An analytical equation of state for molten alkaline and alkaline earth metals:
Prediction from critical point constants
M. H. Mousazadeh 1,*, Z. Mousapoor 2, H. Behnejad 3
1Department of Chemistry, Nuclear Science Research School, Nuclear Science & Technology Research Institute (NSTRI), 11365-3486, Tehran, Iran
2 Chemistry Department, Payamenour University, Tehran, Iran
3Department of Chemistry, University of Tehran, Tehran, Iran
Zahra Batoul . Mousapour –Kotena Chemistry Department, Payamenour University, Tehran, Iran E-mail: [email protected] *Tel.: 0917 361 6577
Abstract
We developed an equation of state based on statistical–mechanical perturbation theory for alkaline and alkaline earth metals. The equation is that the Ihm, Song, and Mason, in which the three temperature dependant parameters are, calculated by means of the corresponding states correlations. In this work it is shown that using the critical constants of fluids as scale constants would correlate the temperature-dependent parameters of the equation of state with sufficient accuracy. We have tested the predicted equation of state against the experimental data for alkaline and alkaline earth metals. Results shows the present correlation is more accurate and covers a much wider range of temperatures and pressures than the previous works.
Introduction
The influence of the attractive forces and the softness of repulsions can be treated by statistical–mechanical perturbation theory. A statistical–mechanical theory has recently been presented to derive a new analytical equation of state (EOS) of fluids by Song and Mason [1].
Theory
In a series of articles Ihm, Song, Mason, (ISM) and Colleagues [1,2,3] have derived
∫
2
2 c r r r r
a simple analytic equation of state for nonpolar fluids:
∞ −βu 2
B2 = 2π
∫
(1 − e)r dr(1)
0
α (T ) = 2π ∞[1 − e−u0 (r ) / kT ]r2 dr
0
b(T ) ≡ 2
πσ 3 =α + T dα
= 2π
∫
rm [1 − (1 + u0 ) e−u0/ kT
]r2 dr
(2) (3)
3 dT 0 kT
B2 is the second virial coefficient, that depend on the intermolecular potential, u. a(T) and b(T) are two parameters that depend on the molecular repulsive forces ( u0 ). T is the temperature, q is the density, and kT is the thermal energy. Z obtained best-fit values. The product hb is analogous to the van der Waals excluded volume. When intermolecular potential and the experimental B2 values are not available, there are several correlation schemes, usually based on the corresponding-states principle, by which second virial coefficients can be calculated [4,5].
Correlation
The parameters a and b are rather insensitive to the details of the potential energy curve and depend primarily on its well-depth, ‹; and the position of minimum, rm [1].
The parameters B2, a, and b can be calculated by rescaling the empirical formulas by Song and Mason [6] in terms of critical point constants (Tc, qc), i.e.
B
(
T)
ρ = 0.3892 − 0.7310(
T)
−1 −1.0309(
T)
−2 + 0.3280(
T)
−3 −0.07148
(
T)
−4(4)
α ρ = a exp
(
− c T)
+ a{
1 − exp(
− c T−0.25
)}
(5)
c 1 1 r 2 2 r
b ρ = a
(
1 − cT
)
exp(
− c T)
+a
{
1 −(
1 + 0.25c T −0.25)}
exp(
−c T −0.25
)
(6)
c 1 1 r 1 r 2 2 r 2 r
where, Tr (=T / Tc), a1 = -0.05265, a2 = 1.00472, c1 = 0.98635, c2 = 1.17728. We will examine this equation of state for the prediction of PVT properties of alkaline and alkaline earth metals in the following sections.
Results and discussion
We used Eqs. (4-6) to calculate B2(T), b(T) and a(T) at each temperature. Tc, qc, were
4
found in Refs. [7]. Z obtained best-fit values. Once the value of h is determined, the densities of liquids at different temperatures and pressures can be predicted by means of Eq. (1). The results are compared with the experimental data [7]. It is clear from the results tabulated in Ta bles 1 that the agreement is quite good. In order to check the predictive power of the present correlation with other previous works [6,7], the. Comparison of the predicted results shows that the present correlation is more accurate and covers a much wider range of temperatures and pressures than the previous works [6,7].
Table 1. The comparison absolute average deviation (%AAD) between this work with other works [13,14] for alkaline metals.
Substance bT %AAD %AAD %AAD
(bHv, q)tr (o, q)b (T, q)c This work
Li 600-2200 3.22 5.85 3.89
Na 600-2200 17.36 14.17 5.69
K 550-2000 13.77 15.61 5.41
Rb 450-2000 17.92 27.74 5.45
Cs 500-1800 24.60 17.56 5.92
Reference:
1. Y. Song, E.A. Mason, J. Chem. Phys. 91 (1989) 7840-7853.
2. G. Ihm, Y. Song, E.A. Mason, J. Chem. Phys. 94 (1991) 3839-3848.
3. Y. Song, E.A. Mason, Phys. Rev. A 42 (1990) 4749-4755.
4. A. Boushehri, E.A. Mason, Int. J. Thermophys. 14 (1993) 685-697.
5. C.Tsonopoulos, Fluid Phase Equilib. 21 (2003) 35-49.
6. Y. Song, E.A. Mason, Fluid Phase Equilib. 75 (1992) 105-115.
7. N.B. Vargaftik, Handbook of Physical Properties of Liquids and Gases, Hemisphere, Washington, DC, 1975.