most accurate vapor pressure values with the least bias, relative to the referent data set from the CRC Handbook of Chemistry and Physics. Temperatures predicted from 3 different Antoine sources across a range of vapor pressures for 59 chemicals are compared to the reference source. The results show that temperatures predicted using Antoine parameters from the 3 sources are not statistically significantly different, indicating that all 3 sources could be useful. However, the Yaws dataset will be used in the SDM 2.0 because the data is readily available and robust.

What’s Important About This Paper?

Vapor pressure is an important parameter in many models of occupational exposures, and can be predicted by the Antione equation. This study demonstrates that 3 possible sources of data for use in the Antione equation gave similar results for estimating vapor pressure, and agreed well with reference data. While all data are equally suitable for estimating vapor pressures at non-ambient temperatures, data from the Yaws Handbook of Vapor Pressure are recommended because they contain the greatest number of relevant chemicals.

Background

The Antoine equation is a widely used method for estimating the vapor pressure of a pure substance as a function of temperature. This equation relates the logarithm of the vapor pressure of a substance to its temperature and 3 constants known as the Antoine parameters. The accuracy of the vapor pressure predictions obtained with the Antoine equation depends on the quality of the Antoine parameter values used for a given substance. In the literature, several sources of Antoine parameter values can be found, each based on different experimental data and estimation methods. As a result, discrepancies may arise among the predictions obtained with different sources of Antoine parameter values, particularly at extreme temperatures.

To evaluate the performance of different sources of Antoine parameter values for predicting vapor pressures, a study was conducted comparing 3 sources of Antoine parameter values to a reference source of vapor pressure data across a range of temperatures. The 3 sources of Antoine parameters included in the study were selected to represent different types of parameter estimation methods, including experimental measurements, regression analyses, and group contribution methods. The reference source of vapor pressure data used in the study was obtained from a reliable experimental database. The study focused on a diverse set of pure substances, including organic compounds, inorganic compounds, and mixtures. The temperature range covered by the study extended from sub-ambient temperatures to high temperatures.

The goal of this study was to determine the optimal source of Antoine equation parameters that can be utilized in the SDM 2.0 (Arnold et al. 2022) for the estimation of vapor pressures with high precision and minimal bias across a broad range of process-related temperatures, for a wide range of chemical compounds.

The results of the study were analyzed and discussed in terms of the level of agreement between the predicted vapor pressures and the reference vapor pressure data, as well as the overall performance of each source of Antoine parameter values. The study findings may provide insights into the selection of the most appropriate source of Antoine parameter values for specific applications in thermodynamics, process design, and other related fields.

Introduction

Vapor pressure is a critical determinant of airborne exposure. Since vapor pressure is temperature dependent, and many processes in workplaces operate at temperatures other than normal, the need for accurate vapor pressure values across a range of temperatures is also critically important. Vapor pressure refers to the force per unit area exerted by the molecules of a vapor that is in equilibrium with a liquid or solid at a given temperature. A higher vapor pressure indicates a greater propensity for the chemical to vaporize, leading to a higher concentration of the substance in the air and thus a greater risk of exposure via inhalation. Vapor pressure values can be measured or estimated. They are key inputs for several exposure assessment algorithms and heuristics which are used in tools that are based on physical–chemical properties for which vapor pressure plays an integral part of the exposure assessment, including the checklist (Arnold et al. 2016) and its successor, the structured deterministic model (SDM)2.0 (Arnold et al. 2022). Exposure judgments based on these tools were shown to predict the true exposure control category (ECC) 64% of the time, and within +/- one category 81% of the time (Arnold et al. 2016). Vapor pressure is also used in exposure assessment tools recommended for use under European Chemicals Agency (ECHA) guidance R.14 in guidelines such as the Stoffenmanager®v4.5 and Advanced REACH Tool ((ECHA) 2010). A study evaluating the performance of the tools found that exposure assessments were highly influenced by the input parameters in the tools, vapor pressure being among the most important (Lee et al. 2018). Accurate values for vapor pressure are also important for informing engineering control designs and specifications.

In occupational settings, vapor pressure is dependent on the process temperature of the chemicals generating the vapor (Stauffer et al. 2008). Chemicals are commonly used in industrial processes at temperatures and pressures that deviate from normal temperature and pressure (NTP) (20 °C and 1 atmosphere). Since the range of possible temperatures is a continuum, there is a need for VP values across a range of relevant temperatures. When measured values are not available at the desired process temperatures, it is necessary to estimate them with a relatively high degree of confidence in their accuracy. The need to calculate vapor pressures at various temperatures is common to many manufacturing sites and methods to conduct these calculations are well established (Sanjari 2013). For example, the Antoine equation (Equation 3), derived from the Clausius–Clapeyron equation (Equation 1). The equation is useful for estimating VP for industrial applications.

The Clausius Clapeyron equation expresses a thermodynamic relationship between the inverse temperature of a chemical, expressed in absolute temperature and the corresponding vapor pressure (VP) at that temperature. This equation assumes the chemical is behaving as an ideal gas. That is, all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces. The equation has 2 forms. The first equation is expressed as follows:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{L}\mathrm{n}\left(P_{\mathrm{v}\mathrm{a}\mathrm{p}}\right)=(-\mathrm{\Delta }{H}_{\mathrm{V}\mathrm{a}\mathrm{p}}/R)\ast \left(1/T\right)+C}\end{array}}$$ (1)

where:

P _vap is the vapor pressure of the chemical expressed in mm Hg,

ΔH_Vap is the enthalpy of vaporization of the liquid at temperature T expressed in kJ/mol⁻¹

R is a constant equal to 8.3145 J/K mol,

T is the absolute temperature expressed in °K

C is a constant associated with the specific chemical of interest

If the independent variable 1/T is plotted against the dependent variable Ln (P_vap), a linear relationship should be observed with the slope of the line equal to −ΔH_Vap/R and an intercept C. However, this would be based on the assumption that there is constant ΔH_Vap, which is only true for narrow temperature ranges (Brozena et al. 2016b). For process temperatures less than 300 °K, the ΔH_Vap drops about 10% per 100 °K. Equation 1 can be integrated between 2 pressure–temperature endpoints to obtain a simple relationship shown in Equation 2.

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{L}\mathrm{n}\left(P_1/P_2\right)=(-\mathrm{\Delta }{H}_{\mathrm{V}\mathrm{a}\mathrm{p}}/R)\ast (\left(1/T_2\right)-\left(1/T_1\right))}\end{array}}$$ (2)

Equation (2) can be used when the VP is known at 2 or more temperatures, such as a standard reference temperature and the boiling point, where the VP is equal to atmospheric pressure (e.g., 760 mm Hg).

The Antoine Equation (3) mimics the Clausius–Clapeyron equation and is easier to use when populating a database with a large number of chemicals.

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{L}\mathrm{n}\left(P\right)=A-\left(B/(T+C)\right)}\end{array}}$$ (3)

where:

P is the vapor pressure expressed in mm Hg,

T is the temperature expressed in °C,

A, B, and C are empirically determined Antoine parameters.

It should be noted that some sources of Antoine’s constants use P expressed in Pascals (Pa) and temperature in °K. The SDM2.0 tool generates VP expressed in mm of Hg, so a unit conversion is necessary. For chemicals included in the SDM 2.0 database, this is done automatically.

An important limitation of the Antoine equation is that the equation parameters were empirically derived using measured VP for a specific temperature range. The use of the equation outside of this range could lead to an over- or underestimation of the VP. In these cases, the predicted airborne concentration resulting from vaporization of chemical contaminants and related inhalation exposure would have more uncertainty. Accurately predicted vapor pressures at normal temperatures is of particular concern when only high-temperature data are available. Additionally, Antoine parameters are specific to a particular chemical and cannot be used to predict the vapor pressure of other chemicals resulting in a need for larger libraries for each chemical. Furthermore, the accuracy of the Antoine equation depends on the quality of the data used to derive the parameters, and the accuracy of the parameters can vary significantly between different sources, hence the need to evaluate accuracy across different sources.

Context for how vapor pressure is used in the structured deterministic model (SDM 2.0)

The SDM 2.0 uses heuristics based on physical–chemical principles that were refined empirically over many years (Arnold et al. 2016). Vapor pressure is a critical model input of these heuristics. The expanded functionality in the SDM 2.0 to assess chemical mixtures is also highly dependent on accurate vapor pressure values at the relevant liquid or solid source temperature. When assessing an exposure scenario involving only pure chemicals, the vapor pressure associated with the temperature of the liquid chemical, expressed in mm Hg is used, along with the relevant occupational exposure limit (OEL) to estimate the vapor hazard ratio (VHR):

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{V}\mathrm{H}{\mathrm{R}}_{\mathrm{A}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}}=\mathrm{V}{\mathrm{P}}_{\mathrm{A}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}}/\mathrm{O}\mathrm{E}{\mathrm{L}}_{\mathrm{A}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}}}\end{array}}$$ (4)

where:

VHR_Agent is the vapor hazard ratio of agent,

VP_Agent is the vapor pressure of pure agent,

OEL_Agent is the occupational exposure limit of agent.

When assessing a scenario involving chemical mixtures, the vapor pressures are dependent on both the temperature of the liquid and on the composition of the mixture. However, the need for accurate vapor pressure values at the relevant temperature remains.

The VHR is used to determine the required level of control. In the case of chemical mixtures, the vapor pressures of components in the mixture and the mole fraction of each component are used to calculate the adjusted VP for each chemical, using Raoult’s Law (as the default) or, alternatively, Henry’s Law for polar or dilute aqueous mixtures. The current version of the SDM 2.0 does not include activity factors to account for intermolecular forces in mixtures and the consequent impact on the components’ VP, but the inclusion of these factors is under consideration. The adjusted vapor pressure is used to identify the controlling compound, defined as the chemical with the highest adjusted VHR, and signifying the chemical with the greatest potential to exceed its OEL. The controlling compound is then used to determine the required level of control for that mixture. A salutary benefit of this heuristic is the ability to estimate the exposure level for all the components in the mixture if the exposure level of any component of the mixture is known by using the ratio of the components’ VHR’s (known/unknown) and their respective OELs. There are several scenarios where estimating exposure through this approach requires additional steps, such as where mixture compositions are reported in ranges. The SDM 2.0 Support File provides additional direction and includes a simplified Raoult’s law or Henry’s law spreadsheet to facilitate this application. The use of VHR has been shown to estimate an exposure control category for pure chemicals that are similar to that predicted with exposure measurement data (Arnold et al. 2016).

To optimize the ease and efficiency of the tool and its heuristics, a database of chemicals and their chemical properties must be readily available to the user. The objective of this work was to identify the best source of Antoine equation parameters for use in the SDM 2.0 to estimate vapor pressures with a reasonable degree of accuracy and the least amount of bias, for the greatest number of chemicals across a range of process-relevant temperatures.

In our evaluation of Antoine parameter sources, we implemented a rigorous process guided by specific criteria. Our focus revolved around several key considerations. First, we assessed the sources’ alignment with empirical data across a wide temperature spectrum, emphasizing consistency in predictions. Second, the validity of each source’s theoretical foundation was critically assessed. We favored sources rooted in established physical chemical principles and supported by prior research, as well as those that were empirically validated whenever possible. This evaluation involved a thorough investigation of the sources to ensure they adhered to sound scientific principles.

Methodology

Directly comparing vapor pressure values estimated from the Antoine parameters across a range of temperatures would require vapor pressure values at each of the temperatures selected for the study. Since only a subset of these values were available from the referent source, the Handbook of Chemistry and Physics 102nd Edition (Rumble 2021) an alternate approach was necessary. The Antoine equation can be rearranged to solve for temperature (T) instead of vapor pressure (P). The equation to predict temperature from vapor pressure using the Antoine equation is:

$${\displaystyle \begin{array}{l}{\displaystyle T=(B-C\ast (\mathrm{l}\mathrm{n}\left(P\right)+A))/C}\end{array}}$$ (5)

where:

P is the vapor pressure expressed in mm Hg,

T is the temperature expressed in °C,

A, B, and C are empirically determined Antoine parameters.

To evaluate the Antoine equation parameters sourced from 4 distinct references, our methodology necessitated a modification in the equation’s structure. Rather than solving for vapor pressure, our approach involved rearranging the equation to derive temperature values. This adjustment was prompted by the available data, where temperatures across a defined range of vapor pressure values were accessible, while vapor pressure values at specific temperature points were lacking. Consequently, to ensure comprehensive parameter assessment, we employed a strategy of estimating temperatures across a range of vapor pressures. This method allowed for a comparative analysis of predicted temperatures based on the Antoine parameters obtained from diverse sources.

Data sets

A data set of 59 commonly used industrial chemicals was compiled using online sources. The chemicals were selected to represent a broad range of chemical groups and chemical properties. These chemicals have different functional groups and include aldehydes, ketones, amines, alcohols, and ethers. A complete list of chemicals and related properties is provided in Supplementary Table S1.

Temperature data for these chemicals at 6 different vapor pressures (0.00751 mm Hg, 0.0751 mm Hg, 0.751 mm Hg; 7.51 mm Hg, 75.1 mm Hg, and 751 mm Hg) were extracted from the Handbook of Chemistry and Physics 102nd Edition (Rumble 2021). In our study, where the accuracy and reliability of data sources play a pivotal role, our choice to utilize the ‘Handbook of Chemistry and Physics 102nd Edition’ as a reference is underpinned by its well-established reputation as a gold standard in providing comprehensive, rigorously validated data on various substances’ physical properties. It includes vapor pressure versus temperature values with a wide range of magnitudes, suggesting a mixture of experimentally derived and potentially extrapolated or interpolated data. This handbook stands as an indispensable resource, renowned for its meticulous curation and verification processes, ensuring the highest quality and precision in the data it presents. By anchoring our study on this trusted source, we establish a solid framework for evaluating and comparing the reliability and accuracy of other parameter sources.

Antoine parameter data were extracted for each chemical from 3 different sources: Yaws (Yaws 2015b, 2015a), Hansen (Hansen 2007), and an internal customized database (Lara 2017). The customized data set was created to support the development of software that allows the selection of best chemical protective materials(Lara 2017). These 3 sources have temperature and vapor pressure values for which experimental data is available and tabulating methods are used to provide parameter estimates. The method involves measuring the vapor pressure of a chemical at several different temperatures, and then using this data to fit the Antoine equation to the data using regression analysis. The Antoine parameters (A, B, and C) are then estimated from the regression results. Equation 3 parameters, (A, B, and C) and their range of application, Tmin, and Tmax, are compiled, which establish temperature limits for each equation, meaning the temperature range used in fitting the equation.

The temperatures were estimated from the Antoine equation (Equation 3) using the Antoine parameters from each source Yaws (Yaws 2015b), Hansen (Hansen 2007), customized at each of the 6 pre-defined VP from the data source. Temperatures that fell outside the set T_min and T_max were removed from the dataset to ensure that comparable data was used in the analysis. Supplementary Table S2 shows the 59 substances included in this study, and the predicted temperatures from the different Antoine parameters at the defined vapor pressures, against the reference temperatures, from the CRC Handbook of Chemistry and Physics.

Data analysis

To evaluate the accuracy and bias of the 3 data sets, 2-way comparisons between predicted temperatures and empirically derived temperatures listed in the CRC Handbook of Chemistry and Physics (102nd Ed.) (Rumble 2021), (referent temperature) were conducted using the Bland–Altman linear regression analysis (Bland Jm Fau—Altman and Altman). Briefly, this method is used to assess the degree of agreement between each predicted value and experimental value. The Bland–Altman comparison is used to describe agreement between 2 quantitative measurements of the same variable using a graphical method. Bland–Altman plots are widely used for comparing and visualizing the similarities/differences between 2 datasets. A variation of the Bland–Altman method is used to determine if analytical methods used to measure exposures have acceptable bias and precision. For this analysis, a scatterplot was generated for each 2-way comparison in which the X-axis represents the average of the predicted temperature and referent temperature and the Y-axis represents the difference between referent temperature values and the predicted temperature values. For example the Bland–Altman plot comparing Yaws’ predicted temperature and CRC reference temperatures, for acrolein at 751 mmHg, the predicted temperature is 56.5 °C while the reference temperature is 52.8 °C. Therefore, the average temperature (x-value) is 54.65 °C and the difference (y-value) is 1.07 °C.

Mean bias and limits of agreement are calculated using the mean of the difference of measurement values and its SD obtained from the one-sample t-test. The data points can be restricted using ±2 SD to demonstrate a 95% CI; precisely defined: mean ± 1.96 SDs of distributed data. The limits of agreement in this data were calculated as ±1.96 SDs from the bias. The critical difference was calculated as a 1.96 SD of the difference (i.e., half the difference between the lower and upper limits of agreement). Further evaluation of the accuracy of the different Antoine parameters was performed by calculating average absolute relative deviation (AARD%), and root mean square deviation (RMSD) of the values obtained different Antoine parameters and the reference values. For comparison purposes, only chemicals with all 3 Antoine parameter values are included (n = 38 chemicals).

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{A}\mathrm{A}\mathrm{R}\mathrm{D}=\frac{1}{N}{\displaystyle \sum _{I=1}^n}\frac{T_{i\left(\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\right)}-T_{i\left(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\right)}}{T_{i\left(\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\right)}}x100}\end{array}}$$ (6)

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}=\frac{1}{N}{\displaystyle \sum _{I=1}^n}{\left(\frac{T_{i\left(\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\right)}-T_{i\left(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\right)}}{T_{i\left(\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\right)}}\right)}^{2}x100}\end{array}}$$ (7)

where:

T _{i (reference)} is the reference temperature (°C),

T _{i (calculated)} is the temperature predicted by the model °C.

Following the initial analysis, the dataset was divided into solid, liquid, and ‘Whole Data’ subsets to assess temperature prediction models (Yaws, Daniel, and Hansen) across substance states. Bland–Altman comparisons were performed separately for each subset, generating scatterplots and determining mean bias, limits of agreement, average absolute relative deviation (AARD%), and root mean square deviation (RMSD). This approach enabled a nuanced evaluation of model performance across different substance states. Moreover, complementary analyses included linear regression and 2-sample Welch t-test methodologies. The linear regression assessed the relationship between predicted and reference temperatures, uncovering insights into their association’s strength and direction. Additionally, the 2-sample Welch t-test facilitated detailed comparisons between datasets, considering sample size and variance discrepancies.

Results

Results from the Bland–Altman analysis revealed a close agreement between predicted temperatures derived from Yaws (Yaws 2015b, 2015a), Hansen (Hansen 2007), and customized against CRC Handbook of Chemistry and Physics (Rumble 2021). The 3-panel figure (Figure 1) displays these comparisons for the whole dataset, solids, and liquids, demonstrating consistent agreement between predicted and referent temperatures across different substance categories. “An aggregate comparison is shown in Fig. 2, demonstrating this close agreement as predicted temperatures largely overlap. Therefore, all the 3 sources are useful. In the whole dataset, only 2.54% of the Yaws’ data lie outside the confidence interval, while there is 2.82% for Hansen and 1.72% for customized data.

Fig. 1.

Nine-panel Bland–Altman plots illustrating the agreement between predicted and reference temperature values across various subsets (whole, solids, and liquids) based on Yaws, Hansen, and custom models.

Fig. 2.

Predicted temperatures generated from Antoine equation coefficients sourced from custom, Yaws in whole data (N = 266), Daniel in whole data (N = 194), Hansen in whole data (N = 266) Yaws in solids (N = 22), custom in solids (N = 18), Hansen in solids (N = 216).

Notably, a higher degree of variability was evident in the solid-state substances, as evidenced by a greater number of points lying outside the limits of agreement. This discrepancy suggests potential challenges or limitations in predicting temperatures for solid substances compared to liquids and gases. Further investigation is warranted to explore the underlying reasons contributing to this increased variability in the solid-state predictions.

Our analysis using t-tests initially displayed differences between the CRC data and predictions from the Yaws and Hansen models in the complete dataset (Table 1). However, upon a closer examination by segregating the dataset into solid and liquid phases, contrasting the CRC data temperatures with predictions from these models revealed interesting outcomes. Notably, while differences were observed in the complete dataset, these models demonstrated consistent and close alignment with the CRC data for both solid and liquid phases individually (Table 1). This indicates that the Hansen and Yaws models show closer agreement with the CRC data when considered separately for solid and liquid phases, despite the differences observed in the complete dataset. Moreover, the customized data set consistently exhibited minimal disparities when compared to the CRC data across different substance states, implying its reliability in predicting temperatures for various phases.

Table 1.

Statistical comparison using Welch t-test of Antoine parameter-predicted temperatures against CRC reference across different subsets ¹(Yaws and Satyro, 2015b, 2015a); ²(Hansen, 2007).

	t-Value	Degrees of freedom	P-value
Whole data
CRC_Reference versus Yaws	2.062	579.66	0.040
CRC_Reference versus Custom	1.267	506.66	0.206
CRC_Reference versus Hansen	1.990	581.71	0.047
Solids
CRC_Reference versus Yaws	−0.346	49.171	0.731
CRC_Reference versus Custom	−0.182	47.815	0.857
CRC_Reference versus Hansen	0.197	49.632	0.844
Comparison
CRC_Reference versus Yaws	0.590	456.89	0.555
CRC_Reference versus Custom	−0.095	395.36	0.924
CRC_Reference versus Hansen	0.454	460.24	0.650

In Table 2, a comparative analysis of temperature prediction model performance across substance states is presented, employing mean absolute error (MAE) and root mean square error (RMSE) metrics. The findings highlight distinct performance variations among predictive models. Notably, in the ‘Whole Data’ set, the Yaws model yielded an MAE of 2.01 and an RMSE of 3.81, while the Daniel model had an MAE of 1.89 and an RMSE of 3.57. Comparatively, the Hansen model exhibited an MAE of 2.57 and an RMSE of 4.90. When examining predictions for solids, the Yaws and Daniel models showed similar performance, with an MAE of 1.95 and an RMSE of 3.54, whereas the Hansen model demonstrated notably higher values, with an MAE of 5.36 and an RMSE of 6.77. Similarly, for liquid phases, the Yaws and Daniel models displayed close MAE values of 1.90 and 1.77, respectively, with corresponding RMSE values of 3.74 and 3.46. However, the Hansen model exhibited an MAE of 2.16 and an RMSE of 4.52. The linear regression analysis, summarized in Supplementary Table S2, indicates a strong correlation between the CRC reference and temperature predictions across the Yaws, Hansen, and custom dataset models for whole, solids, and liquids subsets, exhibiting coefficients ranging from approximately 1.004–1.008 and high R-squared values surpassing 0.99, further corroborating the robust associations observed between these variables. These metrics provide essential insights into the predictive accuracy of models across various substance states. Overall, the data points signifying a unique chemical, from predicted temperature vs. vapor pressure from the 4 sources overlap in the scatter plot and cluster together for each point, suggesting that the Antoine equation coefficients from each source are producing similar results for the given substance (Fig. 2). The cumulative frequency graph (Fig. 3), illustrating the proportion of the data points that are below each AARD value shows the accuracy of each parameter source in predicting temperatures at various vapor pressures.

Table 2.

Comparative analysis of temperature prediction model performance using mean absolute error (MAE) and root mean square deviation (RMSD) metrics across different subsets (Yaws and Satyro, 2015b, 2015a); ²(Hansen, 2007).

Subsets	Source	MAE	RMSD
Whole data	Yaws	2.011	3.805
Whole data	Custom	1.892	3.566
Whole data	Hansen	2.574	4.902
Solid	Yaws	1.946	3.538
Solid	Custom	1.946	3.542
Solid	Hansen	5.364	6.767
Liquid	Yaws	1.9	3.744
Liquid	Custom	1.766	3.461
Liquid	Hansen	2.163	4.519

Fig. 3.

Cumulative frequency graph illustrating the distribution of absolute average relative deviation (AARD) values in the dataset. X-axis represents AARD values (difference between predicted and actual temperatures), while Y-axis shows cumulative proportion of observations at or below specific AARD values.

About 80% of the predicted temperatures are similar for all data sources. Almost all the data points are below 1% AARD, which reflects the small deviation from the reference temperature values.

The percent AARD of temperature calculated from the Antoine equation parameters relative to the referent temperatures from the CRC Handbook for Chemistry and Physics (Rumble 2021) was relatively similar across all 3 data sources. The percent AARD for each chemical ranged from ≤0.6% for Yaws and custom data to ≤1.12% for Hansen. Overall, across all the chemicals, the customized Antoine parameters had the lowest AARD% compared to the other parameters with the exception of a few chemicals, mainly aniline, epichlorohydrin, and diethyl ketone. However, these chemicals do not share similar chemical properties so further conclusions on why the model performed differently cannot be made.

Discussion

The objective of this analysis was to compare Antoine parameter data from 3 independent data sources to identify the source that generates the most accurate vapor pressure values with the least bias, relative to the referent data set from the CRC Handbook of Chemistry and Physics (Rumble 2021). Most literature focuses on the comparison of various vapor pressure prediction methods rather than the accuracy of the Antoine equation parameters (An and Yang 2012; Sanjari 2013; Szczotok et al. 2019). Ghasemi et al. (2019) compared 3 Antoine parameter prediction methods and found that some linear fitting methods provided better prediction values than others. Several researchers have also shown that constraining fit constants can result in different vapor pressure predictions (Brozena et al. 2016b, 2016a) is therefore important to check the accuracy of Antoine parameters before applying them to predict vapor pressures.

The most direct approach to compare Antoine parameters from the 3 data sources would have been using the temperatures for each chemical in Equation 3 to estimate vapor pressure. However, because the temperature values are different for each chemical, it would be difficult to have common vapor pressure input values across the different sources. Our analysis of predicted temperatures using the Antoine equation parameters from 3 distinct models demonstrated no significant deviations on average when compared with reference temperatures. In our comprehensive evaluation of predicted temperatures using Antoine equation parameters across various substance states, intriguing observations surfaced when subdividing the data into solids, liquids, and the complete dataset. Notably, the Bland–Altman analysis revealed fewer data points falling outside the limits of agreement when examining solids and liquids as distinct subsets. This finding suggests a higher degree of agreement between predicted and reference temperatures in these specific substance states. The reduced dispersion of data points beyond the limits of agreement within the solids and liquids subsets implies a comparatively stronger alignment between predicted and reference temperatures. This pattern challenges previous assertions and underscores the importance of considering substance states when evaluating predictive models. It suggests a more consistent performance of the Antoine parameters concerning specific substance phases, indicating their potential reliability and applicability within these contexts.

Across all Bland–Altman plots, a consistent pattern emerged, revealing greater point dispersion around the mean at lower temperatures and vapor pressures, while variability decreased as temperatures and vapor pressures increased. As the predicted and referent temperature increased, there was also less variability. Thus, the models appear to perform better at high vapor pressures than low vapor pressures. Notably, higher vapor pressures often align with a higher likelihood of substances existing in the liquid state under specific conditions. Tentatively, this could suggest a relatively stronger predictive performance of the evaluated Antoine parameters for liquids compared to solids. However, we exercise caution in making conclusive assertions, recognizing the need for a more granular examination of substances categorized as liquids and solids to better comprehend potential performance differences across these states of matter.

The limited representation of substances existing as solids in our vapor pressure study potentially restricts the generalizability of these observations. Including a more diverse dataset with a broader representation of solid-state substances in future studies could provide a more comprehensive understanding and validation of these observed patterns and conclusions. If too many points lie outside the limits of agreement, then the reference temperature and the predicted temperature would be considered highly uncertain, indicating that the model is not a good fit for the data. In these scenarios where an excessive number of data points fall outside the confidence interval, caution should be exercised. This situation could arise due to several factors, including discrepancies between the Antoine equation and the actual vapor pressure–temperature relationship of the specific chemical, substantial measurement errors, or substantial variability within the experimental data. It is crucial to emphasize that such situations do not necessarily imply a fundamental inadequacy of the Antoine equation for all cases, but rather underscore that its effectiveness may be diminished in specific contexts where these conditions prevail. To obtain more accurate results, additional data, or alternative methods (such as the modified Antoine method) may need to be used. In the context of exposure assessment, accurate prediction of vapor pressure plays a pivotal role. If vapor pressure estimations are not precise, they can significantly impact exposure assessments in various practical applications. For example, inaccurate vapor pressure predictions may lead to erroneous estimations of airborne concentrations, potentially resulting in flawed risk assessments. Therefore, ensuring the accuracy of vapor pressure predictions is paramount to mitigating potential errors in exposure assessments and maintaining the reliability of risk evaluation processes.

Slight differences in RMSD were seen with specific chemicals but there were no obvious patterns. For example, large RMSD values were observed for isobutanol (23.44), carbon tetrachloride (5.58), 2-butoxyethanol (9.51) acetic acid (8.95), and methyl n-butyl ketone (15.28). A large RMSD value is a sign of poor accuracy and reliability of the Antoine parameters and suggests that the Antoine equation may not be appropriate for modeling the vapor pressure-temperature relationship for the chemical in question. The cumulative frequency analysis demonstrated that while all the models’ predicted temperatures with a reasonable degree of accuracy, there are some slight differences. A possible explanation of these differences may be experimental errors encountered in the compilation of each Antoine parameter. The notable high coefficients and strong R-squared values observed in the linear regression analysis reaffirm the substantial relationship between the CRC reference and predicted temperatures across the Yaws, Hansen, and Custom dataset models for the whole, solids, and liquids subsets, underscoring the reliability of the Antoine parameter variable in predicting temperatures, which holds implications for predictive modeling and data interpretation in similar contexts.

The limitation of this study is that experimental data was compiled from different sources and therefore the accuracy of the Antoine parameters will depend on the accuracy of the experimental data used to derive them. In some cases, the accuracy of the data may be limited by the precision of the measurement instrument or by the experimental conditions. Additionally, all analyses were based on secondary data, thus there was reduced analytic flexibility. This bias was reduced by performing nonhypothesis driven analysis. A routine update of Antoine parameters as more experimental high vapor pressure data becomes available is recommended, since the accuracy of Antoine parameters may be limited by several factors, including the precision of the measurement instrument, the accuracy of the experimental data, and the limitations of the Antoine equation itself. For chemicals with high vapor pressure, it may be necessary to use alternative methods, such as cubic equations or more sophisticated models, to accurately estimate vapor pressure. In some cases, experimental data may need to be obtained specifically for the high vapor pressure conditions of interest to validate the accuracy of the Antoine parameters.

Conclusion

Analysis of Antoine parameter values from all 3 data sources resulted in similar predicted temperatures suggesting that any of the 3 sources of Antoine parameter values to predict vapor pressure in the SDM tool for qualitative exposure assessment would be acceptable. The Yaws dataset will be used in the SDM 2.0 because the data are readily available and robust. However, due to restrictive properties of the Antoine Equation, users should confirm that the temperatures at which the substance is used are within the range of the Antoine parameters, especially at extreme temperatures.

Funding

Funding for this project was provided in part by NIEHS Grant 5R25ES033035-02.

Conflict of Interest

The authors declare no conflict of interest relating to the material presented in this Article. Its contents, including any opinions and/or conclusions expressed, are solely those of the authors.

Data availability

Data used in this Article are available upon request