A number of corrections are required to the original article,1 as detailed below.

  1. In Eq. (27), the factor of −3 should be replaced by +3; the correct form of the equation is as follows:

fA2=2TθD,026+3b2TθD,02+b22TθD,041+b2TθD,023.
(27)
  1. Equation (29), for the cold-curve contribution to internal energy U, should read:
    U0=A0,
    (29)
    where A0 is the cold-curve Helmholtz energy defined in Eq. (9).
  1. The second term on the right-hand side of Eq. (49) is missing a factor of 2; the correct form of the equation is as follows:

pDVT=γDγDVmpγD2RθDγDVm2D3θDT.
(49)
  1. In Eq. (57), Tt/T was written instead of T/Tt. The equation should read:

psub=ptexpTtTe11TTt+e21TTt32+e31TTt5.
(57)

Additionally, we note that Eq. (50), while not incorrect in general, is potentially confusing as presented, since it becomes indeterminate for those terms with γi = 0 (as specified in Table 3). A more useful form of this equation, valid for all values of γi, is as follows:

piVT=piVqi1+γiθiTexpθiTexpθiT11.
(50)

Most errors are typographical and do not affect the equation of state performance. However, the error in Eq. (27) causes changes in properties related to 2AT2N,V, including cv, cp, and γ. Along the sublimation curve, correcting Eq. (27) (with no changes to the model parameters) causes the isobaric heat capacity to change by zero at T = 0 K to 2.92 J mol−1 K−1 at the triple point (278.67 K), with the relative changes ranging from 0% to 2.27%. Along the melting curve where there are no experimental data, the change in cp continues to increase, reaching 10.88 J mol−1 K−1 (6.48%) at 470 K. At a fixed temperature, correcting Eq. (27) has little impact if the pressure is varied: at 300 K, the relative change in cp ranges from 2.91% to 3.27% when the pressure is varied from (206.6–1600) MPa. Similar numbers in relative changes apply to calculations of cv and γ.

A partial refit of the model is required. Use of the correct equations meant that the melting points measured at the highest available temperature (466 K) could be included in the corrected model’s regression. The resulting best-fit parameters for the re-tuned model are listed in Table 6.

TABLE 6.

Best-fit parameter values for the optimized solid benzene equation of state

ParameterValue
c1 (MPa) 7 360 
c2 (MPa) 29 960 
c3 (MPa) 118 283 
θD,0 101.9 
θ1,0(lib) 131.2 
θ2,0(lib) 122.9 
Sm(g, Tt, pt) (J mol−1 K−1292.06 
qD −5 
qθ1(lib) −5 
qθ2(lib) 
γD,0 4.70 
γθ1,0(lib) 0.214 
γθ2,0(lib) 7.18 
b1 −0.004 23 
b2 0.042 
b3 3.73 
ParameterValue
c1 (MPa) 7 360 
c2 (MPa) 29 960 
c3 (MPa) 118 283 
θD,0 101.9 
θ1,0(lib) 131.2 
θ2,0(lib) 122.9 
Sm(g, Tt, pt) (J mol−1 K−1292.06 
qD −5 
qθ1(lib) −5 
qθ2(lib) 
γD,0 4.70 
γθ1,0(lib) 0.214 
γθ2,0(lib) 7.18 
b1 −0.004 23 
b2 0.042 
b3 3.73 

The retuned model’s representations of the experimental data are similar to those presented in the original manuscript, although the distributions of deviations are slightly different in some cases. For completeness, revised versions of Tables 710 and Figs. 616 from the original manuscript calculated using the updated model are presented here.

TABLE 7.

Summary for the experimental data and the corresponding fitted results [root-mean-square (RMS) deviations and absolute average relative deviations (AAD)] along the sublimation curve. The subscript “all” refers to all the reported data in the literature, and “tuned” refers to the data that were fitted in this work

PropertySourceYearT (K)NallNtunedRMSall (%)AADall (%)RMStuned (%)AADtuned (%)
Vm Andrew and Eades2  1953 78.2, 270.2 0.74 0.61 ⋯ ⋯ 
Andrews and Ubbelohde3  1955 278.7 0.22 0.22 0.22 0.22 
Bacon et al.4  1964 138, 218 0.92 0.92 0.92 0.92 
Biltz et al.5  1930 90.2, 194.2 0.13 0.12 0.13 0.12 
Cox6  1932 251 2.98 2.98 ⋯ ⋯ 
Cox et al.7  1958 270.2 0.65 0.65 0.65 0.65 
Dunitz and Ibberson8  2008 5.5–274 15 15 0.32 0.27 0.32 0.27 
Ferche9  1891 278.5 0.74 0.74 ⋯ ⋯ 
Fortes and Capelli10  2018 10–275 12 12 0.29 0.25 0.29 0.25 
Heuse11  1930 20 0.77 0.77 0.77 0.77 
Heydweiller12  1897 270.2–276.6 0.19 0.17 0.19 0.17 
Higgins et al.13  1964 239.9–269.9 0.42 0.42 0.42 0.42 
Kozhin14  1954 78.2 0.81 0.81 ⋯ ⋯ 
McConville et al.15  2020 100–263 30 28 0.29 0.26 0.29 0.26 
Ziegler and Ditzel16  1929 273.2 0.10 0.10 0.10 0.10 
cp,m Ahlberg et al.17  1937 3.8–93.2 19 19 8.05 4.94 8.05 4.94 
Andrews et al.18  1926 255.2–278.5 9.47 8.04 ⋯ ⋯ 
Aoyama and Kanda19  1935 82.2–223.9 5.44 5.00 ⋯ ⋯ 
Brucksch, Jr. and Ziegler20  1942 15–270 28 28 2.75 1.97 2.75 1.97 
Dewar21  1913 50 9.04 9.04 9.04 9.04 
Diedrich22  2005 180.2–270.2 19 3.28 3.04 ⋯ ⋯ 
Hahnenkamp23  2008 190–270 17 17 0.88 0.62 0.88 0.62 
Huffman et al.24  1930 92.6–259.5 16 16 1.46 1.15 1.46 1.15 
Maass and Waldbauer25  1925 93.2–273.2 10 7.64 6.70 ⋯ ⋯ 
Nan and Tan26,a 2004 78.4–265.0 55 55 1.26 1.00 1.26 1.00 
Nernst27  1911 24.4–200.9 12 12 5.54 4.24 5.54 4.24 
Oliver et al.28  1948 13–278.7 92 92 3.58 2.71 3.58 2.71 
Stull29,b 1937 90–270 19 19 1.49 1.17 1.49 1.17 
KS Brunel30  1979 270 3.15 3.15 3.15 3.15 
Heseltine et al.31  1964 170–250 1.90 1.38 1.90 1.38 
ΔHm,sub de Boer32  1936 273.15 1.04 1.04 1.04 1.04 
Deitz33  1933 192 45.48 45.48 ⋯ ⋯ 
De Kruif and Van Ginkel34,c 1977 193–298 45.49 30.27 2.29 2.29 
Hessler35  1984 278 0.56 0.56 0.56 0.56 
Jackowski36  1974 278.69 NAd NAd ⋯ ⋯ 
Jones37  1960 229–261 6.97 6.39 ⋯ ⋯ 
Milazzo38,39 1956 279 NAd NAd ⋯ ⋯ 
Mündel40  1913 226 8.92 8.92 ⋯ ⋯ 
Růžička et al.41  2014 150–270 13 13 0.59 0.47 0.59 0.47 
Stephenson and Malanowski42,43 1987 264 0.78 0.78 0.78 0.78 
Stull44  1947 282 NAd NAd ⋯ ⋯ 
psube Barker45  1910 195.7 23.59 23.59 ⋯ ⋯ 
Choi and Brown46  1966 227.7–273.2 1.66 1.23 0.71 0.61 
De Kruif and Van Ginkel34  1977 183.0–197.0 10 2.79 2.02 ⋯ ⋯ 
De Kruif47  1980 183.4–196.7 10 1.86 1.75 ⋯ ⋯ 
Deitz33  1933 184.3–200.2 5.67 4.47 1.76 1.50 
Ferche9  1891 272.3–278.5 30 26 4.79 2.30 1.09 0.72 
Ha et al.43  1976 228.7–273.2 14 14 0.48 0.30 0.48 0.30 
Jackowski36  1974 220.8–278.7 21 14 3.27 2.08 0.87 0.72 
Kiss48  1972 234.3–277.2 17 15 1.65 1.24 1.02 0.89 
Liu and Dickhut49  1994 257.8–268.2 16.00 14.36 ⋯ ⋯ 
Milazzo38  1956 195.2–273.1 10 10 1.34 1.03 1.34 1.03 
Milazzo50  1956 195.2–257.2 1.38 1.01 1.38 1.01 
Miljevic et al.51  1977 202.1–278.5 39 35 1.17 0.46 0.25 0.21 
Mündel40  1913 214.6–238.1 8.63 8.51 ⋯ ⋯ 
Radulescu and Alexa52  1938 273.2–277.1 7.89 5.58 1.88 1.68 
Rastogi et al.53  1967 260.2–273.2 20.21 19.21 ⋯ ⋯ 
Růžička et al.41  2014 233.2–260.7 36 36 0.53 0.51 0.53 0.51 
Stull44  1947 236.5–270.5 1.11 0.96 1.11 0.96 
Young54  1910 263.2–273.2 23.69 22.18 ⋯ ⋯ 
PropertySourceYearT (K)NallNtunedRMSall (%)AADall (%)RMStuned (%)AADtuned (%)
Vm Andrew and Eades2  1953 78.2, 270.2 0.74 0.61 ⋯ ⋯ 
Andrews and Ubbelohde3  1955 278.7 0.22 0.22 0.22 0.22 
Bacon et al.4  1964 138, 218 0.92 0.92 0.92 0.92 
Biltz et al.5  1930 90.2, 194.2 0.13 0.12 0.13 0.12 
Cox6  1932 251 2.98 2.98 ⋯ ⋯ 
Cox et al.7  1958 270.2 0.65 0.65 0.65 0.65 
Dunitz and Ibberson8  2008 5.5–274 15 15 0.32 0.27 0.32 0.27 
Ferche9  1891 278.5 0.74 0.74 ⋯ ⋯ 
Fortes and Capelli10  2018 10–275 12 12 0.29 0.25 0.29 0.25 
Heuse11  1930 20 0.77 0.77 0.77 0.77 
Heydweiller12  1897 270.2–276.6 0.19 0.17 0.19 0.17 
Higgins et al.13  1964 239.9–269.9 0.42 0.42 0.42 0.42 
Kozhin14  1954 78.2 0.81 0.81 ⋯ ⋯ 
McConville et al.15  2020 100–263 30 28 0.29 0.26 0.29 0.26 
Ziegler and Ditzel16  1929 273.2 0.10 0.10 0.10 0.10 
cp,m Ahlberg et al.17  1937 3.8–93.2 19 19 8.05 4.94 8.05 4.94 
Andrews et al.18  1926 255.2–278.5 9.47 8.04 ⋯ ⋯ 
Aoyama and Kanda19  1935 82.2–223.9 5.44 5.00 ⋯ ⋯ 
Brucksch, Jr. and Ziegler20  1942 15–270 28 28 2.75 1.97 2.75 1.97 
Dewar21  1913 50 9.04 9.04 9.04 9.04 
Diedrich22  2005 180.2–270.2 19 3.28 3.04 ⋯ ⋯ 
Hahnenkamp23  2008 190–270 17 17 0.88 0.62 0.88 0.62 
Huffman et al.24  1930 92.6–259.5 16 16 1.46 1.15 1.46 1.15 
Maass and Waldbauer25  1925 93.2–273.2 10 7.64 6.70 ⋯ ⋯ 
Nan and Tan26,a 2004 78.4–265.0 55 55 1.26 1.00 1.26 1.00 
Nernst27  1911 24.4–200.9 12 12 5.54 4.24 5.54 4.24 
Oliver et al.28  1948 13–278.7 92 92 3.58 2.71 3.58 2.71 
Stull29,b 1937 90–270 19 19 1.49 1.17 1.49 1.17 
KS Brunel30  1979 270 3.15 3.15 3.15 3.15 
Heseltine et al.31  1964 170–250 1.90 1.38 1.90 1.38 
ΔHm,sub de Boer32  1936 273.15 1.04 1.04 1.04 1.04 
Deitz33  1933 192 45.48 45.48 ⋯ ⋯ 
De Kruif and Van Ginkel34,c 1977 193–298 45.49 30.27 2.29 2.29 
Hessler35  1984 278 0.56 0.56 0.56 0.56 
Jackowski36  1974 278.69 NAd NAd ⋯ ⋯ 
Jones37  1960 229–261 6.97 6.39 ⋯ ⋯ 
Milazzo38,39 1956 279 NAd NAd ⋯ ⋯ 
Mündel40  1913 226 8.92 8.92 ⋯ ⋯ 
Růžička et al.41  2014 150–270 13 13 0.59 0.47 0.59 0.47 
Stephenson and Malanowski42,43 1987 264 0.78 0.78 0.78 0.78 
Stull44  1947 282 NAd NAd ⋯ ⋯ 
psube Barker45  1910 195.7 23.59 23.59 ⋯ ⋯ 
Choi and Brown46  1966 227.7–273.2 1.66 1.23 0.71 0.61 
De Kruif and Van Ginkel34  1977 183.0–197.0 10 2.79 2.02 ⋯ ⋯ 
De Kruif47  1980 183.4–196.7 10 1.86 1.75 ⋯ ⋯ 
Deitz33  1933 184.3–200.2 5.67 4.47 1.76 1.50 
Ferche9  1891 272.3–278.5 30 26 4.79 2.30 1.09 0.72 
Ha et al.43  1976 228.7–273.2 14 14 0.48 0.30 0.48 0.30 
Jackowski36  1974 220.8–278.7 21 14 3.27 2.08 0.87 0.72 
Kiss48  1972 234.3–277.2 17 15 1.65 1.24 1.02 0.89 
Liu and Dickhut49  1994 257.8–268.2 16.00 14.36 ⋯ ⋯ 
Milazzo38  1956 195.2–273.1 10 10 1.34 1.03 1.34 1.03 
Milazzo50  1956 195.2–257.2 1.38 1.01 1.38 1.01 
Miljevic et al.51  1977 202.1–278.5 39 35 1.17 0.46 0.25 0.21 
Mündel40  1913 214.6–238.1 8.63 8.51 ⋯ ⋯ 
Radulescu and Alexa52  1938 273.2–277.1 7.89 5.58 1.88 1.68 
Rastogi et al.53  1967 260.2–273.2 20.21 19.21 ⋯ ⋯ 
Růžička et al.41  2014 233.2–260.7 36 36 0.53 0.51 0.53 0.51 
Stull44  1947 236.5–270.5 1.11 0.96 1.11 0.96 
Young54  1910 263.2–273.2 23.69 22.18 ⋯ ⋯ 
a

Data were read from a plot and were assigned half of the weighting factor.

b

The apparatus was calibrated based on the data reported by Huffman et al.24 

c

One datum is above our chosen triple-point temperature.

d

Not applicable because the reported temperature is above our chosen triple-point temperature.

e

The fitting for sublimation was calculated as ΔGm,subTi,piGm,subfTi,pi using the experimental (Ti, pi) as inputs, and the results are exhibited in the form of pressure in Fig. 10. As indicated in Sec. 4, for the ith sublimation point, Gm,subfTi,pi refers to the Gibbs energy from the fluid model, Gm,subTi,pi is the fitted Gibbs energy from the solid EOS, and ΔGm,subTi,pi is the Gibbs free energy change at the experimental sublimation conditions computed from Eq. (62).

TABLE 8.

Summary for the experimental data and the corresponding fitted results [root-mean-square (RMS) deviations and absolute average relative deviations (AAD)] along the melting curve. The subscript “all” refers to all the reported data in the literature, and “tuned” refers to the data that were fitted in this work

PropertySourceYearT (K)NallNtunedRMSall (%)AADall (%)RMStuned (%)AADtuned (%)
ΔHm,melt Andrews et al.18  1926 278.7 0.32 0.32 0.32 0.32 
Azreg-Aïnou et al.55,56,a 2006 284.6–306.7 10 5.84 5.37 2.71 2.36 
Bridgman57  1914 278.6–477.4 12 11 3.14 2.30 3.28 2.48 
Huffman et al.24  1930 278.6 NAb NAb ⋯ ⋯ 
Maass and Waldbauer25  1925 278.6 NAb NAb ⋯ ⋯ 
Oliver et al.28  1948 278.7 0.32 0.32 0.32 0.32 
Osugi et al.58  1965 288.2–298.2 7.66 7.23 ⋯ ⋯ 
Rossini et al.59  1953 278.7 0.32 0.32 0.32 0.32 
Schäfer and Lax60  1961 278.7 0.02 0.02 0.02 0.02 
Smith61  1979 279.1 5.81 5.81 ⋯ ⋯ 
Stratton and Partington62  1924 279 0.55 0.55 0.55 0.55 
Timmermans63,64 1950 278.7 1.12 1.12 1.12 1.12 
Tschamler65  1948 278.6 NAb NAb ⋯ ⋯ 
Watanabe et al.66  1999 278 NAb NAb ⋯ ⋯ 
Xu et al.67  2007 306.4–466 10 10 3.33 3.07 3.33 3.07 
Ziegler and Andrews68  1942 278.7 0.82 0.82 0.82 0.82 
pmeltc Azreg-Aïnou et al.55,56 2006 284.6–306.7 10 10 0.58 0.48 0.58 0.48 
Block69,70,d 1913 298.2–324.2 0.12 0.09 0.12 0.09 
Bridgman57  1914 305.7–463.7 10 10 0.18 0.15 0.18 0.15 
Bridgman71  1949 297.2 0.16 0.16 0.16 0.16 
Deffet72  1935 283.2–305.2 0.68 0.38 0.68 0.38 
Deffet and Vlerick73  1942 293.1–300.6 0.08 0.08 0.08 0.08 
Domańska and Morawski74  2005 293.2–353.2 0.16 0.15 0.16 0.15 
Domańska and Morawski75  2007 328.2–363.2 0.09 0.09 0.09 0.09 
Easteal et al.76  1985 281.3–310.3 12 10 1.19 0.64 0.33 0.23 
Figuière et al.77  1978 298.2–324.2 0.23 0.20 0.23 0.20 
Fruhling78  1951 306.5–331.2 0.29 0.29 0.29 0.29 
Ghelfenstein and Szwarc79  1975 298.2–324.2 0.07 0.06 0.07 0.06 
Hulett80  1899 279.2–290.7 22 12 62.86 14.69 0.49 0.40 
Makita and Takagi81  1968 283.2–323.2 0.66 0.33 0.15 0.10 
Nagaoka and Makita82  1987 283.2–323.2 0.26 0.20 0.26 0.20 
Osugi et al.83  1968 288.2–298.2 1.57 1.29 0.35 0.35 
Pruzan84  1976 301.7–324.2 0.30 0.27 0.30 0.27 
Pruzan et al.85  1979 302–325 0.46 0.45 0.46 0.45 
Sun et al.86  1987 279.6–323.1 11 1.81 0.95 0.17 0.10 
Tanaka and Kawakami87  1996 278.8–323.2 10 35.24 11.51 0.53 0.42 
Xu et al.67  2007 306.4–466.0 10 10 0.18 0.15 0.18 0.15 
Yokoyama et al.88  1993 294.7–329.8 0.40 0.35 0.40 0.35 
PropertySourceYearT (K)NallNtunedRMSall (%)AADall (%)RMStuned (%)AADtuned (%)
ΔHm,melt Andrews et al.18  1926 278.7 0.32 0.32 0.32 0.32 
Azreg-Aïnou et al.55,56,a 2006 284.6–306.7 10 5.84 5.37 2.71 2.36 
Bridgman57  1914 278.6–477.4 12 11 3.14 2.30 3.28 2.48 
Huffman et al.24  1930 278.6 NAb NAb ⋯ ⋯ 
Maass and Waldbauer25  1925 278.6 NAb NAb ⋯ ⋯ 
Oliver et al.28  1948 278.7 0.32 0.32 0.32 0.32 
Osugi et al.58  1965 288.2–298.2 7.66 7.23 ⋯ ⋯ 
Rossini et al.59  1953 278.7 0.32 0.32 0.32 0.32 
Schäfer and Lax60  1961 278.7 0.02 0.02 0.02 0.02 
Smith61  1979 279.1 5.81 5.81 ⋯ ⋯ 
Stratton and Partington62  1924 279 0.55 0.55 0.55 0.55 
Timmermans63,64 1950 278.7 1.12 1.12 1.12 1.12 
Tschamler65  1948 278.6 NAb NAb ⋯ ⋯ 
Watanabe et al.66  1999 278 NAb NAb ⋯ ⋯ 
Xu et al.67  2007 306.4–466 10 10 3.33 3.07 3.33 3.07 
Ziegler and Andrews68  1942 278.7 0.82 0.82 0.82 0.82 
pmeltc Azreg-Aïnou et al.55,56 2006 284.6–306.7 10 10 0.58 0.48 0.58 0.48 
Block69,70,d 1913 298.2–324.2 0.12 0.09 0.12 0.09 
Bridgman57  1914 305.7–463.7 10 10 0.18 0.15 0.18 0.15 
Bridgman71  1949 297.2 0.16 0.16 0.16 0.16 
Deffet72  1935 283.2–305.2 0.68 0.38 0.68 0.38 
Deffet and Vlerick73  1942 293.1–300.6 0.08 0.08 0.08 0.08 
Domańska and Morawski74  2005 293.2–353.2 0.16 0.15 0.16 0.15 
Domańska and Morawski75  2007 328.2–363.2 0.09 0.09 0.09 0.09 
Easteal et al.76  1985 281.3–310.3 12 10 1.19 0.64 0.33 0.23 
Figuière et al.77  1978 298.2–324.2 0.23 0.20 0.23 0.20 
Fruhling78  1951 306.5–331.2 0.29 0.29 0.29 0.29 
Ghelfenstein and Szwarc79  1975 298.2–324.2 0.07 0.06 0.07 0.06 
Hulett80  1899 279.2–290.7 22 12 62.86 14.69 0.49 0.40 
Makita and Takagi81  1968 283.2–323.2 0.66 0.33 0.15 0.10 
Nagaoka and Makita82  1987 283.2–323.2 0.26 0.20 0.26 0.20 
Osugi et al.83  1968 288.2–298.2 1.57 1.29 0.35 0.35 
Pruzan84  1976 301.7–324.2 0.30 0.27 0.30 0.27 
Pruzan et al.85  1979 302–325 0.46 0.45 0.46 0.45 
Sun et al.86  1987 279.6–323.1 11 1.81 0.95 0.17 0.10 
Tanaka and Kawakami87  1996 278.8–323.2 10 35.24 11.51 0.53 0.42 
Xu et al.67  2007 306.4–466.0 10 10 0.18 0.15 0.18 0.15 
Yokoyama et al.88  1993 294.7–329.8 0.40 0.35 0.40 0.35 
a

Data were obtained from fitted correlations.

b

Not applicable because the reported temperature is below our chosen triple-point temperature.

c

The fitting for melting was calculated as ΔGm,meltTi,piGm,meltfTi,pi using experimental (Ti, pi) as inputs, and the results are exhibited in the form of pressure in Fig. 11. As indicated in Sec. 4, for the ith melting data point, Gm,meltfTi,pi refers to the Gibbs energy from the fluid model, Gm,meltTi,pi is the fitted Gibbs energy from the solid EOS, and ΔGm,meltTi,pi is the Gibbs free energy change at the experimental melting conditions computed from Eq. (62) by substituting all the “sub” subscripts with “melt.”

d

Data were obtained from Table 1 of Figuière et al.77 

TABLE 9.

Summary of the experimental data and the corresponding fitted results [root-mean-square (RMS) deviations and absolute average relative deviations (AAD)] for single-phase data measured at high pressures. The subscript “all” refers to all the reported data in the literature, and “tuned” refers to the data that were fitted in this work

PropertySourceYearT (K)p (MPa)NallNtunedRMSall (%)AADall (%)RMStuned (%)AADtuned (%)
Vm Budzianowski and Katrusiak89  2005 296 300–1100 3.39 2.91 1.72 1.69 
Figuière et al.77,a,b 1978 253.2–324.2 0–490 114 114 1.81 1.76 1.81 1.76 
Hofmann and Kuleshova90  2014 138–270 100 2.73 2.37 ⋯ ⋯ 
Katrusiak et al.91,b 2010 295 79.5–1287.9 40 40 1.53 1.22 1.53 1.22 
ΔVmc Bridgman92  1941 323.2 490–981 0.99 0.99 ⋯ ⋯ 
Bridgman93  1942 298.2–348.2 490–981 5.90 5.90 ⋯ ⋯ 
Bridgman71  1949 297.2 245–981 0.45 0.32 ⋯ ⋯ 
cp,m Ross et al.94  1979 300 206.6–1600 6.19 5.34 ⋯ ⋯ 
α Fuchs et al.95  1979 253.6–355.2 0.4–411.1 164 155 15.16 6.11 4.31 3.47 
Pruzan et al.85  1979 302–325 113–657 31 31 4.99 3.92 4.99 3.92 
Pruzan et al.96  1986 268–355 16–423.5 40 20.89 16.8 ⋯ ⋯ 
PropertySourceYearT (K)p (MPa)NallNtunedRMSall (%)AADall (%)RMStuned (%)AADtuned (%)
Vm Budzianowski and Katrusiak89  2005 296 300–1100 3.39 2.91 1.72 1.69 
Figuière et al.77,a,b 1978 253.2–324.2 0–490 114 114 1.81 1.76 1.81 1.76 
Hofmann and Kuleshova90  2014 138–270 100 2.73 2.37 ⋯ ⋯ 
Katrusiak et al.91,b 2010 295 79.5–1287.9 40 40 1.53 1.22 1.53 1.22 
ΔVmc Bridgman92  1941 323.2 490–981 0.99 0.99 ⋯ ⋯ 
Bridgman93  1942 298.2–348.2 490–981 5.90 5.90 ⋯ ⋯ 
Bridgman71  1949 297.2 245–981 0.45 0.32 ⋯ ⋯ 
cp,m Ross et al.94  1979 300 206.6–1600 6.19 5.34 ⋯ ⋯ 
α Fuchs et al.95  1979 253.6–355.2 0.4–411.1 164 155 15.16 6.11 4.31 3.47 
Pruzan et al.85  1979 302–325 113–657 31 31 4.99 3.92 4.99 3.92 
Pruzan et al.96  1986 268–355 16–423.5 40 20.89 16.8 ⋯ ⋯ 
a

Data were read from a plot and were assigned half of the weighting factor.

b

Data were obtained from fitted correlations.

c

Refers to the volume decrement upon compression at a given temperature.

TABLE 10.

Sample calculations of solid benzene thermodynamic properties

PropertySymbolUnitValue
Input: T = 210 K, p = 10.0 MPa 
Molar internal energy Um J mol−1 10 385.8 
Molar entropy Sm J mol−1 K−1 100.847 
Molar volume Vm cm3 mol−1 74.037 
Molar Gibbs energy Gm J mol−1 −10 051.8 
Molar enthalpy Hm J mol−1 11 126.2 
Isochoric molar heat capacity cv,m J mol−1 K−1 74.126 
Isobaric molar heat capacity cp,m J mol−1 K−1 88.651 
Isobaric expansivity α K−1 4.744 × 10−4 
Isothermal compressibility KT MPa−1 2.409 × 10−4 
Isentropic compressibility KS MPa−1 2.014 × 10−4 
Thermal Grüneisen parameter γ 1.967 
Molar Helmholtz energy Am J mol−1 −10 792.1 
Input: T = 330 K, Vm = 70 cm3 mol−1 
Molar internal energy Um J mol−1 20 461.3 
Molar entropy Sm J mol−1 K−1 134.157 
Pressure p MPa 574.6 
Molar Gibbs energy Gm J mol−1 16 408.4 
Molar enthalpy Hm J mol−1 60 680.3 
Isochoric molar heat capacity cv,m J mol−1 K−1 113.624 
Isobaric molar heat capacity cp,m J mol−1 K−1 127.905 
Isobaric expansivity α K−1 2.834 × 10−4 
Isothermal compressibility KT MPa−1 1.299 × 10−4 
Isentropic compressibility KS MPa−1 1.154 × 10−4 
Thermal Grüneisen parameter γ 1.344 
Molar Helmholtz energy Am J mol−1 −23 810.7 
PropertySymbolUnitValue
Input: T = 210 K, p = 10.0 MPa 
Molar internal energy Um J mol−1 10 385.8 
Molar entropy Sm J mol−1 K−1 100.847 
Molar volume Vm cm3 mol−1 74.037 
Molar Gibbs energy Gm J mol−1 −10 051.8 
Molar enthalpy Hm J mol−1 11 126.2 
Isochoric molar heat capacity cv,m J mol−1 K−1 74.126 
Isobaric molar heat capacity cp,m J mol−1 K−1 88.651 
Isobaric expansivity α K−1 4.744 × 10−4 
Isothermal compressibility KT MPa−1 2.409 × 10−4 
Isentropic compressibility KS MPa−1 2.014 × 10−4 
Thermal Grüneisen parameter γ 1.967 
Molar Helmholtz energy Am J mol−1 −10 792.1 
Input: T = 330 K, Vm = 70 cm3 mol−1 
Molar internal energy Um J mol−1 20 461.3 
Molar entropy Sm J mol−1 K−1 134.157 
Pressure p MPa 574.6 
Molar Gibbs energy Gm J mol−1 16 408.4 
Molar enthalpy Hm J mol−1 60 680.3 
Isochoric molar heat capacity cv,m J mol−1 K−1 113.624 
Isobaric molar heat capacity cp,m J mol−1 K−1 127.905 
Isobaric expansivity α K−1 2.834 × 10−4 
Isothermal compressibility KT MPa−1 1.299 × 10−4 
Isentropic compressibility KS MPa−1 1.154 × 10−4 
Thermal Grüneisen parameter γ 1.344 
Molar Helmholtz energy Am J mol−1 −23 810.7 
FIG. 6.

Comparison between the molar volume data, Vm, along the sublimation curve and the EOS description as a function of temperature: (a) Vm measurements and model calculations (black curve); (b) deviations from the experimental data.

FIG. 6.

Comparison between the molar volume data, Vm, along the sublimation curve and the EOS description as a function of temperature: (a) Vm measurements and model calculations (black curve); (b) deviations from the experimental data.

Close modal
FIG. 7.

Comparison between the isobaric heat capacity data, cp, along the sublimation curve and the equation of state description as a function of temperature: (a) cp measurements and model calculations (black curve); (b) deviations from the experimental data.

FIG. 7.

Comparison between the isobaric heat capacity data, cp, along the sublimation curve and the equation of state description as a function of temperature: (a) cp measurements and model calculations (black curve); (b) deviations from the experimental data.

Close modal
FIG. 8.

Comparison between the isentropic bulk modulus data, KS, along the sublimation curve and the equation of state description as a function of temperature: (a) KS measurements and model calculations (in a black curve); (b) deviations from the experimental data.

FIG. 8.

Comparison between the isentropic bulk modulus data, KS, along the sublimation curve and the equation of state description as a function of temperature: (a) KS measurements and model calculations (in a black curve); (b) deviations from the experimental data.

Close modal
FIG. 9.

Comparison between the enthalpy of sublimation (melting) data, ΔH, and the equation of state description as a function of temperature: (a) enthalpy measurements and model calculations; (b) deviations from the experimental data.

FIG. 9.

Comparison between the enthalpy of sublimation (melting) data, ΔH, and the equation of state description as a function of temperature: (a) enthalpy measurements and model calculations; (b) deviations from the experimental data.

Close modal
FIG. 10.

Comparison between the sublimation pressure measurements, psub, and the equation of state description as a function of temperature: (a) sublimation pressure measurements and model calculations (in a black curve) on a logarithmic vertical scale; (b) deviations from the experimental data.

FIG. 10.

Comparison between the sublimation pressure measurements, psub, and the equation of state description as a function of temperature: (a) sublimation pressure measurements and model calculations (in a black curve) on a logarithmic vertical scale; (b) deviations from the experimental data.

Close modal
FIG. 11.

Comparison between the melting pressure measurements, pmelt, and the equation of state description as a function of temperature: (a) pmelt measurements and model calculations (in a black curve); (b) deviations from the experimental data.

FIG. 11.

Comparison between the melting pressure measurements, pmelt, and the equation of state description as a function of temperature: (a) pmelt measurements and model calculations (in a black curve); (b) deviations from the experimental data.

Close modal
FIG. 12.

Comparison between the molar volume data Vm measured for the solid phase at high pressures and the equation of state description as a function of pressure: (a) Vm measurements and model calculations; (b) deviations from the experimental data.

FIG. 12.

Comparison between the molar volume data Vm measured for the solid phase at high pressures and the equation of state description as a function of pressure: (a) Vm measurements and model calculations; (b) deviations from the experimental data.

Close modal
FIG. 13.

Comparison between the thermal expansivity data, α, measured for the solid phase at high pressure and the equation of state description as a function of pressure: (a) α measurements and model calculations; (b) deviations from the experimental data.

FIG. 13.

Comparison between the thermal expansivity data, α, measured for the solid phase at high pressure and the equation of state description as a function of pressure: (a) α measurements and model calculations; (b) deviations from the experimental data.

Close modal
FIG. 14.

Thermodynamic properties calculated from the equation of state to examine the thermodynamic surface behavior. (a) p-Vm curves; (b) isentropic bulk modulus KS; (c) isobaric expansivity α; (d) thermal pressure coefficient β.

FIG. 14.

Thermodynamic properties calculated from the equation of state to examine the thermodynamic surface behavior. (a) p-Vm curves; (b) isentropic bulk modulus KS; (c) isobaric expansivity α; (d) thermal pressure coefficient β.

Close modal
FIG. 15.

Molar heat capacities cm as functions of temperature along the saturation (sublimation and melting) curve.

FIG. 15.

Molar heat capacities cm as functions of temperature along the saturation (sublimation and melting) curve.

Close modal
FIG. 16.

Grüneisen parameter γ curves for isotherms and saturated conditions as a function of molar volume.

FIG. 16.

Grüneisen parameter γ curves for isotherms and saturated conditions as a function of molar volume.

Close modal

The re-adjusted parameters also impact the simulation results comparisons shown in Table S1 to S3 of the supplementary material and the sample calculation Excel sheet. The updated versions of these files are attached here as the revised supplementary material. A software implementation of this equation of state can also be accessed through the freely available web application ThermoFAST Web, available at the URL: https://thermofastweb.azurewebsites.net/.

See supplementary material for (i) initial values used for the solid benzene EOS regression together with comparisons between this solid model and simulation-based literature data and (ii) sample calculations made with this solid model, the extrapolated fluid EOS, and the auxiliary functions.

The authors are grateful to Dr. Peter Falloon (University of Western Australia) for pointing out these errors, and for assisting with the testing of the revised corrected model.

1.
X.
Xiao
,
J. P. M.
Trusler
,
X.
Yang
,
M.
Thol
,
S. Z. S.
Al Ghafri
,
D.
Rowland
, and
E. F.
May
,
J. Phys. Chem. Ref. Data
50
,
043104
(
2021
).
2.
E. R.
Andrew
and
R. G.
Eades
,
Proc. R. Soc. London, Ser. A
218
,
537
(
1953
).
3.
J. N.
Andrews
and
A. R. J. P.
Ubbelohde
,
Proc. R. Soc. London, Ser. A
228
,
435
(
1955
).
4.
G. E.
Bacon
,
N. A.
Curry
, and
S. A.
Wilson
,
Proc. R. Soc. London, Ser. A
279
,
98
(
1964
).
5.
W.
Biltz
,
W.
Fischer
, and
E.
Wünnenberg
,
Z. Phys. Chem.
151A
,
13
(
1930
).
6.
E. G.
Cox
,
Proc. R. Soc. London, Ser. A
135
,
491
(
1932
).
7.
E. G.
Cox
,
D. W. J.
Cruickshank
, and
J. A. S.
Smith
,
Proc. R. Soc. London, Ser. A
247
,
1
(
1958
).
8.
J. D.
Dunitz
and
R. M.
Ibberson
,
Angew. Chem., Int. Ed.
47
,
4208
(
2008
).
10.
A. D.
Fortes
and
S. C.
Capelli
,
Phys. Chem. Chem. Phys.
20
,
16736
(
2018
).
11.
W.
Heuse
,
Z. Phys. Chem.
147A
,
266
(
1930
).
12.
A.
Heydweiller
,
Ann. Phys.
297
,
527
(
1897
).
13.
P. F.
Higgins
,
R. A. B.
Ivor
,
L. A. K.
Staveley
, and
J. J. d. C.
Virden
,
J. Chem. Soc.
1964
,
5762
.
14.
V.
Kozhin
,
Zh. Fiz. Khim.
28
,
566
(
1954
).
15.
C. A.
McConville
,
Y.
Tao
,
H. A.
Evans
,
B. A.
Trump
,
J. B.
Lefton
,
W.
Xu
,
A. A.
Yakovenko
,
E.
Kraka
,
C. M.
Brown
, and
T.
Runčevski
,
Chem. Commun.
56
,
13520
(
2020
).
16.
K.
Ziegler
and
F.
Ditzel
,
Justus Liebigs Ann. Chem.
473
,
194
(
1929
).
17.
J. E.
Ahlberg
,
E. R.
Blanchard
, and
W. O.
Lundberg
,
J. Chem. Phys.
5
,
539
(
1937
).
18.
D. H.
Andrews
,
G.
Lynn
, and
J.
Johnston
,
J. Am. Chem. Soc.
48
,
1274
(
1926
).
19.
S.
Aoyama
and
E.
Kanda
,
Sci. Rep. Tohoku Imp. Univ. Ser.
24
,
116
(
1935
).
20.
W. F.
Brucksch
, Jr.
and
W. T.
Ziegler
,
J. Chem. Phys.
10
,
740
(
1942
).
21.
J.
Dewar
,
Proc. R. Soc. London, Ser. A
89
,
158
(
1913
).
22.
A.
Diedrichs
, “
Optimization of a dynamic differential scanning calorimeter for the experimental determination of heat capacity
,” M.S. thesis,
University of Oldenburg
,
Germany
,
2005
.
23.
I.
Hahnenkamp
, “
Experimental and theoretical studies on the solubility of drugs in solvents
,” M.S. thesis,
University of Oldenburg
,
Germany
,
2008
.
24.
H. M.
Huffman
,
G. S.
Parks
, and
A. C.
Daniels
,
J. Am. Chem. Soc.
52
,
1547
(
1930
).
25.
O.
Maass
and
L. J.
Waldbauer
,
J. Am. Chem. Soc.
47
,
1
(
1925
).
26.
Z.
Nan
and
Z.-C.
Tan
,
Thermochim. Acta
419
,
275
(
2004
).
28.
G. D.
Oliver
,
M.
Eaton
, and
H. M.
Huffman
,
J. Am. Chem. Soc.
70
,
1502
(
1948
).
29.
D. R.
Stull
,
J. Am. Chem. Soc.
59
,
2726
(
1937
).
31.
J. C. W.
Heseltine
,
D. W.
Elliott
, and
O. B.
Wilson
, Jr.
,
J. Chem. Phys.
40
,
2584
(
1964
).
32.
J. H.
De Boer
,
Trans. Faraday Soc.
32
,
10
(
1936
).
33.
V. R.
Deitz
,
J. Am. Chem. Soc.
55
,
472
(
1933
).
34.
C. G.
De Kruif
and
C. H. D.
Van Ginkel
,
J. Chem. Thermodyn.
9
,
725
(
1977
).
35.
W.
Hessler
,
Wiss. Z. Wilhelm-Pieck-Univ. Rostock
33
,
9
(
1984
).
36.
A. W.
Jackowski
,
J. Chem. Thermodyn.
6
,
49
(
1974
).
37.
A. H.
Jones
,
J. Chem. Eng. Data
5
,
196
(
1960
).
38.
G.
Milazzo
,
Ann. Chim.
46
,
1105
(
1956
).
39.
G.
Milazzo
,
Chem. Abstr.
51
,
7791b
(
1957
).
40.
C. F.
Mündel
,
Z. Phys. Chem.
85U
,
435
(
1913
).
41.
K.
Růžička
,
M.
Fulem
, and
C.
Červinka
,
J. Chem. Thermodyn.
68
,
40
(
2014
).
42.
R. M.
Stephenson
and
S.
Malanowski
,
Handbook of the Thermodynamics of Organic Compounds
(
Elsevier
,
New York
,
1987
).
43.
H.
Ha
,
J. A.
Morrison
, and
E. L.
Richards
,
J. Chem. Soc., Faraday Trans. 1
72
,
1051
(
1976
).
44.
D. R.
Stull
,
Ind. Eng. Chem.
39
,
517
(
1947
).
45.
J. T.
Barker
,
Z. Phys. Chem.
71U
,
235
(
1910
).
46.
S. U.
Choi
and
H. C.
Brown
,
J. Am. Chem. Soc.
88
,
903
(
1966
).
47.
C. G.
De Kruif
,
J. Chem. Thermodyn.
12
,
243
(
1980
).
48.
I.
Kiss
,
G. Y.
Jakli
, and
H.
Illy
,
Acta Chim. Acad. Sci. Hung.
71
,
59
(
1972
).
49.
K.
Liu
and
R. M.
Dickhut
,
Chemosphere
29
,
581
(
1994
).
50.
G.
Milazzo
,
Chem. Ing. Tech.
28
,
646
(
1956
).
51.
N. R.
Miljevic
,
Z. V.
Knezevic
,
V. R.
Dokic
, and
J. D.
Pupezin
,
Glas. Hem. Drus. Beograd
42
,
243
(
1977
).
52.
D.
Radulescu
and
M.
Alexa
,
Bull. Soc. Chim. Romania
20A
,
89
(
1938
).
53.
R. P.
Rastogi
,
J.
Nath
, and
J.
Misra
,
J. Phys. Chem.
71
,
1277
(
1967
).
54.
S.
Young
,
Sci. Proc. R. Dublin Soc.
12
,
374
(
1910
).
55.
M.
Azreg-Aïnou
,
A.
Hüseynov
, and
B.
İbrahimoğlu
,
J. Chem. Phys.
124
,
204505
(
2006
).
56.
M.
Azreg-Aïnou
,
A.
Hüseynov
, and
B.
İbrahimoğlu
,
J. Chem. Phys.
125
,
099901
(
2006
).
57.
P. W.
Bridgman
,
Phys. Rev.
3
,
153
(
1914
).
58.
J.
Osugi
,
K.
Shimizu
, and
A.
Onodera
,
Rev. Phys. Chem. Jpn.
34
,
97
(
1965
).
59.
F. D.
Rossini
,
K. S.
Pitzer
,
R. L.
Arnett
,
R. M.
Braun
, and
G. C.
Pimentel
,
Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds
(
Carnegie Press
,
Pittsburgh
,
1953
).
60.
K.
Schäfer
and
E.
Lax
,
Kalorische Zustandsgrößen (Landolt-Börnstein: Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik, 2/4)
(
Springer-Verlag
,
Berlin
,
1961
).
61.
G. W.
Smith
,
Mol. Cryst. Liq. Cryst.
49
,
207
(
1979
).
62.
K.
Stratton
and
J. R.
Partington
,
Lond. Edinb. Dublin Philos. Mag. J. Sci.
48
,
1085
(
1924
).
63.
J.
Timmermans
,
Physico-Chemical Constants of Pure Organic Compounds
(
Elsevier Publishing
,
Amsterdam
,
1950
).
64.
J.
Timmermans
,
Physico-Chemical Constants of Pure Organic Compounds Vol. II
(
Elsevier
,
New York
,
1965
).
65.
H.
Tschamler
,
Monatsh. Chem.
79
,
162
(
1948
).
66.
A.
Watanabe
,
T.
Iiyama
, and
K.
Kaneko
,
Chem. Phys. Lett.
305
,
71
(
1999
).
67.
W.
Xu
,
R.
Zhu
,
Y.
Tian
,
H.
Li
, and
H.
Li
,
J. Chem. Eng. Data
52
,
1975
(
2007
).
68.
W. T.
Ziegler
and
D. H.
Andrews
,
J. Am. Chem. Soc.
64
,
2482
(
1942
).
69.
E. A.
Block
,
Z. Phys. Chem.
82U
,
403
(
1913
).
70.
G.
Tammann
,
Kristallisieren und Schmelzen
(
University of Leipzig
,
Leipzig
,
1903
).
71.
P. W.
Bridgman
,
Proc. Am. Acad. Arts Sci.
77
,
129
(
1949
).
72.
L.
Deffet
,
Bull. Soc. Chim. Belg.
44
,
41
(
1935
).
73.
L.
Deffet
and
G.
Vlerick
,
Bull. Soc. Chim. Belg.
51
,
237
(
1942
).
74.
U.
Domańska
and
P.
Morawski
,
J. Chem. Thermodyn.
37
,
1276
(
2005
).
75.
U.
Domańska
and
P.
Morawski
,
Green Chem.
9
,
361
(
2007
).
76.
A. J.
Easteal
,
L. A.
Woolf
, and
F. L.
Wilson
,
Int. J. Thermophys.
6
,
275
(
1985
).
77.
P.
Figuière
,
A. H.
Fuchs
,
M.
Ghelfenstein
, and
H.
Szwarc
,
J. Phys. Chem. Solids
39
,
19
(
1978
).
79.
M.
Ghelfenstein
and
H.
Szwarc
,
Chem. Phys. Lett.
32
,
93
(
1975
).
80.
G. A.
Hulett
,
Z. Phys. Chem.
28U
,
629
(
1899
).
81.
T.
Makita
and
T.
Takagi
,
Rev. Phys. Chem. Jpn.
38
,
41
(
1968
).
82.
K.
Nagaoka
and
T.
Makita
,
Int. J. Thermophys.
8
,
415
(
1987
).
83.
J.
Osugi
,
K.
Shimizu
,
K.
Yasunami
,
M.
Moritoki
, and
A.
Onodera
,
Rev. Phys. Chem. Jpn.
38
,
90
(
1969
).
84.
P.
Pruzan
, Ph.D. thesis,
These de Doctorat es-Sciences
,
Paris
,
1976
.
85.
P.
Pruzan
,
L.
Ter Minassian
, and
A.
Soulard
, in
High-Pressure Science and Technology
, edited by
K. D.
Timmerhaus
and
M. S.
Barber
(
Springer; University of Colorado
,
Boulder, CO
,
1979
), Vol. 1, pp.
368
378
.
86.
T. F.
Sun
,
P. J.
Kortbeek
,
S. N.
Biswas
,
N. J.
Trappeniers
, and
J. A.
Schouten
,
Ber. Bunsenges. Phys. Chem.
91
,
1013
(
1987
).
87.
Y.
Tanaka
and
M.
Kawakami
,
Fluid Phase Equilib.
125
,
103
(
1996
).
88.
C.
Yokoyama
,
T.
Ebina
, and
S.
Takahashi
,
Fluid Phase Equilib.
84
,
207
(
1993
).
89.
A.
Budzianowski
and
A.
Katrusiak
,
Acta Crystallogr., Sect. B: Struct. Sci.
62
,
94
(
2006
).
90.
D. W. M.
Hofmann
and
L. N.
Kuleshova
,
Cryst. Growth Des.
14
,
3929
(
2014
).
91.
A.
Katrusiak
,
M.
Podsiadło
, and
A.
Budzianowski
,
Cryst. Growth Des.
10
,
3461
(
2010
).
92.
P. W.
Bridgman
,
J. Chem. Phys.
9
,
794
(
1941
).
93.
P. W.
Bridgman
,
Proc. Am. Acad. Arts Sci.
74
,
399
(
1942
).
94.
R. G.
Ross
,
P.
Andersson
, and
G.
Bäckström
,
Mol. Phys.
38
,
527
(
1979
).
95.
A. H.
Fuchs
,
P.
Pruzan
, and
L.
Ter Minassian
,
J. Phys. Chem. Solids
40
,
369
(
1979
).
96.
P.
Pruzan
,
D. H.
Liebenberg
, and
R. L.
Mills
,
J. Phys. Chem. Solids
47
,
949
(
1986
).

Supplementary Material