New Vapor-Pressure Prediction with Improved Thermodynamic

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New Vapor Pressure Prediction with Improved Thermodynamic Consistency using the Riedel Equation Joseph Hogge, Neil F. Giles, Richard L. Rowley, Thomas Allen Knotts, and Wade Vincent Wilding Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03748 • Publication Date (Web): 16 Nov 2017 Downloaded from http://pubs.acs.org on November 20, 2017

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Industrial & Engineering Chemistry Research

New Vapor Pressure Prediction with Improved Thermodynamic Consistency using the Riedel Equation Joseph W. Hogge*, Neil F. Giles, Richard L. Rowley, Thomas A. Knotts IV, W. Vincent Wilding Department of Chemical Engineering, Brigham Young University, Provo, UT, 84602, USA

* [email protected]

Keywords: Multi-property optimization Riedel correlation Vapor pressure Heat capacity

Abstract Vapor pressure, heat of vaporization, liquid heat capacity, and ideal gas heat capacity for pure compounds between the triple point and critical point are important properties for process design and optimization. These thermophysical properties are related to each other through temperature derivatives of thermodynamic relationships stemming from a temperature-dependent vapor pressure correlation. The Riedel equation has been considered an excellent and simple choice among vaporpressure correlating equations 1 but requires modification of the final coefficient to provide thermodynamic consistency with thermal data.2 New predictive correlations with final coefficients in integer steps from 1 to 6 have been created for compounds with limited or no vapor pressure data, based on the methodology used originally by Riedel.3 Liquid heat capacity was predicted using these vapor pressure correlations, and best final coefficient values were chosen based on the ability to simultaneously represent vapor pressure and liquid heat capacity. This procedure improves the fit to liquid heat capacity data by 5-10% (average absolute deviation), while maintaining the fit of vapor pressure data similar to other prediction methods. Additionally, low-temperature vapor pressure predictions were improved by relying on liquid heat capacity data.

Introduction Vapor pressure ( ) is an important property for a variety of chemical processes, especially since other properties such as enthalpy of vaporization (Δ ) and liquid heat capacity ( ) can be derived from a temperature-dependent vapor pressure correlation. However, few experimental data exist for many industrially important compounds. Therefore, predicting vapor pressure well as a function of temperature could drastically improve the reliability, safety, and profitability of chemical processes. The Riedel equation 3 for fitting and predicting the temperature dependence of vapor pressure is 1 = exp + + ln + where is vapor pressure, is temperature, and − are fitting coefficients. Recently, it has been found that changing the value – the 5th parameter – can improve both the vapor pressure fit 4 and the 1

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thermodynamic consistency with other properties—especially enthalpy of vaporization and liquid and ideal gas heat capacities.2 The Riedel equation was originally published with a method to predict vapor pressure for compounds without experimental data.3 However, this method restricted the ability of the prediction of liquid heat capacity by keeping at 6. This study created a series of new predictive vapor pressure equations with =1,2,3,4,5, and 6 values to improve the flexibility of the correlation and increase thermodynamic consistency with liquid heat capacity data while increasing the training set to include a more diverse group of compounds. Then, a larger set of compounds was tested to see which value works best for each chemical family. Previous publications found best values for alkane, alkene, aldehyde, aromatic, ether, and ketone families2. This report expanded this analysis into the alkyne, ester, gas, halogenated, multifunctional, ring alkane, silane, and sulfide families as well. Using this prediction along with methods established in the theory section of this paper extends the prediction into enthalpy of vaporization and liquid heat capacity as well, effectively turning one predictive correlation into three. The rest of the paper will go as follows. First, existing vapor pressure prediction methods will be introduced from the literature. Then, the theory linking vapor pressure, enthalpy of vaporization, and ideal gas and liquid heat capacities will be explained along with the theory behind this new predictive method. After that, the steps used to create the new method are outlined along with the compounds used to train the method. Finally, the new method is compared to other prediction methods and tested using a different set of compounds with favorable results.

Overview of Vapor Pressure Prediction and Correlation Methods

Several prediction methods are in use today. Riedel’s original prediction method, established in 1957 3, has the form: 2 ln = + + ln + where and are reduced temperature and vapor pressure, respectively, − are fitting parameters, and of Equation 1 is set to 6. This value of was chosen because it appeared to give a favorable shape towards the critical point compared to enthalpic data. Coefficients − are found using the following three constraints: 1. force Equation 2 to give 1 atm at the normal boiling point ( ) 2. force Equation 2 to give the critical pressure ( ) at the critical temperature ( ) 3. set the slope of the Riedel parameter to zero at the critical point The Riedel parameter, !, is defined as: #ln 3 !≡ = − + + 6 #ln The last coefficient is found from fitting to experimental data. Specifically, the parameter is constrained to relate to the Riedel parameter (α), evaluated at the critical point, according to: 4 = −%& − ! where % and & were found to be 0.0838 and 3.758, respectively, from a fit of data for the compounds listed in Table 2-1.

2


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Unfortunately, this formulation, where the parameter is fixed at 6, has been shown to cause the liquid heat capacity ( ) curve obtained from the so-called Derivative Method (See Section 3 and Reference 2) to dip too low at the triple point temperature ('( ).2 To correct for this, the Derivative Method can be paired with a corresponding states method (explained below) to predict .5 The former is used at higher temperatures and the latter at lower temperatures. Unfortunately, this does not ensure thermodynamic consistency between and . Table 1: Compounds originally investigated by Riedel, grouped by chemical family

Training Set Family

Number of Compounds

Alkane

4

Alkene

2

Ester

5

Ether

1

Gas

5

Halogenated

3

Lee and Kesler introduced a corresponding states prediction method (Lee-Kesler) in 1975 6 of the form: 5 ln = ln) + *ln+ where * is the acentric factor predicted using the normal boiling point, and ln) and ln+ are empirically fit equations with the same form as the Riedel predictive method with = 6. This method was proposed as a thermodynamically robust correlation and drew favorable comparisons to previous methods. Citing a need for a better predictive form, Vetere published a method using the Wagner correlation (Vetere-Wagner) in 1991.7 This approach related the temperature to the vapor pressure according to: 1 6 ln = -1 − + 1 − +./ + 1 − 0 1 where the coefficients − were calculated using the following constraints: 1. Forcing Equation 6 to give 1 atm at the normal boiling point 2. Fitting a predicted Riedel parameter, defined in Equation 3, at the critical point 3. Fitting a predicted Riedel parameter, also defined in Equation 3, at the normal boiling point These Riedel parameters were predicted using correlations from an empirical study of different chemical families. The authors admitted, at the time of its publication, that this method was only an initial attempt to make the Wagner correlation fully predictive,7 but no other major improvements have been made since. Seeking to improve on the Riedel predictive method, Vetere 8 kept Equation 2 with at 6 but changed how the empirical constant % was calculated. Instead of using only vapor pressure data to fit Equation 4, he used the normal boiling point, critical point, acentric factor, reduced temperature, and chemical family. Initially, the selected families were nonpolar compounds, acids, alcohols, glycols, and other polar compounds.8 Fifteen years later in 2006, Vetere improved upon his previous method by segregating the 3



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chemical families further and increasing the mathematical complexity of the % parameter calculations.9 This Vetere-Riedel method gave better results when predicting the vapor pressures of saturated and branched hydrocarbons, olefins and aromatic compounds, alcohols, and other compounds 9 compared to the original Riedel prediction and the Lee-Kesler prediction. These methods predict vapor pressure well. However, none of them could adequately replicate liquid heat capacity data below the normal boiling point, where the majority of heat capacity data exist. At these temperatures, few experimental vapor pressure data typically exist, and when they do, they suffer from high uncertainties.10-11 Therefore, liquid heat capacity data can be used to formulate a better vapor pressure prediction at low temperatures. A new prediction is necessary to achieve thermodynamic consistency by matching both and data.

Theory Thermodynamic Relationships and Consistency of Vapor Pressure, Enthalpy of Vaporization, Liquid Heat Capacity, and Ideal Gas Heat Capacity

In order to overcome the limitations explained in the previous section, a new prediction method was created in order to predict both and correctly. is related to through rigorous thermodynamic equations involving the first and second temperature derivatives of . The first derivative of the correlation is connected to the enthalpy of vaporization via the Clapeyron equation according to 4 7 Δ = Δ2 3 5 4 where 6 is the enthalpy of vaporization and ∆2 is the difference between the saturated vapor and liquid volumes. For this process, vapor volume for each compound was calculated using the SoaveRedlich-Kwong (SRK) equation of state,12 and the liquid volume for each compound was calculated using the corresponding DIPPR liquid density correlation.13 Although alternative models exist for vapor volume calculation, the SRK equation of state balances accuracy with ease-of-use. The second derivative of vapor pressure is needed when predicting the liquid heat capacity with what is called the “Derivative Method,”2, 14 which requires the derivative of the enthalpy of vaporization – and therefore the second derivative of vapor pressure – according to 89

(

= − : 3 )

# ; 2 4Δ 4 #Δ2 + ?Δ2 − @3 5 4 − 5 ; # 4 # ( 4 (

8

89

Here, is the liquid isobaric heat capacity, and is the ideal gas isobaric heat capacity. This refers

to A

BC D , B' (

the slope of the enthalpy with temperature at constant pressure P, which is the saturation

pressure at a given temperature. This is not to be confused with E , which is A

BC D , B' FG

or the slope of the

enthalpy with temperature along the saturation curve with changing P. The volume terms in Equation 8 serve as corrections from ideal gas to vapor, and from saturated to isobaric heat capacities.5 These terms are nearly zero below the normal boiling point for each compound, the temperature region where 4


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most data are present. Therefore, the vapor volume calculation model does little to affect the ability of Equation 8 to fit liquid heat capacity data, and the SRK equation of state remained an accurate and reasonable choice. The ideal gas heat capacities in Equation 8 were calculated from vibrational frequencies generated by ab initio methods with appropriate scaling factors for the basis sets and levels of theory used.15 Equations 7 and 8 show how to rigorously connect vapor pressure and liquid heat capacity.

New Predictive Method

The new predictive method for was created using the same methodology laid out by Riedel with one crucial exception: the exponent on in the final term, of Equation 1, was changed from 6 to other integer values. Specifically, the general Riedel equation was used in the linearized and reduced form 9 ln = + + ln + where is the reduced pressure, is the reduced temperature, − are fitting parameters, and is changed from 1 to 6 in integer steps. Changing has been found to improve the ability of the correlation to successfully predict using the Derivative Method for compounds with and data in a previous study.2 Therefore, the best will be determined by the predictive correlation’s ability to fit and data. Four constraints are necessary to calculate coefficients − . Three constraints are the exact same as Riedel’s original prediction method: 1. force Equation 9 to give 1 atm at the normal boiling point ( ) 2. force Equation 9 to give the critical pressure ( ) at the critical temperature ( ) 3. set the slope of the Riedel parameter to zero at the critical point (See Equation 12) This third constraint comes from an analysis of experimental data done by Plank and Riedel,16 and has been used in several vapor pressure correlations.3, 8-9 This means that the second derivative of the vapor pressure curve is not mathematically indeterminate, contrary to modern theory. However, as noted by Vetere 8 and Ambrose,17 it is not important for correlations to be analytic at the critical point. Solving − in terms of gives: ln ln 10 ln = lnHIJ 3 5 + MN − NKL 3 5O lnKL lnKL where ; 11 N = ; − 1 − − + 1 ln + and HIJ and KL are the reduced atmosphere the reduced normal boiling point, respectively. To obtain the fourth constraint, the Riedel parameter ! is defined: #ln 12 !≡ = − + + #ln which is the same as Riedel’s predictive method, except it has been generalized for all values of . At the critical point, the Riedel parameter reduces to: 13 ! = − + + 5



or

lnHIJ NKL 14 − lnKL lnKL when and are substituted using Constraints 1-3. is found following Riedel’s original methodology by fitting Equation 4 to experimental data to find % and & . This corresponding states process is further described in the Supporting Information. Substituting Equation 14 into Equation 4 and rearranging allows to be rewritten in terms of % and & according to: lnHIJ + %& NKL 15 = %M −& O lnKL + %NKL Substitution of Equation 15 into Equation 10 yields the desired predictive correlation: ln ln = lnHIJ 3 5 lnKL 16 lnHIJ + %& NKL ln +%M − & O MN − NKL 3 5O lnKL + %NKL lnKL where each value of requires a unique set of %, & , and N. The parameter N is defined as: ; ; 17 N = − 1 − − + 1 ln + ! =

Methods

Regression of % and &P as a Function of

Recall (See Section 2) that Riedel set = 6 and found by fitting Equation 10 to experimental data. He then found % and & (0.0838 and 3.758, respectively) that satisfied Equation 4 so that the method could be predictive. The same procedure was followed here using Equation 10 with = 1, 2, 3, 4, 5, and 6.

The training set in this study, used to fit the coefficient, consisted of 37 compounds from the eight chemical families listed in Table 2. These compounds were selected because: 1) they contained considerable data over a wide temperature range, and 2) they were a chemically diverse set. Summaries of the data used for these compounds are given in the Supplementary Material. In each case, the coefficient was fit to experimental data for a training set of compounds and the corresponding % and &P were found. Thus, six sets % and &P values were obtained—one set for each value. Table 3 contains the results. Notice that % decreases rapidly (orders of magnitude) with increasing values. & P also decreases with increasing , but much more slowly. The dependence of % and & P on is displayed graphically in Figure 1. The behavior of % is exponential in nature, while that of &P is quadratic. Thus, the behavior of each can be fit reliably (Q ; = 0.9990 for both) to the following correlations (displayed as lines on the figure). 18 % = 4.96465 V;.;W+W/ ; 19 & = 4.14524 − 0.0818433 + 0.00310685 With these equations non-integer values of may be chosen to fit vapor pressure data when the 6


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and data quantity and quality merit. A sample calculation using this predictive method is given in the Supplementary Information. The last two rows of Table 3 are significant. These show the % and &P values for = 6 for the new method and Riedel’s original method. Notice that when = 6, % and & for the new method are nearly identical to those from Riedel’s original work. Even though the training set for the new method contains nearly double the number of compounds as Riedel’s original training set, the results are similar. Table 2: Training set of compounds for the new predictive Pvap correlation, grouped by chemical families

Training Set Family

Number

Alkane

11

Alkene

1

Aromatic

5

Ester

3

Ether

1

Gas

5

Halogenated

4

Ketone

7

Table 3: Summary of prediction constants for the original Riedel Pvap prediction and this new Pvap prediction

Source

This Work

Riedel 3

[ 1 2 3 4 5 6 6

K 5.0398 1.0048 0.3931 0.2045 0.1244 0.0836 0.0838

Xc 4.065 3.996 3.928 3.867 3.812 3.767 3.758

7



6

4.10

5

4.05

3.95 3 3.90 2

Xc

4.00

4 K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3.85

1

3.80

0

3.75 1

2

3

4

5

6

E Value Figure 1 Fitting parameters K ( ) and Xc ( ) as functions of E

Training Method

As is apparent from Figure 1, must be selected before the vapor pressure can be predicted. It was hypothesized that an optimal value could be found for families of compounds. This was tested using a set of 106 compounds from the families listed in Table 4 which includes the 36 compounds used to create the data in Figure 1. Summaries of the data used for these compounds are given in the Supplementary Material. This was done by minimizing the average absolute deviation () of the predictions from the experimental data for and simultaneously. Enthalpy of vaporization could also have been used in this optimization, but most Δ data in the literature were derived from vapor pressure fits using the Clapeyron equation. When calorimetric data were present in the literature, they were usually at temperatures where vapor pressure data were plentiful with low uncertainty. Therefore, Δ was not used to find the best in this analysis. Specifically, for the property was defined as: 1 _8 − _`, 8 = ] ^ ^ 20 \ _8 8 where _8 is the experimental value for the property (vapor pressure or liquid heat capacity), _`, 8 is the model prediction for the property for a certain and temperature 8 , and \ is the number of experimental points for that property. The “best” value of was defined as the value that minimized the average of and for each compound. Although propyl formate was used to create this prediction method, it was omitted from this analysis because it did not have sufficient experimental data. Only one alkyne and one silane were included because few compounds in those families contained and data. Of noticeable omission are the alcohols and acids, families where self-association has been well documented.18-21 For these compounds, more data are necessary along with a separate analysis scheme that may require a different correlation form and a modified equation of state to calculate the vapor volumes. Due to these 8


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complications, strongly hydrogen bonding families will not be discussed here, but may be the topic of future analyses.

Method Summary

In summary, the coefficients − for Equation 9 are calculated using , , and ! as follows: 21 = ; − 1 ; 22 = − 23 = ! − + 1 The coefficient and the Riedel parameter at the critical point ! are calculated from the reduced normal boiling point (KL ), %, & , and : lnHIJ + %& NKL , 24 = %M −& O lnKL + %NKL , lnHIJ + %& NKL , 25 ! = lnKL + %NKL , where HIJ is the reduced atmosphere, KL is the reduced normal boiling point, and N is given as: ; 26 NKL , = ; − 1 − − + 1 lnKL + KL KL Equations 18 and 19 are used to calculate % and & , respectively. Equations 21-26 provide a set of predictive correlations with values from 1 to 6 in integer steps.

Results

Optimized by Family

The last three columns of Table 4 contain the average ’s for all of the compounds tested in each chemical family using the best , which was found by minimizing the average of the and for each compound. Lowering the value from 6 steepens the low temperature slope of the vapor pressure curve without greatly affecting the values predicted from the vapor pressure curve. Therefore, the properties derived from the slope of the vapor pressure curve (i.e. heat of vaporization and heat capacity) are affected more than the vapor pressure curve. The for Δ was not used in the optimization scheme because only 74 of the compounds contained experimental Δ data (see Supplementary Material). When Δ data did exist, changing the value did not change the ability of the correlation to follow the experimental Δ values. The new predictive method predicts for all families below 5% , and predicts for all but the gas family below 3% on average. Members of the gas family include many small compounds that behave much differently, and that difference is reflected in the large .

9



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Table 4 Average cdef and gih for the compounds in each chemical family using the best [ value for each compound

Family

Number of Compounds

Avg cdef (%)

Avg jkdef (%)

1.42

1.35

Avg gih (%)

Aldehyde

4

4.21

Alkane

15

1.06

1.72

3.23

Alkene

9

4.43

0.91

4.64

Alkyne

1

1.52

0.92

2.84

Amine

8

3.18

0.82

2.18

Aromatic

12

4.73

0.52

2.58

Ester

6

4.46

0.45

1.64

Ether

5

1.97

1.26

3.70

Gas

5

0.59

1.43

12.20

Halogenated

8

1.65

0.95

3.62

Ketone

8

1.52

1.15

2.64

Multifunctional

14

2.26

0.44

2.43

Ring Alkane

5

2.41

2.22

2.10

Silane

1

2.49

--

2.38

Sulfide

5

0.95

1.27

2.78

Figure 2 shows the distribution of best values for the 106 test compounds separated into chemical families. The majority of compounds were best optimized using = 3 or = 4, though amines seem to do best with = 2 and aromatics seem to favor = 4 and above.

10


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50

Aldehyde (4)

45

Alkane (15)

40

Alkene (9) Alkyne (1)

Number of Compounds

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


35

Amine (8)

30

Aromatic (12)

25

Ester (6) Ether (5)

20

Gas (5)

15

Halogenated (8) Ketone (8)

10

Multifunctional (14)

5

Ring Alkane (5) Silane (1)

0 1

2

3

Best E

4

5

6

Sulfide (5)

Figure 2 Distribution of best [ values using the new predictive Riedel cdef method on the test compounds

The best integer value for each family was chosen by finding the integer that best fits the most compounds in each family. These recommended values are summarized in Table 5. The five light gases showed a spread in best value, with a much larger average than the other families. In this case, a value of was chosen as it was closest to the average best value. For compounds outside of the families listed, should be set to 3 or 4, unless the compounds strongly self-associate. As was explained in Section 4 above, alcohols and acids were not included.

11



Table 5 Recommended [ values for each of the chemical families tested

Best [

Family

Aldehyde

4

Alkane

3

Alkene

3

Alkyne

3

Amine

2

Aromatic

4

Ester

4

Ether

3

Gas

5

Halogenated

4

Ketone

3

Multifunctional

3 or 4

Ring Alkane

4

Silane

3

Sulfide

3

In general, the best values used to predict are 3 or 4 for all but the amine and gas families. Contrast these results to a previous study that found that a value of 2 should be used for these same exact families.2 The key difference here is that the previous article did not use the Riedel parameter constraint (Constraint 3), while this prediction does. Adding an additional constraint at the critical point changed the shape of the curve in a way that the best correlations required a larger value. For compounds with extensive data, Constraint 3 is not needed. In those cases, a smaller value works better. However, for compounds with limited or no data, an value one integer larger works better.

Comparison to Other Predictive Methods The new prediction method was compared to Riedel’s original method,3 the Lee-Kesler corresponding states method,6 Vetere’s Wagner correlation prediction method,7 and Vetere’s newest Riedel correlation prediction method.9 The recommended values in Table 5 were applied to each of the 106 compounds analyzed. The distribution of the test compounds is given in Figure 3 for the five prediction methods. This bar chart shows the number of compounds on the y-axis for each range of on the x-axis and each prediction method, shown with different color bars. The red, orange, yellow, and green bars show the results for the Riedel, Lee-Kesler, Vetere-Wagner, and Vetere-Riedel methods, respectively. The blue dotted bars show the results for this new predictive method. For example, this chart shows that 70 compounds were predicted with a in the range 0-2% using this new predictive method. This new method performs at least as well as the other methods since the 12


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distribution for this method is bunched slightly more toward zero than the other prediction methods. The distribution for the new method is not discernably different from the best of other methods (Riedel). Therefore, this new method retains but does not noticeably improve the fit of where the data are located. is greater than 10% using this work for three compounds: 1,3-trans-pentadiene, pyrene, and octyl acetate. In these cases, the majority of data are under 1000 Pa where uncertainty in measurements grows.10-11 80 70

Number of Compounds

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60 50 40 30 20 10 0 0-2%

2-4%

4-6%

6-8%

8-10%

10-12% 12-14% 14-16% 16-18% 18-20%

vap

>20%

Figure 3 The distribution of cdef llm for 106 test compounds using cdef prediction methods: Riedel’s original method ( ), Lee-Kesler ( ), Vetere’s Wagner method ( ), Vetere’s Riedel method ( ), and this work ( )

The Δ distribution of the test compounds is given in Figure 4. Notice that the x-axis extends to 10% instead of 20% as shown in Figure 3. Although the scale is different, the message is the same: there’s no clear winner among these prediction methods when looking at Δ data. However, the Δ distributions for the Riedel prediction and this prediction (solid red and blue dotted bars, respectively) are slightly bunched closer to zero than the other three methods. Although looking towards Δ does not give much new information, the Δ distribution indicates that the original Riedel prediction and this work may do slightly better for Δ .

13



35 30 Number of Compounds

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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25 20 15 10 5 0 0-1%

1-2%

2-3%

3-4%

4-5%

Δvap 5-6%

6-7%

7-8%

8-9%

9-10%

>10%

Figure 4 The distribution of jkdef llm for 106 test compounds using cdef prediction methods: Riedel’s original method ( ), Lee-Kesler ( ), Vetere’s Wagner method ( ), Vetere’s Riedel method ( ), and this work ( )

The distribution of the test compounds is given in Figure 5 and demonstrates how the new method excels. This distribution for this new method (again, the blue dotted bars) is bunched much closer to 0% than any of the other prediction methods. Only one compound has an greater than 12% for the new method compared to 11 for Vetere-Wagner, 18 for Vetere-Riedel, 21 for the original Riedel method, and 24 for Lee-Kesler. The method developed in this work does a much better job predicting thermodynamically consistent and data than the other prediction methods found in the literature.

14


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50 45

Number of Compounds

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 35 30 25 20 15 10 5 0 0-2%

2-4%

4-6% gih

6-8%

8-10% 10-12% 12-14% 14-16% 16-18% 18-20%

Cpl

>20%

Figure 5 The distribution of llm for the test compounds using cdef prediction methods within the Derivative method: Riedel’s original method ( ), Lee-Kesler ( ), Vetere’s Wagner method ( ), Vetere’s Riedel method ( ), and this work ( )

Testing the Methods with Compound Not Used in the Training Set To fully test the applicability of the method, five sets of 40 compounds were randomly chosen from the 70 compounds not used to train the % and & values, and the average , Δ , and ’s were found for the five prediction methods (four previous methods and that of this work). Only 70 compounds were used for this analysis because the 36 other compounds were used to fit the correlations, and including these would bias the results in favor of the new method. The average , Δ , and ’s of the five sets of compounds for the five prediction methods are given in Figure 6 with 95% confidence intervals. The values are ordered on the X-axis by chronological order of publication. The new method significantly decreased from around 7% to 3% without significantly increasing and Δ . Since almost all of the data in the literature are just above the triple point temperature ('( ) and most of the and Δ data in the literature are bunched around for each compound, an improvement in corresponds with an improvement in the and Δ correlations near the '( where it is difficult to measure accurately.10-11 Therefore, this new method predicts vapor pressure at lower temperatures with lower uncertainty than any other prediction method tested. Lowering the value of changes the vapor pressure curve at low temperatures.

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Figure 6 Average cdef (

Lee-Kesler (1975) ), jkdef (

Vetere-Wagner (1991) Vetere-Riedel (2006)

vap Prediction Method

This Work (2017)

), and gih ( ) llm for five sets of 40 compounds using different cdef prediction methods

Conclusions Predictive methods for the Riedel equation with different final coefficient values were created. These correlations were compared to experimental vapor pressure, enthalpy of vaporization, and liquid heat capacity data. Vapor pressure and liquid heat capacity data for aldehydes, alkanes, alkenes, alkynes, amines, aromatics, esters, ethers, gases, halogenated compounds, ketones, multifunctional compounds, ringed alkanes, silanes, and sulfides were used to determine the selection of the Riedel coefficient for vapor pressure via the Derivative Method for predicting liquid heat capacity. In this procedure, families such as alcohols and acids were not included because they may require modifications to the vapor pressure correlation form and the equation of state to account for self-association. This analysis showed that an of 2-5 fits these families much better than the traditional of 6. This method fit vapor pressure as well as other predictive methods, but it fit liquid heat capacity much better than other vapor pressure predictive methods using randomized samples of the compounds with data. Since most liquid heat capacity data are at temperatures close to the triple point, this method improves the shape of the and Δ curves at low temperatures where they are difficult to measure. Therefore, this method predicts thermodynamically consistent vapor pressure temperature-dependent correlations over wide temperature ranges.

Acknowledgements The authors gratefully acknowledge the financial support of DIPPR Project 801.

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data and application of a two-state model to pure liquids and mixtures. J. Phys. Chem. B 2007, 111 (5), 1119-1128.

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