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.
$\begin{array}{l}\frac{\text{\rho}\left(T,P\right)}{\text{\rho}\left(T,{P}_{\text{ref}}\right)}=\frac{1}{1-C\left(T\right)\mathrm{ln}\left\{\frac{B\left(T\right)+P}{B\left(T\right)+{P}_{\text{ref}}}\right\}},\\ C\left(T\right)={a}_{1}+{b}_{1}\left(T\text{/K}\right)+{c}_{1}{\left(T\text{/K}\right)}^{2},\\ B\left(T\right)\text{/MPa}={a}_{2}+{a}_{3}\left(T\text{/K}\right)+{a}_{4}{\left(T\text{/K}\right)}^{2}+{a}_{5}{\left(T\text{/K}\right)}^{3}+{a}_{6}{\left(T\text{/K}\right)}^{4}\end{array}$ (1)
Parameters a_{i} (i = 1,...,6) and b_{1} are given in the second row of each entry in the table below. Parameter c_{1} 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.
$\text{\rho}\left(T,{P}_{\text{ref}}\right)/\left(\text{kg}\text{\hspace{0.17em}}{\text{m}}^{-3}\right)={\displaystyle \sum _{i=1}^{{N}_{\text{p}}}{a}_{i}{\left(T\text{/K}\right)}^{(i-1)}}$ (2)
$\text{\rho}\left(T,{P}_{\text{ref}}\right)={\text{\rho}}_{\text{c}}\left[1+{\displaystyle \sum _{i=1}^{{N}_{\text{p}}}{a}_{i}{\left(1-{T}_{\text{r}}\right)}^{\left(i/3\right)}}\right],\text{\hspace{1em}}{T}_{\text{r}}=T/{T}_{\text{c}}$ (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 C_{1} to C_{10} and n-alkanes C_{5} to C_{16} 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).
$RMSD/\%=100{\left\{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left(\frac{{\text{\rho}}_{\mathrm{exp}}-{\text{\rho}}_{\text{calc}}}{{\text{\rho}}_{\mathrm{exp}}}\right)}^{2}}\right\}}^{1/2}$
where N is the number of experimental values included in the fit.
Wagner equation: If the maximum temperature 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
$\begin{array}{c}{P}_{\text{sat}}\left(T\right)=\\ {P}_{\text{c}}\mathrm{exp}\left[\frac{{a}_{\text{1}}\left(1-{T}_{\text{r}}\right)+{a}_{2}{\left(1-{T}_{\text{r}}\right)}^{1.5}+{a}_{3}{\left(1-{T}_{\text{r}}\right)}^{2.5}+{a}_{4}{\left(1-{T}_{\text{r}}\right)}^{5}}{{T}_{\text{r}}}\right]\end{array}$ (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/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.
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