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Chemical ordering and thermodynamic properties of HgNa liquid alloys
Journal of Non-Crystalline Solids 357 (2011) 2892–2896
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Chemical ordering and thermodynamic properties of HgNa liquid alloys D. Adhikari a,⁎, B.P. Singh a, I.S. Jha b, B.K. Singh a a b
Univ. Dept. of Physics, T.M.Bhag. University, Bhagalpur, Bihar, India Dept. of Physics, M.M.A.M. Campus (Tribhuvan University), Biratnagar, Nepal
a r t i c l e
i n f o
Article history: Received 11 November 2010 Received in revised form 22 March 2011 Available online 14 April 2011 Keywords: Binary alloys; Entropy; Activity; Asymmetry
a b s t r a c t We have studied the thermodynamic properties and microscopic structures of HgNa liquid alloy at 673 K on the basis of regular associated solution model. The concentrations of ApB type complex in a regular associated solution of Hg and Na have been determined. We have then used the concentration of complex to calculate the free energy of mixing, enthalpy of mixing, entropy of mixing, activity, concentration fluctuations in long wavelength limit SCC(0) and the Warren–Cowley short-range parameter α1. The analysis suggests that heterocoordination leading to the formation of complex Hg4Na is likely to exist in the liquid and is of a strongly interacting nature. The theoretical analysis reveals that the pairwise interaction energies between the species depend considerably on temperature and the alloy is more ordered towards Hg-rich region. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The mixing properties of liquid alloys are important for understanding process metallurgy and material preparation. Structural and thermodynamic properties of the initial melt play an important role in the formation of alloy. Thus determination of thermodynamic and structural functions for alloys has been the subjects of active research for many years. Moreover, investigating the physicochemical properties of binary liquid alloys is of major significance in connection with their extensive application in various branches of science and novel engineering. HgNa alloy (sodium amalgam) has been used in organic chemistry as a powerful reducing agent which is safer to handle than sodium itself. An example of its use is in the Emde degradation. A sodium amalgam is also used in the design of the high pressure sodium lamp providing sodium to produce the proper color, and mercury to tailor the electrical characteristics of the lamp. In this paper, we will study the chemical ordering and thermodynamic properties of HgNa alloy in liquid state. The thermodynamic properties of liquid HgNa alloy in molten state exhibit anomalous behaviour as a function of concentration. The melting temperature against concentration curve (called liquidus lines) is S-shaped [1]. The alloy also exhibits large volume contractions [2] on alloying, but unlike free energy of mixing (ΔG) and heat of mixing (ΔH), the volume of mixing is symmetrical around equiatomic composition. This behaviour is very much in contrast to other similar strongly
⁎ Corresponding author. E-mail address: [email protected] (D. Adhikari). 0022-3093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.03.029
interacting systems such as NaPb or LiPb. Asymmetry in various properties of mixing of HgNa alloys in molten state is noticed around equiatomic composition. The size factor (ΩNa/ΩHg = 1.6, Ω being the atomic volume) and electronegativity difference (EHg – ENa = 1.07) are not large enough to account for the anomalous behaviour of mixing properties. These interesting aspects led the authors to the theoretical investigations of HgNa alloys. Theoreticians have worked with various compound formation models [3–6] to understand the alloying behaviour of compound forming binary alloys in the molten state. In present paper, we have used regular associated solution model to investigate the thermodynamic properties (free energy of mixing, heat of mixing, entropy of mixing and activity) and microscopic properties (concentration fluctuation in long wavelength limit, SCC(0) and Warren–Cowley short range parameter , α1). In the regular associated solution model, specific interactions such as hydrogen bonding, acid–base association, and charge transfer do not occur, but the intermolecular forces are no longer equal. That is, the energies associated with A–A and B–B interactions are not equal to the A–B interactions. We assume strong interactions between the constituents of the alloy are present in the molten state. Because of the existence of the strong interactions, these alloys, in the solid, form intermetallic compounds at one or more stoichiometric compositions. Large negative excess free energy of mixing, entropy of mixing and heat of mixing with sharp change in slope near the compound-forming concentrations are the characteristics [7,8] of the compound-forming alloys. In phase diagram of HgNa alloy, Hg4Na, Hg2Na, HgNa and Hg2Na3 intermetallic compounds are indicated [1]. Out of these phases, we have found that Hg and Na atoms are energetically favoured to form chemical compound Hg4Na in the molten state of HgNa alloy. Theoretical formalism is given in Sections 2 and 3 which deals with the numerical results and discussion. Conclusion is provided in Section 4.
D. Adhikari et al. / Journal of Non-Crystalline Solids 357 (2011) 2892–2896
2893
Now using the equations listed above the integral excess free energy ΔG is given by
2. Theory Consider one mole of binary solution of HgNa alloy comprising of x1 mole of A(=Hg) atoms and x2 moles of B(=Na) atoms. The presence of ApB (=Hg4Na) type complex in the solution results in a depletion of concentration of free atoms of the components of A and B. The liquid solution is thus composed of three species namely free atoms A and B and the complex ApB. As a result of associations, the thermodynamic behaviour of the components A and B is governed by the true mole fractions xA and xB rather than the gross mole fraction x1 and x2. Thus it is convenient to operate with two frames of references, one referring to gross mole fractions x1 and x2 and other referring to actual mole fractions of each species (xA, xB and xApB). Further, it is assumed that there are n1 moles of species A, n2 moles of species B and n3 moles of species ApB per mole of the binary solution. From the conservation of mass, the two frames of reference can be interrelated as follows: n1 = x1 pn3 ; n2 = x2 n3 and n = n1 þ n2 þ n3 = 1 pn3 n1 n1 n2 n2 xA = = ; xB = = n1 þ n2 þ n3 1 pn3 n1 þ n2 þ n3 1 pn3
ΔG =
1 RT ðx x ω þ xA xApB ω13 þ xB xApB ω23 Þ þ ð1 þ pxApB Þ A B 12 ð1 þ pxApB Þ × ðxA ln xA þ xB ln xB þ xApB ln xApB Þ þ
ð8Þ
Once the expressions for ΔG is obtained, other thermodynamic and microscopic functions follow readily. Enthalpy of mixing, entropy of mixing and concentration fluctuations in the long-wavelength limit are related to ΔG through standard thermodynamic relations ∂ΔG ΔH ¼ ΔG T ∂T ;P
ð9Þ
ΔH ΔG T
ð10Þ
ΔS ¼
2 1
2
ð1Þ
xApB RT ln k ð1 þ pxApB Þ
SCC ð0Þ ¼ RTð∂ ΔG = ∂C ÞT;P
ð11Þ
SCC ð0Þ = ð1 CÞa1 ð∂a1 = ∂CÞ1 T;P
ð12Þ
= Ca2 ð∂a2 = ∂ð1 CÞÞ1 T;P
and xApB
n3 n3 = = n1 þ n2 þ n3 1 pn3
ð2Þ
Here,
ΔH =
1 1 pn3 ¼ 1þ ¼ 1 þ pxApB = 1 pn3 n 1 pn3
ð3Þ
For the sake of convenience one or more of these frames of reference may be used. Now xA, xB and xApB can be inter-related with each other as follows by the relations xA ¼ x1 px2 xApB
ð4Þ
xB = x2 ð1 px2 ÞxApB :
ð5Þ
In regular associated solution, the gross chemical potentials of components 1 and 2 are equal to the chemical potentials of the monomeric species A and B [9]. The activity coefficients γA, γB and γApB of monomers and complex can be expressed in terms of pairwise interaction energies through [10] 2
2
RT ln γA = xB ω12 þ xApB ω13 þ xB xApB ðω12 ω23 þ ω13 Þ 2
2
RT ln γB = xApB ω23 þ xA ω12 þ xA xApB ðω23 ω13 þ ω12 Þ 2
2
RT ln γA pB = xA ω13 þ xB ω23 þ xB xApB ðω13 ω12 þ ω23 Þ
ð6aÞ ð6bÞ ð6cÞ
where ω12, ω13 and ω23 are interaction energies for the species A, B; A, ApB and B, ApB respectively, T the temperature and R stands for the universal gas constant. The equilibrium constant in a regular associated can be obtained [11] as ln k ¼ ln þ
where C (= xHg) is concentration of A component in the alloy. Eq. (8) is used in Eqs. (9) and (11), we obtained expressions for ΔH and SCC(0) as
! xPA xB ω ω þ 12 ½pxB ð1 xB Þ þ xA þ 13 ½pxApB ð1 xA Þ xA xApB RT RT
ω23 ½x ð1 pxB Þ xB RT ApB
ð7Þ
1 T ðx x ω þ xA xApB ω13 þ xB xApB ω23 Þ ð1 þ pxApB Þ A B 12 ð1 þ pxApB Þ ∂ω12 ∂ω13 ∂ω23 × xA xB þ xA xApB þ xB xApB ð13Þ ∂T ∂T ∂T
xApB 2 d ln k RT ð1 þ pxApB Þ dT
f
SCC ð0Þ ¼
½
1 2 = = = = = = ðx x ω þ xA xApB ω13 þ xB xApB ω23 Þ ð1 þ pxApB Þ RT A B 13 =2 ! −1 =2 =2 xApB x x þ A þ B þ xA xB xApB
g
Here;
ð14Þ
∂2 ΔG ∂ΔG N 0 for ¼ 0 ∂C ∂C2
where prime denotes the differentiations with respect to concentration and x/A and x/B are determined by using Eq. (5) [6]. x/ApB is d ln k determined using the Eq. (5) and the condition ¼ 0 [6]. It dC may be noted that the factor (1+ pxApB)-1 which appears as a coefficient of all terms containing xA, xB and xApB in the above Equations, is a result of the change in the basis for expressing mole fractions of species A, B and ApB from that used for x1 and x2. Experimental determination of SCC(0) possesses more difficulty. It can be determined from measured activity data following Eq. (12) [12]. This is usually considered as the experimental value. In order to fit the degree of order in the liquid alloy, Warren– Cowley short-range parameter α1 [13,14] can be estimated from the knowledge of concentration–concentration structure factor SCC(q) and the number–number structure factor SNN(q). However, in most diffraction experiments these quantities are not easily measurable for all kinds of binary liquid alloy [15,16]. On the other hand α1 can be estimated from the knowledge of Scc(0) [17,18] α1 =
S1 S ð0Þ ; ; S ¼ CC SðZ 1Þ þ 1 Sid CC ð0Þ
id
SCC ¼ CðC 1Þ
ð15Þ
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where Z is coordination number and Z = 10 is taken for our calculation. We note that varying the value of Z does not have any effect on the position of the minima of α1; the effect is to vary the depth while the overall feature remains unchanged. The pairwise interaction energies and equilibrium constant are determined by the following method: In a regular associated solution x1γ1 = xAγA and x2γ2 = xBγB, where γ1 and γ2 are respective gross activity coefficients of components 1 and 2. Thus ln γ1 ¼ ln γA þ ln
xA x1
ð16aÞ
xB x2
ð16bÞ
and ln γ2 ¼ ln γB þ ln
the pairwise interaction energies, the equilibrium constants and the activity coefficients at infinite dilution can be written as [11] 0
ln γ1 =
ω12 RT
ð17aÞ
k expðω13 =RTÞ
o o γ γ ¼ o1 2o γ1 γ2
ð17bÞ
where γo1 and γo2 are activity coefficients of component A and that of B at zero concentrations. Solving Eqs. (6a) and (6b) we obtain ω13 = RT
xB ln
ω23 = RT
xA ln
a2 a ω þð1 xB Þ ln 1 xB ð1 xB Þ 12 xB xA RT x2ApB
ð18Þ
a1 a ω þð1 xA Þ ln 2 xA ð1 xA Þ 12 xA xB RT x2ApB
ð19Þ
Using Eqs. (7) and (17) we can derive ln kþ
ω13 = RT
! ! ap a 1 þ xA a x a ω ln 1 þ B ln 2 12 þ ln 1 2 ð20Þ xApB xA xApB xB RT xApB
3. Results To calculate the free energy of mixing, we require mole fraction of various species, pairwise interaction energies between the species formed in the molten state and equilibrium constant. The mole fraction xHg4Na of complex Hg4Na is determined using experimental data of activity [1] and Eqs. (17) and (20) employing the iterative procedure. The compositional dependence of various species (Fig. 1) shows that the maximum association occurs at 80 at. pct. of Hg. At this composition and 673 K, about 33 mol pct. of the liquid alloy is associated. The equilibrium constant and pairwise interaction energies are determined from the Eqs. (7), (17a), (18), (19) and the computed values are slightly adjusted using observed data of integral excess free energy of mixing [1]. The values of pairwise interaction energies and the equilibrium constant are slightly adjusted using the expression for free energy of mixing as a function of concentration and the experimental values of ΔG. The best fit values of equilibrium constant and interaction energies for the alloy Hg4Na in liquid state at 673 K are found to be k = 9:5 × 10
Fig. 1. Upper part: compositional dependence of mole fractions xA (A = Hg), xB (B = Na) and xApB (ApB = Hg4Na) versus xHg (concentration of Hg); lower part: Integral excess free energy of mixing (ΔGxs/RT) versus xHg in the liquid HgNa solution at 673K; (―) theory(○○○) experiment [1].
−4
; ω12 = −34750 J mol
and ω23 = −120240 J mol
−1
mixing is −13,982 Jmol− 1 at xHg = 0.6. Fig. 1 shows an excellent agreement between the experimental and calculated integral excess free energies. The uncertainty in the experimental data in integral excess free energy of mixing is ±627 Jmol− 1 at xHg = 0.5[1]. The remaining experimental values of free energy of mixing were calculated from the selected values using Gibbs–Duhem relation [1]. Activity is a very important thermodynamic function because it is one of the fortune functions which is obtained directly from experiment and it can be used to obtain other thermodynamic functions. The graph between the experimental and theoretical values of ln a with respect to xHg is shown in Fig. 2. It is clear from the graph that theoretical values of ln a are in a good agreement with experimental values. We have observed that if the interaction energies are supposed to ∂ω12 = 0 etc., then ΔH and ΔS be independent of temperature, i.e., ∂T so obtained are in very poor agreement with experimental data. This simply suggests importance of the dependence of interaction energies on temperature. We have thus assumed that the pairwise interaction energies ωij are temperature dependent. On using Eq. (9) and observed values of ΔH [1], the best fit values of heat of dis∂ln k and other temperature dependent parameters sociation R T2 ∂T are found to be ∂ω12 ∂ω13 −1 −1 ∂ω23 −1 −1 = 0; = −5:5 Jmol K ; = 20 Jmol K ∂T ∂T ∂T ∂ ln k 1 and R T2 ¼ 75000 1400 J mol ∂T
−1
; ω13 = −14780 J mol
−1
Theoretical calculation of excess free energy of mixing for HgNa liquid alloy shows that the minimum value of excess free energy of
Fig. 2. Activity (lna) of Hg and Na in liquid HgNa solution (673 K) versus xHg; (―) theory, (○○○) experiment [1].
D. Adhikari et al. / Journal of Non-Crystalline Solids 357 (2011) 2892–2896
∂ln k lies within the uncertainty of ∂T ±1400 Jmol . The dependence of energy parameters on temperature can be observed from the study of ΔH and ΔS. It is found from the analysis that the heat of mixing is negative at all concentration. Our theoretical values of ΔH agree well with the experimental values obtained from direct reaction calorimetry methods [1]. Our calculation shows that the minimum value of the heat of mixing is − 20,749 Jmol− 1 at xHg = 0.6 which almost matches with the experimental value [1]. The uncertainty in the experimental value of ΔH is ±104 Jmol− 1 at xHg = 0.5 [1]. The remaining experimental values of heat of mixing were calculated from the selected values using Gibbs–Duhem relation [1]. Further it is observed that the concentration dependence of asymmetry in ΔH can be explained only when one considers the temperature dependence of the pairwise interaction energies. The agreement between the calculated and experimental values is also good. The free energy of mixing is found to be −18,231 Jmol− 1 at xHg = 0.6 which almost matches with the experimental values. The uncertainty in the experimental value of free energy of mixing is ±627 Jmol− 1 xHg = 0.5 [1]. The calculated values of free energy of mixing and heat of mixing are used to calculate the entropy of mixing. The entropy of mixing is found to be S-shaped which is in agreement with its experimental behaviour. The asymmetries in ΔG, ΔH and ΔS are well explained. Neale et al. [19] and Ishiguro et al. [20] also observed such asymmetry in liquid HgNa alloy when they carried out electromotive measurements (Fig. 3). Fig. 4 shows the computed and experimental values of Scc(0) as well as ideal values. The calculated values for Scc(0) shows a very good agreement with the experimental values. We have found that the calculated value of Scc(0) is less than the ideal value of Scc(0) at all concentrations except xHg = 0.1. At xHg = 0.1, the Scc(0) is slightly greater than ideal value. Plot of chemical short range order (α1) with respect to concentration of Hg (=xHg) is displayed in Fig. 4 which shows that α1is negative throughout the concentration range except at xHg = 0.1 ng.
2895
The best fit value of R T2 −1
4. Discussion All the interaction energies are negative and show that unassociated-Hg and unassociated-Na atoms are attracted to each other and to the complex Hg4Na. Theoretical calculation of excess free energy of mixing for HgNa liquid alloy shows that HgNa alloy in liquid state is strongly interacting system. The deviations from ideal behaviour of the alloys can be incorporated into activity. The measurement of activities within a class of similar system can be expected to provide, at least, a basis for correlation of the behaviour, which can then be used for extrapolation of the behaviour of more complex system. In general, heat of dissociation is more sensitive to errors. Any observed fit between calculated and observed activities for a given set
Fig. 4. Upper part: concentration fluctuations in long wavelength limit (Scc(0)) versus xHg of liquid HgNa solution (673 K); lower part: short range ordering parameter(α1) of liquid HgNa solution (673 K) versus xHg ;(―) theory, (○○○) experiment , (−−−−) ideal values.
of k and the interaction energies cannot be taken as sufficient until the observed heats of formation are also explained by the same set of parameters. It is clear from the result that ω12 through which the interaction between left-over Hg and Na atoms is expressed is independent of temperature or negligibly temperature dependent. The other two parameters ω13 and ω23 through which the respective interaction between left-over Hg atoms and complex Hg4Na and leftover Na atoms and complex Hg4Na are expressed are found to be considerably dependent on temperature. This means that the interaction between unassociated Hg atoms and the Na atoms is almost independent of temperature while the interaction between unassociated Hg atoms and complex will become more attractive with increase in temperature. Similarly the interaction between unassociated Na atoms and the complex will become more repulsive with increasing in temperature. The Scc(0) can be used to understand the nature of atomic order in the binary liquid alloys [21,22]. At a given composition, if Scc(0) b id Sid CC(0), ordering in liquid alloy is expected and if Scc(0) N SCC(0), there is tendency of segregation. Our theoretical analysis shows that the order exists for HgNa alloy in the liquid state at whole concentration range. At equiatomic composition, α1 lies in between − 1 and + 1. It is well known that α1 = − 1 implies complete ordering of unlike atoms paring at nearest neighbours α1 = + 1 implies total segregation leading to the phase separation and α1 = 0 corresponds to a random distribution of atoms. Our analysis shows that heterocoordination leading to the formation of complex Hg4Na is likely to exist in the liquid HgNa alloy. 5. Conclusion In the present work, we have used regular associated solution model to obtain equilibrium constant, pairwise interaction energies for Hg4Na liquid alloy at 673 K. The knowledge of nature and extent of interaction energies between the unassociated components and complex have been used for the estimation of thermodynamic properties of Hg4Na alloy in liquid state. Computed results suggest that there is a tendency of unlike atoms pairing (Hg–Na) at all concentrations but this tendency is greater in Hg-rich concentration. The analysis also shows that the HgNa system is strongly interacting heterocoordination system. The pairwise interaction energies of this system depend strongly on temperature. Acknowledgement
Fig. 3. Upper part: entropy of mixing (ΔS/R) versus xHg, lower part: heat of mixing (ΔH/ RT) versus xHg of liquid HgNa solution (673 K); (―) theory, (○○○) experiment [1].
D. Adhikari is thankful to the University Grant Commission (UGC), Nepal, for providing financial support to pursue the research.
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References [1] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metal Park, 1973. [2] T. Ishiguro, S. Takeda, S. Tamaka, J. Phys. F. 12 (1982) 845. [3] A.B. Bhatia, W.H. Hargrove, Phys. Rev. B10 (1974) 3186. [4] K. Hoshino, W.H. Young, J. Phys. F10 (1980) 1365. [5] R.N. Singh, F. Sommer, Z. Metall. 83 (1992) 553. [6] B.P. Singh, D. Adhikari, I.S. Jha, J. NonCrys. Solids 356 (2010) 1730. [7] N.H. March, Liquid Metals, Concept and Theory, Cambridge University Press, Cambridge, 1990. [8] F. Sommer, J. NonCryst. Solids 117 (1990) 505. [9] Prigogine, R. Defay, Chem. Thermodynamics, Longmans Green and Co, London, 1974, p. 257.
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
A.S. Jordan, Metall. Trans. 1 (1970) 239. S. Lele, P. Ramchandra Rao, Metall. Trans. 12 B (1981) 659. T.S. Yih, J.C. Thompson, J. Phys. F. Met. Phys. 12 (1982) 1625. B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading MA, 1969 p. 227. J.M. Cowley, Phys. Rev. 77 (1950) 667. S. Steeb, F. Falch, P. Lamparter, Z. Metallkd. 75 (1984) 599. C.N.J. Wagner, Rapidly Quenched Metals, in: S. Steeb, H. Warlemont (Eds.), NorthHolland, Amsterdam, 1985, p. 4095. R.N. Singh, Can. J. Phys. 65 (1987) 309. K. Hoshino, W.H. Young, J. Phys. F10 (1980) 1365. F.E. Neale, N.E. Cusack, E.D. Crozier, J. Phys. F 16 (1981) 3509. T. Ishiguro, S. Takeda, S. Tamaki, J. Phys. F. 12 (1982) 845. R.N. Singh, I.K. Mishra, V.N. Slngh, J. Phys. Condens. Matter 2 (1990) 8445. L.C. Prasad, R.N. Singh, Phys. Chem. Liquid 22 (1990) 1.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Chemical ordering and thermodynamic properties of HgNa liquid alloys D. Adhikari a,⁎, B.P. Singh a, I.S. Jha b, B.K. Singh a a b
Univ. Dept. of Physics, T.M.Bhag. University, Bhagalpur, Bihar, India Dept. of Physics, M.M.A.M. Campus (Tribhuvan University), Biratnagar, Nepal
a r t i c l e
i n f o
Article history: Received 11 November 2010 Received in revised form 22 March 2011 Available online 14 April 2011 Keywords: Binary alloys; Entropy; Activity; Asymmetry
a b s t r a c t We have studied the thermodynamic properties and microscopic structures of HgNa liquid alloy at 673 K on the basis of regular associated solution model. The concentrations of ApB type complex in a regular associated solution of Hg and Na have been determined. We have then used the concentration of complex to calculate the free energy of mixing, enthalpy of mixing, entropy of mixing, activity, concentration fluctuations in long wavelength limit SCC(0) and the Warren–Cowley short-range parameter α1. The analysis suggests that heterocoordination leading to the formation of complex Hg4Na is likely to exist in the liquid and is of a strongly interacting nature. The theoretical analysis reveals that the pairwise interaction energies between the species depend considerably on temperature and the alloy is more ordered towards Hg-rich region. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The mixing properties of liquid alloys are important for understanding process metallurgy and material preparation. Structural and thermodynamic properties of the initial melt play an important role in the formation of alloy. Thus determination of thermodynamic and structural functions for alloys has been the subjects of active research for many years. Moreover, investigating the physicochemical properties of binary liquid alloys is of major significance in connection with their extensive application in various branches of science and novel engineering. HgNa alloy (sodium amalgam) has been used in organic chemistry as a powerful reducing agent which is safer to handle than sodium itself. An example of its use is in the Emde degradation. A sodium amalgam is also used in the design of the high pressure sodium lamp providing sodium to produce the proper color, and mercury to tailor the electrical characteristics of the lamp. In this paper, we will study the chemical ordering and thermodynamic properties of HgNa alloy in liquid state. The thermodynamic properties of liquid HgNa alloy in molten state exhibit anomalous behaviour as a function of concentration. The melting temperature against concentration curve (called liquidus lines) is S-shaped [1]. The alloy also exhibits large volume contractions [2] on alloying, but unlike free energy of mixing (ΔG) and heat of mixing (ΔH), the volume of mixing is symmetrical around equiatomic composition. This behaviour is very much in contrast to other similar strongly
⁎ Corresponding author. E-mail address: [email protected] (D. Adhikari). 0022-3093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.03.029
interacting systems such as NaPb or LiPb. Asymmetry in various properties of mixing of HgNa alloys in molten state is noticed around equiatomic composition. The size factor (ΩNa/ΩHg = 1.6, Ω being the atomic volume) and electronegativity difference (EHg – ENa = 1.07) are not large enough to account for the anomalous behaviour of mixing properties. These interesting aspects led the authors to the theoretical investigations of HgNa alloys. Theoreticians have worked with various compound formation models [3–6] to understand the alloying behaviour of compound forming binary alloys in the molten state. In present paper, we have used regular associated solution model to investigate the thermodynamic properties (free energy of mixing, heat of mixing, entropy of mixing and activity) and microscopic properties (concentration fluctuation in long wavelength limit, SCC(0) and Warren–Cowley short range parameter , α1). In the regular associated solution model, specific interactions such as hydrogen bonding, acid–base association, and charge transfer do not occur, but the intermolecular forces are no longer equal. That is, the energies associated with A–A and B–B interactions are not equal to the A–B interactions. We assume strong interactions between the constituents of the alloy are present in the molten state. Because of the existence of the strong interactions, these alloys, in the solid, form intermetallic compounds at one or more stoichiometric compositions. Large negative excess free energy of mixing, entropy of mixing and heat of mixing with sharp change in slope near the compound-forming concentrations are the characteristics [7,8] of the compound-forming alloys. In phase diagram of HgNa alloy, Hg4Na, Hg2Na, HgNa and Hg2Na3 intermetallic compounds are indicated [1]. Out of these phases, we have found that Hg and Na atoms are energetically favoured to form chemical compound Hg4Na in the molten state of HgNa alloy. Theoretical formalism is given in Sections 2 and 3 which deals with the numerical results and discussion. Conclusion is provided in Section 4.
D. Adhikari et al. / Journal of Non-Crystalline Solids 357 (2011) 2892–2896
2893
Now using the equations listed above the integral excess free energy ΔG is given by
2. Theory Consider one mole of binary solution of HgNa alloy comprising of x1 mole of A(=Hg) atoms and x2 moles of B(=Na) atoms. The presence of ApB (=Hg4Na) type complex in the solution results in a depletion of concentration of free atoms of the components of A and B. The liquid solution is thus composed of three species namely free atoms A and B and the complex ApB. As a result of associations, the thermodynamic behaviour of the components A and B is governed by the true mole fractions xA and xB rather than the gross mole fraction x1 and x2. Thus it is convenient to operate with two frames of references, one referring to gross mole fractions x1 and x2 and other referring to actual mole fractions of each species (xA, xB and xApB). Further, it is assumed that there are n1 moles of species A, n2 moles of species B and n3 moles of species ApB per mole of the binary solution. From the conservation of mass, the two frames of reference can be interrelated as follows: n1 = x1 pn3 ; n2 = x2 n3 and n = n1 þ n2 þ n3 = 1 pn3 n1 n1 n2 n2 xA = = ; xB = = n1 þ n2 þ n3 1 pn3 n1 þ n2 þ n3 1 pn3
ΔG =
1 RT ðx x ω þ xA xApB ω13 þ xB xApB ω23 Þ þ ð1 þ pxApB Þ A B 12 ð1 þ pxApB Þ × ðxA ln xA þ xB ln xB þ xApB ln xApB Þ þ
ð8Þ
Once the expressions for ΔG is obtained, other thermodynamic and microscopic functions follow readily. Enthalpy of mixing, entropy of mixing and concentration fluctuations in the long-wavelength limit are related to ΔG through standard thermodynamic relations ∂ΔG ΔH ¼ ΔG T ∂T ;P
ð9Þ
ΔH ΔG T
ð10Þ
ΔS ¼
2 1
2
ð1Þ
xApB RT ln k ð1 þ pxApB Þ
SCC ð0Þ ¼ RTð∂ ΔG = ∂C ÞT;P
ð11Þ
SCC ð0Þ = ð1 CÞa1 ð∂a1 = ∂CÞ1 T;P
ð12Þ
= Ca2 ð∂a2 = ∂ð1 CÞÞ1 T;P
and xApB
n3 n3 = = n1 þ n2 þ n3 1 pn3
ð2Þ
Here,
ΔH =
1 1 pn3 ¼ 1þ ¼ 1 þ pxApB = 1 pn3 n 1 pn3
ð3Þ
For the sake of convenience one or more of these frames of reference may be used. Now xA, xB and xApB can be inter-related with each other as follows by the relations xA ¼ x1 px2 xApB
ð4Þ
xB = x2 ð1 px2 ÞxApB :
ð5Þ
In regular associated solution, the gross chemical potentials of components 1 and 2 are equal to the chemical potentials of the monomeric species A and B [9]. The activity coefficients γA, γB and γApB of monomers and complex can be expressed in terms of pairwise interaction energies through [10] 2
2
RT ln γA = xB ω12 þ xApB ω13 þ xB xApB ðω12 ω23 þ ω13 Þ 2
2
RT ln γB = xApB ω23 þ xA ω12 þ xA xApB ðω23 ω13 þ ω12 Þ 2
2
RT ln γA pB = xA ω13 þ xB ω23 þ xB xApB ðω13 ω12 þ ω23 Þ
ð6aÞ ð6bÞ ð6cÞ
where ω12, ω13 and ω23 are interaction energies for the species A, B; A, ApB and B, ApB respectively, T the temperature and R stands for the universal gas constant. The equilibrium constant in a regular associated can be obtained [11] as ln k ¼ ln þ
where C (= xHg) is concentration of A component in the alloy. Eq. (8) is used in Eqs. (9) and (11), we obtained expressions for ΔH and SCC(0) as
! xPA xB ω ω þ 12 ½pxB ð1 xB Þ þ xA þ 13 ½pxApB ð1 xA Þ xA xApB RT RT
ω23 ½x ð1 pxB Þ xB RT ApB
ð7Þ
1 T ðx x ω þ xA xApB ω13 þ xB xApB ω23 Þ ð1 þ pxApB Þ A B 12 ð1 þ pxApB Þ ∂ω12 ∂ω13 ∂ω23 × xA xB þ xA xApB þ xB xApB ð13Þ ∂T ∂T ∂T
xApB 2 d ln k RT ð1 þ pxApB Þ dT
f
SCC ð0Þ ¼
½
1 2 = = = = = = ðx x ω þ xA xApB ω13 þ xB xApB ω23 Þ ð1 þ pxApB Þ RT A B 13 =2 ! −1 =2 =2 xApB x x þ A þ B þ xA xB xApB
g
Here;
ð14Þ
∂2 ΔG ∂ΔG N 0 for ¼ 0 ∂C ∂C2
where prime denotes the differentiations with respect to concentration and x/A and x/B are determined by using Eq. (5) [6]. x/ApB is d ln k determined using the Eq. (5) and the condition ¼ 0 [6]. It dC may be noted that the factor (1+ pxApB)-1 which appears as a coefficient of all terms containing xA, xB and xApB in the above Equations, is a result of the change in the basis for expressing mole fractions of species A, B and ApB from that used for x1 and x2. Experimental determination of SCC(0) possesses more difficulty. It can be determined from measured activity data following Eq. (12) [12]. This is usually considered as the experimental value. In order to fit the degree of order in the liquid alloy, Warren– Cowley short-range parameter α1 [13,14] can be estimated from the knowledge of concentration–concentration structure factor SCC(q) and the number–number structure factor SNN(q). However, in most diffraction experiments these quantities are not easily measurable for all kinds of binary liquid alloy [15,16]. On the other hand α1 can be estimated from the knowledge of Scc(0) [17,18] α1 =
S1 S ð0Þ ; ; S ¼ CC SðZ 1Þ þ 1 Sid CC ð0Þ
id
SCC ¼ CðC 1Þ
ð15Þ
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where Z is coordination number and Z = 10 is taken for our calculation. We note that varying the value of Z does not have any effect on the position of the minima of α1; the effect is to vary the depth while the overall feature remains unchanged. The pairwise interaction energies and equilibrium constant are determined by the following method: In a regular associated solution x1γ1 = xAγA and x2γ2 = xBγB, where γ1 and γ2 are respective gross activity coefficients of components 1 and 2. Thus ln γ1 ¼ ln γA þ ln
xA x1
ð16aÞ
xB x2
ð16bÞ
and ln γ2 ¼ ln γB þ ln
the pairwise interaction energies, the equilibrium constants and the activity coefficients at infinite dilution can be written as [11] 0
ln γ1 =
ω12 RT
ð17aÞ
k expðω13 =RTÞ
o o γ γ ¼ o1 2o γ1 γ2
ð17bÞ
where γo1 and γo2 are activity coefficients of component A and that of B at zero concentrations. Solving Eqs. (6a) and (6b) we obtain ω13 = RT
xB ln
ω23 = RT
xA ln
a2 a ω þð1 xB Þ ln 1 xB ð1 xB Þ 12 xB xA RT x2ApB
ð18Þ
a1 a ω þð1 xA Þ ln 2 xA ð1 xA Þ 12 xA xB RT x2ApB
ð19Þ
Using Eqs. (7) and (17) we can derive ln kþ
ω13 = RT
! ! ap a 1 þ xA a x a ω ln 1 þ B ln 2 12 þ ln 1 2 ð20Þ xApB xA xApB xB RT xApB
3. Results To calculate the free energy of mixing, we require mole fraction of various species, pairwise interaction energies between the species formed in the molten state and equilibrium constant. The mole fraction xHg4Na of complex Hg4Na is determined using experimental data of activity [1] and Eqs. (17) and (20) employing the iterative procedure. The compositional dependence of various species (Fig. 1) shows that the maximum association occurs at 80 at. pct. of Hg. At this composition and 673 K, about 33 mol pct. of the liquid alloy is associated. The equilibrium constant and pairwise interaction energies are determined from the Eqs. (7), (17a), (18), (19) and the computed values are slightly adjusted using observed data of integral excess free energy of mixing [1]. The values of pairwise interaction energies and the equilibrium constant are slightly adjusted using the expression for free energy of mixing as a function of concentration and the experimental values of ΔG. The best fit values of equilibrium constant and interaction energies for the alloy Hg4Na in liquid state at 673 K are found to be k = 9:5 × 10
Fig. 1. Upper part: compositional dependence of mole fractions xA (A = Hg), xB (B = Na) and xApB (ApB = Hg4Na) versus xHg (concentration of Hg); lower part: Integral excess free energy of mixing (ΔGxs/RT) versus xHg in the liquid HgNa solution at 673K; (―) theory(○○○) experiment [1].
−4
; ω12 = −34750 J mol
and ω23 = −120240 J mol
−1
mixing is −13,982 Jmol− 1 at xHg = 0.6. Fig. 1 shows an excellent agreement between the experimental and calculated integral excess free energies. The uncertainty in the experimental data in integral excess free energy of mixing is ±627 Jmol− 1 at xHg = 0.5[1]. The remaining experimental values of free energy of mixing were calculated from the selected values using Gibbs–Duhem relation [1]. Activity is a very important thermodynamic function because it is one of the fortune functions which is obtained directly from experiment and it can be used to obtain other thermodynamic functions. The graph between the experimental and theoretical values of ln a with respect to xHg is shown in Fig. 2. It is clear from the graph that theoretical values of ln a are in a good agreement with experimental values. We have observed that if the interaction energies are supposed to ∂ω12 = 0 etc., then ΔH and ΔS be independent of temperature, i.e., ∂T so obtained are in very poor agreement with experimental data. This simply suggests importance of the dependence of interaction energies on temperature. We have thus assumed that the pairwise interaction energies ωij are temperature dependent. On using Eq. (9) and observed values of ΔH [1], the best fit values of heat of dis∂ln k and other temperature dependent parameters sociation R T2 ∂T are found to be ∂ω12 ∂ω13 −1 −1 ∂ω23 −1 −1 = 0; = −5:5 Jmol K ; = 20 Jmol K ∂T ∂T ∂T ∂ ln k 1 and R T2 ¼ 75000 1400 J mol ∂T
−1
; ω13 = −14780 J mol
−1
Theoretical calculation of excess free energy of mixing for HgNa liquid alloy shows that the minimum value of excess free energy of
Fig. 2. Activity (lna) of Hg and Na in liquid HgNa solution (673 K) versus xHg; (―) theory, (○○○) experiment [1].
D. Adhikari et al. / Journal of Non-Crystalline Solids 357 (2011) 2892–2896
∂ln k lies within the uncertainty of ∂T ±1400 Jmol . The dependence of energy parameters on temperature can be observed from the study of ΔH and ΔS. It is found from the analysis that the heat of mixing is negative at all concentration. Our theoretical values of ΔH agree well with the experimental values obtained from direct reaction calorimetry methods [1]. Our calculation shows that the minimum value of the heat of mixing is − 20,749 Jmol− 1 at xHg = 0.6 which almost matches with the experimental value [1]. The uncertainty in the experimental value of ΔH is ±104 Jmol− 1 at xHg = 0.5 [1]. The remaining experimental values of heat of mixing were calculated from the selected values using Gibbs–Duhem relation [1]. Further it is observed that the concentration dependence of asymmetry in ΔH can be explained only when one considers the temperature dependence of the pairwise interaction energies. The agreement between the calculated and experimental values is also good. The free energy of mixing is found to be −18,231 Jmol− 1 at xHg = 0.6 which almost matches with the experimental values. The uncertainty in the experimental value of free energy of mixing is ±627 Jmol− 1 xHg = 0.5 [1]. The calculated values of free energy of mixing and heat of mixing are used to calculate the entropy of mixing. The entropy of mixing is found to be S-shaped which is in agreement with its experimental behaviour. The asymmetries in ΔG, ΔH and ΔS are well explained. Neale et al. [19] and Ishiguro et al. [20] also observed such asymmetry in liquid HgNa alloy when they carried out electromotive measurements (Fig. 3). Fig. 4 shows the computed and experimental values of Scc(0) as well as ideal values. The calculated values for Scc(0) shows a very good agreement with the experimental values. We have found that the calculated value of Scc(0) is less than the ideal value of Scc(0) at all concentrations except xHg = 0.1. At xHg = 0.1, the Scc(0) is slightly greater than ideal value. Plot of chemical short range order (α1) with respect to concentration of Hg (=xHg) is displayed in Fig. 4 which shows that α1is negative throughout the concentration range except at xHg = 0.1 ng.
2895
The best fit value of R T2 −1
4. Discussion All the interaction energies are negative and show that unassociated-Hg and unassociated-Na atoms are attracted to each other and to the complex Hg4Na. Theoretical calculation of excess free energy of mixing for HgNa liquid alloy shows that HgNa alloy in liquid state is strongly interacting system. The deviations from ideal behaviour of the alloys can be incorporated into activity. The measurement of activities within a class of similar system can be expected to provide, at least, a basis for correlation of the behaviour, which can then be used for extrapolation of the behaviour of more complex system. In general, heat of dissociation is more sensitive to errors. Any observed fit between calculated and observed activities for a given set
Fig. 4. Upper part: concentration fluctuations in long wavelength limit (Scc(0)) versus xHg of liquid HgNa solution (673 K); lower part: short range ordering parameter(α1) of liquid HgNa solution (673 K) versus xHg ;(―) theory, (○○○) experiment , (−−−−) ideal values.
of k and the interaction energies cannot be taken as sufficient until the observed heats of formation are also explained by the same set of parameters. It is clear from the result that ω12 through which the interaction between left-over Hg and Na atoms is expressed is independent of temperature or negligibly temperature dependent. The other two parameters ω13 and ω23 through which the respective interaction between left-over Hg atoms and complex Hg4Na and leftover Na atoms and complex Hg4Na are expressed are found to be considerably dependent on temperature. This means that the interaction between unassociated Hg atoms and the Na atoms is almost independent of temperature while the interaction between unassociated Hg atoms and complex will become more attractive with increase in temperature. Similarly the interaction between unassociated Na atoms and the complex will become more repulsive with increasing in temperature. The Scc(0) can be used to understand the nature of atomic order in the binary liquid alloys [21,22]. At a given composition, if Scc(0) b id Sid CC(0), ordering in liquid alloy is expected and if Scc(0) N SCC(0), there is tendency of segregation. Our theoretical analysis shows that the order exists for HgNa alloy in the liquid state at whole concentration range. At equiatomic composition, α1 lies in between − 1 and + 1. It is well known that α1 = − 1 implies complete ordering of unlike atoms paring at nearest neighbours α1 = + 1 implies total segregation leading to the phase separation and α1 = 0 corresponds to a random distribution of atoms. Our analysis shows that heterocoordination leading to the formation of complex Hg4Na is likely to exist in the liquid HgNa alloy. 5. Conclusion In the present work, we have used regular associated solution model to obtain equilibrium constant, pairwise interaction energies for Hg4Na liquid alloy at 673 K. The knowledge of nature and extent of interaction energies between the unassociated components and complex have been used for the estimation of thermodynamic properties of Hg4Na alloy in liquid state. Computed results suggest that there is a tendency of unlike atoms pairing (Hg–Na) at all concentrations but this tendency is greater in Hg-rich concentration. The analysis also shows that the HgNa system is strongly interacting heterocoordination system. The pairwise interaction energies of this system depend strongly on temperature. Acknowledgement
Fig. 3. Upper part: entropy of mixing (ΔS/R) versus xHg, lower part: heat of mixing (ΔH/ RT) versus xHg of liquid HgNa solution (673 K); (―) theory, (○○○) experiment [1].
D. Adhikari is thankful to the University Grant Commission (UGC), Nepal, for providing financial support to pursue the research.
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