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,P_ref), at the same temperature T. 

$\begin{array}{l}\frac{\text{ρ}\left(T,P\right)}{\text{ρ}\left(T,P_{\text{ref}}\right)}=\frac{1}{1-C\left(T\right)\ln \left\{\frac{B\left(T\right)+P}{B\left(T\right)+P_{\text{ref}}}\right\}},\\ C\left(T\right)=a_1+b_1\left(T\text{/K}\right)+c_1{\left(T\text{/K}\right)}^2,\\ B\left(T\right)\text{/MPa}=a_2+a_3\left(T\text{/K}\right)+a_4{\left(T\text{/K}\right)}^2+a_5{\left(T\text{/K}\right)}^3+a_6{\left(T\text{/K}\right)}^4\end{array}$  (1)

Parameters a_i (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 P_ref = 0.101325 MPa at temperatures either at or below the normal boiling point temperature (T_nbp) and P_ref = P_sat(T) (saturated vapor pressure) at temperatures T > T_nbp. Ranges of validity of the equation (T_min, T_max, P_max) are derived from ranges of experimental data; the minimal pressure of validity is taken as P_ref, i.e., interpolation between P_ref 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 (T_max – T_min)P_max. If in a specific temperature interval(s) the maximum experimental pressure exceeded the given value of P_max, 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 P_max is given as a ratio where the first value corresponds to T_min and the second one to T_max, 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,P_ref) for use in Eq. 1, one of two smoothing functions is used, both polynomial expansions.

$\text{ρ}\left(T,P_{\text{ref}}\right)/\left(\text{kg}{\text{m}}^{-3}\right)={\displaystyle \sum _{i=1}^{N_{\text{p}}}{a}_i{\left(T\text{/K}\right)}^{(i-1)}}$  (2)

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

where N_p is the number of adjustable parameters, a_i, whose values are given in the third row of each entry, as prefaced by the appropriate equation number. Values of the critical density ρ_c and the critical temperature T_c 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).

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

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

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

$\begin{array}{c}{P}_{\text{sat}}\left(T\right)=\\ P_{\text{c}}\exp \left[\frac{a_{\text{1}}\left(1-T_{\text{r}}\right)+a_2{\left(1-T_{\text{r}}\right)}^{1.5}+a_3{\left(1-T_{\text{r}}\right)}^{2.5}+a_4{\left(1-T_{\text{r}}\right)}^5}{T_{\text{r}}}\right]\end{array}$  (4)

or

$\begin{array}{c}{P}_{\text{sat}}\left(T\right)=\\ P_{\text{c}}\exp \left[\frac{a_{\text{1}}\left(1-T_{\text{r}}\right)+a_2{\left(1-T_{\text{r}}\right)}^{1.5}+a_3{\left(1-T_{\text{r}}\right)}^3+a_4{\left(1-T_{\text{r}}\right)}^6}{T_{\text{r}}}\right]\end{array}$  (5)

where T _r = T/T_c are recorded in the third line for each substance. Values of the critical pressure P_c and critical temperature T_c 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, κ_T = –(1/V)(∂V/∂P)_T = (1/ρ)(∂ρ/∂P)_T , and the isobaric cubic expansion coefficient, α_P = (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, P_ref) equation, respectively. In a very few cases when the lower temperature limit of the Tait equation T_min is greater than 298.15 K, the extrapolated values of isothermal compressibility are given.

The column and row definitions are as follows.

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