Madiematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 67-75 This paper is available online at http://stdb.hnue.edu.vn
EQUATION OF STATE AND MELTING TEMPERATURE FOR N2, CO, CO2 AND N2O MOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang H o c \ Dinh Quang Vinh^ Bui Due Tinh^ and Nguyen Due Hien^
^Faculty of Physics, Hanoi National University of Education
"^Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province Abstract. The equation of state, the absolute stability temperature of crystalline state and the melting temperature for N2, CO, CO2 and N2O molecular cryocrystals under pressure are determined using the statistical moment method and are compared with the experimental data and other calculations.
Keywords: Molecular cryocrystal, statistical moment method, limiting temperature, absolute stability.
1. Introduction
Molecular crystals are characterized by their strong intramolecular forces and much weaker intermolecular forces. High-pressure spectroscopic studies provide useful data for refining the various model potentials which are used to predict the physical properties of such systems as well as the formation of various crystalline phases.
In the most cases, the melting temperature of crystals is described by the empirical Simon equation In ( P -I- a) = clnT -\- b, where a, b and c are constant and P and T, respectively, are the melting pressure and the melting temperature [1]. However, this equation cannot be used for crystals at extremely high pressure.
On the theoretical side, in order to determine the melting temperature we must use the equilibrium condition of the liquid and solid phases. However, a clear expression of the melting temperature has not yet been obtained in this way.
Notice that the limiting temperature of absolute stability for the crystalline state at a determined pressure is not far from the melting temperamre. Therefore, some researchers had identified the melting curve with the curve of absolute stability for the crystalline state. In order to better determine the limiting temperature of absolute stability for the crystalline state, the correlation effects are calculated using the one-particle distribution function method [4, 10]. Because the difference between these two temperatures is large Received January 7, 2014. Accepted September 30, 2014.
Contact Nguyen Quang Hoc, e-mail address: hocnq® hnue.edu. vn
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Nguyen Quang Hoc, Dinh Quang Vinh, Bui Due Tinh and Nguyen Due Hien at high pressure, this approximation is effective only at low pressure. Other researchers have concluded that it is impossible to find the melting temperature using only the limit of absolute stability for the solid phase because the obtained results on the basis of the self-consistent phonon method and the one-particle distribution function method are larger than the corresponding melting temperatures by a factor of 3 to 4 and 1.3 to 1.6, respectively [5].
On the basis of the statistical moment method (SMM) in statistical mechanics, some authors have determined the limiting lemperatiu-e of absolute stability for the crystalline state at various pressures and then they adjust this temperature in order to find the melting temperature [9, 10]. The melting temperature is obtained by this way for low as well as high pressures. The calculated results for the inert gas crystals agree rather well with the experimental data [2],
In the present study, we apply the SMM to investigate the equation of state and the melting temperature of solid N2, CO, CO2 and N2O. We will calculate the pressure dependence of the lattice constant and the melting temperature of these crystals.
2. Content
2.1. Equation of state, limiting temperature of absolute stability and melting temperature for molecular cryocrystals
The equation of state of a crystal with a face-centered cubic (fee) structure can be written in the following form [2]:
-. ^ o ^ ^ l „ = 2 J Ao (|a.!) ,lo = -^-g^^-^ = xcothi.O = kBT,x = ^ ,
- ^ E
g ^ l =mu,',l3 = x,y,z. (2.1) here P is the hydrostatic pressure, v is the volume of the fee lattice, a is the nearestneighbor distance of the fee crystal, di is the vector determining the equilibrium position of the rth particle, 0io is the interaction potential between the jth particle and the 0th particle, 7J is the Gruneisen constant and ks is the Boltzmann constant. From the Hmiting condition of absolute stability for the crystalline state
(f), = ° ' " 0 , = °- (2.2)
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we find the corresponding expression of the limiting temperature Tg as follows:
T,=
SfcBK'g)^-!?]'
(2.3)If we take the values of the parameters a, k, u at the same limiting temperature of absolute stabihty T^ then (2.3) can be transformed into die form [2]
T, 4fc2
In the case of P = 0, it gives 4e
"ifc
]_(8k\
' 2k \daj^
2Pv
a a V y 2fc \da)j.
(2.4)
(2.5) The nearest neighbor distance a is determined by a = ao + u^o, where a^ denotes the distance a at temperature 0 K and is determined from the experimental data. The displacement U^Q of a particle from the equilibrium position is calculated by
V 3 * 3 - * ' ^ - 1 2 ^ duf„ duhduf.^
6 r „st
5^7,^8,7 = I,t/,z,yl = a i + ^ f p - ' (2.6) where a,(i = 1 — 6) is determined in [2]. The equation for calculating the nearest neighbor distances at pressure P and at temperature 0 K has the form [2]
y .1.1948 + 0.17172/* - 0..0087 •/ + 0.0021 / , ! / = P^3 p^Z (2.7) We notice that the nearest neighbor distance a^ corresponding to the melting temperature Tm of the crystal is approximately equal to the nearest neighbor distance a^
corresponding to hmiting temperature Ts. In addition, from (2.1) we see that temperature 7 is a function of nearest neighbor distance a when pressure P is constant, i.e. T = T(a).
Therefore, we can expand temperature Tm according to the distance difference a^ - a^
and keep only the first approximate term [2]
!T, + - knYo
1 duo d^ua\
"^18a^+'''^j'
(2.8)69
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Due Tinh and Nguyen Due Hien where o„ = a ( T „ . F ) , a . = a { r . , F ) , « , = f a M ^ = ( ! ? ) „ „ , , % = ( & ' ) „ . „ ; the Gruneisen parameter 7J is regarded as invariable in the interval from T to T„ because it changes very litfle and 7J = - ( ^ | ^ x coth x)^^^_^^^_,_.
The equation of state of a crystal widi a hexagonal close-packed (hep) stiucture can be written in the following form [9):
p ^ = _ ^ ^ _ £ ^ + 127Se.
4 9a 2 da '°
'm.V^ dc) mkAda^dc X = x coth X, X, = x^cthx,,
, _ l ^ p \(&^
2» ' ^ ' ^ 2fl ' " 2 ^ \dul
X
i ^ ^ l v f ^ ) =mojl (2.9) here a and c are flie lattice constants of the hep crystal, v is flie volume and 7^ is flie
Gruneisen constant.
From the limiting condition of absolute stability of the crystalline state
we find the corresponding expression of the limiting temperature T^ as follows:
Pi; + L 3 V ,
^'= ^ ^ ^ ^ - V T I 4 \ - ^ - (2.11) 12ks[a{^)-^i]
If we take the values of die parameters o, c, k^, k^,^,^,... at the same limiting temperature of absolute stability T^ then (2.11) can be transformed into the form [9]
rV-ta . g^2 '^^ 3c ^^ \ k^ da^ ^ 2k, da^ ) '^^\k^~a^ ^ 2k;~dF)
^'^ fcBa[41f(at + 2 c ^ ) + ^ f t ( a ^ + 2ci^)] ' (2.12) In the case of P = 0, it gives
-^. (2.13)
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The nearest neighbor distance or the lattice constant a is determined by a = o-o + Uxo, where ao denotes the distance a at temperature 0 K and is determined from the experimental data. The displacement Ua;o of a particle from the equilibrium position in direction x or y is calculated from
-Ef^l"..^ \E
+ 6 du^du^^, ^ (2.14) where ai(? = 1 — 6) is determined in [9]. Lattice constant c is determined by c = Co + UzOi where Co denotes the distance c at temperature 0 K and is determined from the experimental data. Displacement U^Q of a particle from tiie equihbrium position in direction z is calculated from\t
1/2' " 12 ^ V dui
'•-4.T
dutdul- 4 E
dui^duiyduf. 9 V . 0 (2.15) here 6j(z = 1 - 6) is determined in [9]. The equation for calculating nearest neighbor distances at pressure P and at temperature 0 K has the form [9]y' = 0.9231 + 0.3188!/* Po-'
- 0.0015 1/= • 0 . 0 3 1 6 / + 0.0007 y'' Dfj3 PD-2 -O.OOOl^^y",
. = ( - ) (2.16) The melting temperature T^ of the hep crystal is approximately equal to [9]
'T.+- r2P«, 1 fdu„
6kB-y'a
where Cm = c(r„.P) ,c, = ciT„P), v.
6A:B7G
r2Pt;, 1
: V!5„2,
9uo d Ua 4 \dc',*°"lk?,
(2.17)
)1
aiCs and other quantities are as in (2.8).
2.2. N u m e r i c a l results a n d d i s c u s s i o n
For Q-CO2 and a - N 2 0 with a fee structure and for ^-N2 and /3-CO with a hep structure, the interaction potential between two atoms is usually used in the form of the Lennard-Jones pair potential
--Ae
^r
/f7\6 (2.18)where a is flie distance in which 0 ( r ) = 0 and e is the depfll of the potential well. The values of the parameters e and o are determined from the experimental data. ^ 218.82
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Due Tinh and Nguyen Due Hien K,cr = 3.829.10"^° m for a-COa, ~ = 235.48 K, CT = 3.802.10"^° m for a-NaO, -^- 95.05 K, f7 = 3.698.10-1'* m for ^-Na^and ^ = lOO.I K, a = 3.769.10-^° m for ^-CO [3].
Therefore, using two coordinate spheres and the results in [2, 9], we obtain the values of the crystal parameters for a-C02 and a- N2O as follows:
-i(Dl-'-cr--'H'-"(Di'
4410,797 -346,172 (2.19)where a is the nearest neighbor distance of the fee crystal at temperature T and the crystal parameters for /?-N2 and /?-C0 are as follows:
fc. = $ ( f ) " [614.6022(^)'' - 162.8535] , fc, = J ( f ) ° [286.3722(1)° - 64.7487] , 7 = - I f (f)" [l61.952(f)° - 24.984] . n = ^(if [6288.912(f)° - 473.6748] ,
|11488.3776('-y-752 5176 ,T3 = - J ( - ) ' ' 8133.888(-)"-737.352 (2.20) Our calculated results for the limiting temperature of absolute stability and the melting temperature of a-C02, a-N20, /3-N2 and /3-CO at different pressures (low pressures) are expressed in Figures 1-4.
-^(Dl^^
500-
400-
J 300- 1-
200'
100-
0-
'
--»-- T, (SMM)
» T „ (SMM) - • - T „ ( E X P T [ 7 1 ) !
SB5 1 •
,
- A
T - •
—'—;
-
p(Pa)
Figure 1. The limiting temperature of absolute stability and the melting temperature at different pressures for Q-CO2
The discrepancy m flie melting temperatiire of a-C02 that exists between our calculated results and the experimental data [7] is 5.3% at P = 0 and increases to 18%
at P = 1000 bar.
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210-
190-
tan-
L I
^ '
-SMM SMM
EXPTml ^
*
^^ .. —
«
.-• -
0 300 600 900
P (bar)
Figure 2. The limiting temperature of absolute stability and the melting temperature at different pressures for a-N^O
The discrepancy in the melting temperature of Q-N2O that exists between our calculated results and the experimental data [7] is 0.04% at P = 0 and increases to 7%
atP= 1000 bar.
- T3 (SMS) - T^ (SMS) - T , „ ( C A L [ 1 3 ] ) - T ^ ( E X P T [ 8 ] ) - T „ (EXPTPI)
- • •
p ( P a )
Figure 3. The limiting temperature of absolute stability and the melting temperature ai different pressures for 0-N2
73
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Due Tinh and Nguyen Due Hien The discrepancy in the melting temperature of 0-1^2 that exists between our calculated results and the experimental data [8] is 2.46% and between our calculated results and the experimental data [3] it is 2.69% at P = 0 and increases to 5% compared with the experimental data [8] at F = 100 bar. Our calculation is better tiian that in [10].
-T3(SMM) - T^ (SMM) - T [ ( E X P T [ 7 ] )
p(Pa)
Figure 4. The limiting temperature of absolute stability and the melting temperature at different pressures for ^-CO
The discrepancy in the melting temperature of ^-CO that exists between our calculated results and the experimental data [8] is 5.75% at P = 0 and increases to 7.68%
at P = 100 bar.
3. Conclusion
From the SMM and the limiting condition of absolute stability for the crystalline state, we find the equation of state, the hmiting temperature of absolute stability for the crystalhne state and the melting temperature for crystals with fee and hep sttuctures at zero pressure and under pressure, the equation for calculating the nearest neighbor distances at pressure P and at temperature 0 K for fee and hep crystals. These results are analytic and general.
Theoretical results are apphed to determine the melting temperature for molecular cryocrystals of nitrogen type (N2, CO, N2O, CO2) witii fee and hep structures in the interval of pressure from 0 to 100 bar for j0-N2, /3-CO and from 0 to 1000 bar for a-COg, a-N20. In general, our numerical calculations are in good agreement with the experimental data [3, 6-8] and other calculation [10], especially for a-N20, /^-No and j5-C0 molecular cryocrystals. Our obtained results can be enlarged to cases in higher pressures.
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