Section: 6 | Temperature and Pressure Dependence of Liquid Density |
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 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# TEMPERATURE AND PRESSURE DEPENDENCE OF LIQUID DENSITY

Ivan Cibulka

This table provides the data to calculate the temperature and pressure dependence of the denisty of 61 organic liquids using two different methods: the Tait equation and the Wagner function.

Tait Equation: The Tait equation (Refs. 1, 2) gives the ratio between the density at pressure P, ρ(T,P), relative to the density at a reference pressure, ρ(T,Pref), at the same temperature T.

$\begin{array}{l}\frac{\text{ρ}\left(T,P\right)}{\text{ρ}\left(T,{P}_{\text{ref}}\right)}=\frac{1}{1-C\left(T\right)\mathrm{ln}\left\{\frac{B\left(T\right)+P}{B\left(T\right)+{P}_{\text{ref}}}\right\}},\\ C\left(T\right)={a}_{1}+{b}_{1}\left(T\text{/K}\right)+{c}_{1}{\left(T\text{/K}\right)}^{2},\\ B\left(T\right)\text{/MPa}={a}_{2}+{a}_{3}\left(T\text{/K}\right)+{a}_{4}{\left(T\text{/K}\right)}^{2}+{a}_{5}{\left(T\text{/K}\right)}^{3}+{a}_{6}{\left(T\text{/K}\right)}^{4}\end{array}$  (1)

Parameters ai (i = 1,...,6) and b1 are given in the second row of each entry in the table below. Parameter c1 is zero for most substances, and therefore its value for heptane, the only organic liquid included in the table for which it is relevant, is given in a footnote.

The reference pressure is Pref = 0.101325 MPa at temperatures either at or below the normal boiling point temperature (Tnbp) and Pref = Psat(T) (saturated vapor pressure) at temperatures T > Tnbp. Ranges of validity of the equation (Tmin, Tmax, Pmax) are derived from ranges of experimental data; the minimal pressure of validity is taken as Pref, i.e., interpolation between Pref and lowest experimental pressure is allowed. The upper limit of application is the freezing line (if not limited by the ranges of validity). To avoid any large-scale extrapolation, the validity ranges are rectangular areas (TmaxTmin)Pmax. If in a specific temperature interval(s) the maximum experimental pressure exceeded the given value of Pmax, then the maximum pressure given in the table is denoted by (r) which means that the validity range given in the table is a rectangular subset of the non-rectangular experimental T, P range. In a few cases Pmax is given as a ratio where the first value corresponds to Tmin and the second one to Tmax, i.e., the validity range has approximately a trapezoidal shape.

Values of parameters were taken from the papers (Refs. 3–10) where detailed information on the fits, experimental data, and application ranges is available. The numerical values of the parameters are different from those reported in papers (Refs. 3–10) because the forms of polynomials C(T) and B(T) differ. Also, the parameters recorded in the table below do not necessarily correspond to those in Refs. 3–10 as some fits were updated using newly published experimental data.

Smoothing function: To determine the density at the reference pressure ρ (T,Pref) for use in Eq. 1, one of two smoothing functions is used, both polynomial expansions.

$\text{ρ}\left(T,{P}_{\text{ref}}\right)/\left(\text{kg}\text{\hspace{0.17em}}{\text{m}}^{-3}\right)=\sum _{i=1}^{{N}_{\text{p}}}{a}_{i}{\left(T\text{/K}\right)}^{\left(i-1\right)}$  (2)

$\text{ρ}\left(T,{P}_{\text{ref}}\right)={\text{ρ}}_{\text{c}}\left[1+\sum _{i=1}^{{N}_{\text{p}}}{a}_{i}{\left(1-{T}_{\text{r}}\right)}^{\left(i/3\right)}\right],\text{ }{T}_{\text{r}}=T/{T}_{\text{c}}$  (3)

where Np is the number of adjustable parameters, ai, whose values are given in the third row of each entry, as prefaced by the appropriate equation number. Values of the critical density ρc and the critical temperature Tc used for the fits using Eq. 3 are also recorded in the table. Parameters were mostly taken from Refs. 3–10, those for 1-alkanols C1 to C10 and n-alkanes C5 to C16 are from Ref. 11. Data used for the fits were predominantly recommended values published in the TRC Thermodynamic Tables (Refs. 12,13), sometimes combined with the original experimental data or, in a few cases, the original experimental data were correlated.

RMSD is a relative root-mean-square deviation (in percent) between experimental values of density and those calculated from the particular function (Tait Eq. 1 or Eqs. 2,3).

$RMSD/%=100{\left\{\frac{1}{N}\sum _{i=1}^{N}{\left(\frac{{\text{ρ}}_{\mathrm{exp}}-{\text{ρ}}_{\text{calc}}}{{\text{ρ}}_{\mathrm{exp}}}\right)}^{2}\right\}}^{1/2}$

where N is the number of experimental values included in the fit.

Wagner equation: If the maximum temperature Tmax of validity of the Tait equation is greater than the normal boiling point temperature Tnbp, then the Wagner equation is applicable in the form of either

$\begin{array}{c}{P}_{\text{sat}}\left(T\right)=\\ {P}_{\text{c}}\mathrm{exp}\left[\frac{{a}_{\text{1}}\left(1-{T}_{\text{r}}\right)+{a}_{2}{\left(1-{T}_{\text{r}}\right)}^{1.5}+{a}_{3}{\left(1-{T}_{\text{r}}\right)}^{2.5}+{a}_{4}{\left(1-{T}_{\text{r}}\right)}^{5}}{{T}_{\text{r}}}\right]\end{array}$  (4)

or

$\begin{array}{c}{P}_{\text{sat}}\left(T\right)=\\ {P}_{\text{c}}\mathrm{exp}\left[\frac{{a}_{\text{1}}\left(1-{T}_{\text{r}}\right)+{a}_{2}{\left(1-{T}_{\text{r}}\right)}^{1.5}+{a}_{3}{\left(1-{T}_{\text{r}}\right)}^{3}+{a}_{4}{\left(1-{T}_{\text{r}}\right)}^{6}}{{T}_{\text{r}}}\right]\end{array}$  (5)

where T r = T/Tc are recorded in the third line for each substance. Values of the critical pressure Pc and critical temperature Tc used in Eqs. 4 and 5 are also recorded in the table as prefaced by the equation number. Values of the critical temperature may differ a little from those recorded for the function, Eq. 3. Parameters of Eqs. 4 and 5 were taken mostly from the papers by McGarry (Ref. 14) and Ambrose and Walton (Ref. 15); in a few cases, the fits were performed using original experimental data or in combination with the recommended values from the TRC Thermodynamic Tables (Refs. 12,13).

The two right-hand-most columns gives values of the isothermal compressibility coefficient, κT = –(1/V)(∂V/∂P)T = (1/ρ)(∂ρ/∂P)T  , and the isobaric cubic expansion coefficient, αP = (1/V)(∂V/∂T)P = –(1/ρ)(∂ρ/∂T)P ,  calculated for T = 298.15 K and P = 0.101325 MPa using Tait Eq. 1 and from the ρ (T, Pref) equation, respectively. In a very few cases when the lower temperature limit of the Tait equation Tmin is greater than 298.15 K, the extrapolated values of isothermal compressibility are given.

The column and row definitions for the table are as follows.

 Column heading Definition Row 1 Entry information Mol. form. Molecular formula of liquid; liquids listed by Hill order Name Liquid name Tnbp Normal boiling point, in K κT Isothermal compressibility coefficient, in units Pa-1 αP Isobaric cubic expansion coefficient, in units kK-1 Ref. Reference Row 2 Tait equations Eq. Tait equation number; see text a1, a2, a3, a4, a5, b1 Tait equation coefficients; for heptane, coefficient c1 is required and given in Footnote b Tmin/Tmax Temperature range of validity for Tait equation, in K Pmax Maximum pressure for validity of Tait equation, in MPa RMSD Relative root-mean-square deviation (in percent) between experimental and calculated (Eqs. 1-3) values of density Rows 3 and 4 Smoothing and Wagner equations Eq. Equation number a1, a2, a3, a4, a5 Equation coefficients; see Footnote a for extra term needed for ethanol calculations Tc Critical temperature, in K Pc Critical pressure, in MPa ρc Density at critical temperature and pressure, in kg m-3

# References

1. Tait, P. G., in Physics and Chemistry of the Voyage of H.M.S. Challenger, Vol. II, Part IV, Thomson, C. W., and Murray, J., Eds., H.M.S.O., London, 1889.
2. Tamman, G., Z. Phys. Chem. 17, 620, 1895. [https://doi.org/10.1007/BF01841600]
3. Cibulka, I., and Ziková, M., J. Chem. Eng. Data 39, 876, 1994. [https://doi.org/10.1021/je00016a055]
4. Cibulka, I., and Hndkovský, L., J. Chem. Eng. Data 41, 657, 1996. [https://doi.org/10.1021/je960058m]
5. Cibulka, I., Hndkovský, L., and Takagi, T., J. Chem. Eng. Data 42, 2, 1997. [https://doi.org/10.1021/je960199o]
6. Cibulka, I., Hndkovský, L., and Takagi, T., J. Chem. Eng. Data 42, 415, 1997. [https://doi.org/10.1021/je960199o]
7. Cibulka, I., and Takagi, T., J. Chem. Eng. Data 44, 411, 1999. [https://doi.org/10.1021/je980278v]
8. Cibulka, I., and Takagi, T., J. Chem. Eng. Data 44, 1105, 1999. [https://doi.org/10.1021/je990140s]
9. Cibulka, I., Takagi, T., and Rika, K., J. Chem. Eng. Data 46, 2, 2001. [https://doi.org/10.1021/je0002383]
10. Cibulka, I., and Takagi, T., J. Chem. Eng. Data 47, 1037, 2002. [https://doi.org/10.1021/je0200463]
11. Cibulka, I., Fluid Phase Equilib. 89, 1, 1993. [https://doi.org/10.1016/0378-3812(93)85042-K]
12. TRC Thermodynamic Tables, Hydrocarbons, Thermodynamics Research Center (TRC), NIST, Thermophysical Properties Division, Boulder, CO.
13. TRC Thermodynamic Tables, Non-Hydrocarbons, Thermodynamics Research Center (TRC), NIST, Thermophysical Properties Division, Boulder, CO.
14. McGarry, J., Ind. Eng. Chem., Process Des. Develop. 22, 313, 1983. [https://doi.org/10.1021/i200021a023]
15. Ambrose, D., and Walton, J., Pure Appl. Chem. 61, 1395, 1989. [https://doi.org/10.1351/pac198961081395]

## Parameters to Calculate Temperature and Pressure Dependence of Density and Calculated Values of Physical Constants for Selected Liquids

 Mol. form. Name Eq. a1 a2 a3 a4 a5 a6 b1 Tmin/TmaxK PmaxMPa TcK PcMPa ρckg m–3 RMSD% Continued on next page... CCl4 Tetrachloromethane Tnbp = 349.9 K κT = 1.074 GPa-1 αP = 1.209 kK-1 Ref. 9 1 9.33340·10–2 1.11363·103 -8.68453 2.80698·10–2 –4.22880·10–5 2.37923·10–8 273/413 51/388 0.04 3 1.58994 2.51946 -5.82313 6.96793 -2.51359 253/554 556.4 557.33 0.042 5 -7.07139 1.71497 -2.8993 -2.49466 250/556 556.4 4.551 CHBr3 Tribromomethane Tnbp = 422.3 K κT = 0.809 GPa-1 αP = 0.907 kK-1 Ref. 9 1 1.03492·10–1 2.64208·102 –4.57399·10–1 323/368 150/343 0.058 2 3.55953·103 -1.96212 –1.08712·10–3 283/403 0.001 CHCl3 Trichloromethane Tnbp = 334.4 K κT = 1.037 GPa-1 αP = 1.274 kK-1 Ref. 9 1 9.57210·10–2 4.79593·102 -1.84011 1.81340·10–3 273/348 100(r) 0.031 3 3.56339 -3.86051 3.35636 213/333 536.4 499.49 0.043 5 -6.95546 1.16625 -2.1397 -3.44421 215/536 536.4 5.366 CH2Cl2 Dichloromethane Tnbp = 313.4 K κT = 1.032 GPa-1 αP = 1.428 kK-1 Ref. 9 1 9.76370·10–2 5.24365·102 -2.06633 2.09494·10–3 293/423 100(r) 0.091 3 3.00368 -2.19763 2.34269 178/383 510 440.07 0.014 5 -7.35739 2.17546 -4.07038 3.50701 233/510 510 6.3 CH3I Iodomethane Tnbp = 315.6 K κT = 1.052 GPa-1 αP = 1.255 kK-1 Ref. 9 1 9.54770·10–2 5.36810·102 -2.25115 2.53188·10–3 253/313 160 0.038 2 3.48981·103 -7.47709 1.83592·10–2 –2.36742·10–5 213/313 0.011 CH3OH Methanol Tnbp = 337.7 K κT = 1.231 GPa-1 αP = 1.201 kK-1 Ref. 3 -5.322·10-5
 aAdditional term a7(1 – Tr)(7/3) is included in Eq. 3 with a7 = –1.04344·102﻿. bCoefficient for additional term in Eq. 1 is c1 = -1.513·10-7﻿.

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