B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.16 (Online Publication: Dec., 2016) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal Temperature dependence of entropy of mixing of liquid alkali alloys B. K. Singh1*, Sudhir Singh2, Golak Kumar Mandal1, Dhiraj Kumar Jha1 1 University Department of Physics, T.M. Bhagalpur University, Bhagalpur 2 Department of Physics, R. D. College Sheikhpura, T.M. Bhag. Univ., Bhagalpur *E-mail : bijay.k.singh@gmail.com Article history: Received 21 May, 2016; Accepted 3 August, 2016 DOI: http://dx.doi.org/10.3126/bibechana.v14i0.15410 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ Abstract A semi-empirical approach has been considered to study the temperature dependence of entropy of mixing, (ΔsM), for various alkai-alkali alloys using hard-sphere model. The most important physical parameters occurring here is the hard-sphere diameter (σ) and the packing fraction (η). For pure liquid metals, this is usually determined empirically from the observed entropy as a function of temperature which in turn are utilised to compute ΔSM for Na-K, K-Rb, Na-Rb, NaCs, Rb-Cs and K-Cs alloys as a function of concentration at five different temperature ranging from 400K-800K. The study reveals that entropy of mixing for Na-K, Na-Rb and K-Rb systems decreases with increasing temperature. But the result for Cs-based alloys exhibit a mixed behaviour. Keywords: Alkali alloys; Semi-empirical approach; Entropy; Helmholtz free energy, Boltzmann Constant; Packing fraction. 1. Introduction Though enormous experimental data exist on entropy of pure liquid metals and alloys [1], the theoretical works lag behind. Recently the hard-sphere (HS) model has been widely used to remedy this lack. The theoretical study of the entropy of mixing is of current interest to unlock the secrets of the structure of liquid alloys on the basis of experimental observations. The most important and basic ingredients are the hard sphere diameter (σ) and the packing fraction (η). Several workers [2-5] have computed the HS parameters by minimising the free energy of the system through first principle estimation. In the present work, a semi-empirical technique, without undergoing a rigorous first principle calculation attempt has B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.17 (Online Publication: Dec., 2016) been made to determine the HS parameters from the observed entropy of pure liquid metals[6], which in turn are utilised to compute the entropy of mixing for Na-K, Na-Rb, K-Rb, Na-Cs, Rb-Cs, and K-Cs alloys [7] as a function of concentration and temperature. These systems are preferred as their densities as a function of composition are experimentally available [8] and also there are typical representation of simple system where the sizes of constituent atoms differ from a factor of 1 to 3. Out of these, Cs-based binary molten of alloys are of particular interest for both theoretical [9-12] and experimental workers [13- 15]. 2. Theory We consider a simple binary liquid alloy with components of atomic concentration C1 and C2 comprising C1N hard spheres with diameter 1 of species 1 and C2N with diameter 2 of species 2. Following Umar et. al., the expression for the entropy of hard sphere mixture can be expressed as S = Sgas + Sc + Sη + Sσ (1) where Sgas is the ideal gas entropy, Sc represents the ideal entropy of mixing, S corresponds to the packing density η and Sσ is the entropy contribution due to mismatch of the hard sphere diameters σ1 and σ2. Explicit expression for the various contributions are 1 2 3/ 2 1 2 2 5 2 2 c c gas B n B S m m K T K               (2) 1 1 2 2( )c B S C nC C nC K     (3) ( 1)( ) B S K        (4) 2 1 2 2 2( )c B S A C C K     (5) The expressions used for Sη and Sσ which are derived by Umar et. al.[16] from the Helmholtz free energy formulae of Mansoori et. al.]17], the latter work itself providing an accurate analytical fit to the computer Simulation data. Here α = (1-η)-1 and 3 3 1 1 2 2( ) 6        is the packing fraction. Ω is the atomic volume of an alloy for given concentration C1 and C2, (C1 + C2 = 1). m1 and m2 are atomic masses of the constituent species n1(= c1/Ω) and n2(= c2/Ω) are partial number densities. KB is the Boltzmann Constant.In Eq. (5), A = [(-1) - n](y1 + y2) + 3(-1)y1 (6) where y1 = (1 + 2) -3 (7) B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.18 (Online Publication: Dec., 2016) y2 = 12(C11 + C22) -6 (8) where 3 3 1/3 1 1 2 2( )C C    is the hard-sphere diameter of the alloy. Following Eq.(1), the entropy of mixing for a binary alloy can be written as. SM = Sgas + Sη + Sσ + Sc (9) with (1) (2) 1 2 alloy gas gas gas gasS S C S C S    (10) (1) (2) 1 2 alloyS S C S C S       (11) Superscript 1 and 2 in Eqs. (10) and (11) refer to pure species 1 and 2 respectively. It is clear that for a given composition and volume of alloy, the entropy expressions contain two unknown parameters namely, hard sphere diameter σ1 and σ2. The packing fraction η is easily derivable from the value of σ. In order to evaluate the value of σ at different temperatures, we reduce the expression (2) to (5) to pure elements. For the corresponding pure liquid metals, we merely use C1 = 1, C2 = 0 and C2 = 1, C1 = 0 above. The formalism is much simplified and, in particular, Sc = 0 and Sσ = 0 in Eq.(1). Thus ( 1,2)i i i gasS S S i   (12) with 3/ 2 2 5 2 2 ii B n i B m K TS K                (13) 2(1 ) {1 (1 ) } i n i i B S K        (14) where 22 i i         is the packing fraction of pure liquid materials. It may be mentioned that the well-known Carnahan and Starling [18-22] formula for the entropy of pure element can be obtained from Eq. (14) by expanding In (1-ηi) and retaining only the terms up to 2 i . The packing fraction and hence the hard sphere the hence the hard sphere diameter has been determind at a given temperature by using Eq. (12) and taking the observed values of entropy from Hultgren et. al., The appropriate density of liquid metals required at a given temperature have been taken from literature [8]. B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.19 (Online Publication: Dec., 2016) 3. Results and Discussion In Table 1. We have listed the computed values of σ and η for Na, K, Rb and Cs at temperatures 400K, 500K, 600K, 700K and 800K along with the available experimental values of entropy and density. The result shows that the values of η vary from metal to metal. The maximum value is obtained for Na and the minimum value is for Cs. The values of η also depend on temperature. As the temperature increases, η decreases. The values of (dη/dT) for different alkali metals are of the order of: (dη/dt)Na = -2.4*10-4(K)-1, (dη/dt)k = -2.7*10-4(K)-1, (dη/dt)Rb = -2.9*10-4(K)-1 and (dη/dt)Cs = -2.9*10- 4(K)-1. Though, these coefficients are very small, but they may affect the properties which are sensitive to temperature. The entropies of the pure alkali metals, S, are determind by adding ideal gas term (Sgas) and the packing density term Sη through Eq. (12). The former is two to four times greater in magnitude and opposite in sign to sη for all the metals under consideration. The basic parameters required in the calculation are taken from Table 1. Hard sphere parameters for pure elements as reported in Table 1, have also been utilised to compute the various contribution i.e. Sgas, Sc, Sη and Sη and Sσ respectively from Eqs. (2) to (5) for alloys of interest. The relevant data for the various contribution to the entropy for alloys are listed in Tables 2 to 7 and hence ΔSM can easily be evaluated through Eq. (9) for these systems. The computation of the various contribution to entropy are very important to understand the gross behaviour of alkali-alkali alloys. First we note that the absolute value of Sgas are much larger than Sη but after alloying ΔSη becomes effective in comparison to ΔSgas. This shows that in alloying the packing of constituent atoms are more important than the behaviour of individual atom in a given volume element. For investigating the effect of temperature on ΔSM for all the six systems, the values of entropy of mixing are displayed in Figs. (1) to (6). A perusal of Figs: (1) to (3) reveals that ΔSM of Na-K, Na-Rb and K-Rb alloys decreases with increasing temperature. But the result of Na-Cs, Rb-Cs and K-Cs systems exhibit a mixed behaviour. Near the melting point i.e. between 400K to 600K for Na-Cs ΔSM increases with increasing temperature and above 600K it starts falling towards the ideal entropy of mixing. But, ΔSM for Rb-Cs and K-Cs alloys, increases with increasing temperature between 400 to 500 and above 500K it starts falling similarly to Na-Cs. Though the experimental values of temperature dependence of ΔSM for the above mentioned systems are not available in literature for an effective comparison, yet good accord with experiment for Na-K and Na- Cs at melting point confirms the validity of results obtained for these system at different temperatures. Thus, our work provides an additional confidence in the hard sphere formalism for calculating the entropies of liquid metals and alloys. This is the most valuable contribution of semi-empirical approach to study ΔSM of alkali-alkali alloys as a function of concentration and temperature. This is not easily accessible from any other source and such investigation to the best of our knowledge are completely lacking. B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.20 (Online Publication: Dec., 2016) In conclusion it may be asserted that this paper outlines the general theory of ΔSM and the numerical values obtained for alloys form a good set of reliable data. Table 1: The hard sphere parameters () entropy (S/KB) and density of liquid alkali metals at different temperatures. Pure liquid Metals Temp. (K) Hard-Sphere parameter Entropy (S/KB) expt. Density (a.u) expt.Hard-sphere diameter () Packing factor() Sodium 400 6.052 0.4143 8.061 0.92038 500 5.954 0.3835 8.895 0.89438 600 5.872 0.3572 9.557 0.86838 700 5.804 0.3346 10.103 0.83238 800 5.725 0.3111 10.606 0.81638 Potassium 400 7.460 0.4027 9.724 0.812155 500 7.330 0.3710 10.556 0.788655 600 7.216 0.3434 11.226 0.765155 700 7.118 0.3194 11.782 0.741655 800 7.030 0.2980 12.261 0.718155 Rubidium 400 7.919 0.3928 11.259 1.447653 500 7.760 0.3578 12.119 1.401553 600 7.628 0.3287 12.796 1.355453 700 7.519 0.3041 13.351 1.309353 800 7.432 0.2833 13.821 1.263253 Caesium 400 8.416 0.3725 12.452 1.77885 500 8.185 0.3322 13.340 1.72385 600 8.042 0.3049 13.969 1.6685 700 7.954 0.2853 14.460 1.61385 800 7.886 0.2686 14.884 1.55885 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.21 (Online Publication: Dec., 2016) Table 2 : Relevant data for Sgas/KB, Sη/KB and Sσ/KB used in the calculation. Alloys Temp (K) Cna Sgas/KB -S/KB S/KB Na-K 400 0.8 11.6540 3.3480 0.0520 0.7017 11.8033 3.3445 0.0632 0.5001 12.1004 3.3064 0.0650 0.2997 12.3844 3.2459 0.0475 0.1999 12.5214 3.2153 0.0339 500 0.8 12.0158 2.8909 0.0434 0.7017 12.1649 2.8862 0.0523 0.5001 12.4624 2.8477 0.0542 0.2997 12.7471 2.7914 0.0396 0.1999 12.8842 2.7647 0.0283 600 0.8 12.3171 2.5463 0.0369 0.7017 12.4659 2.5406 0.0449 0.5001 12.7639 2.5015 0.0461 0.2997 13.0494 2.4478 0.0336 0.1999 13.1866 2.4237 0.0240 700 0.8 12.5769 2.2765 0.0318 0.7017 12.7255 2.2697 0.0387 0.5001 13.0240 2.2294 0.0398 0.2997 13.3103 2.1770 0.0289 0.1999 13.4476 2.1546 0.0207 800 0.8 12.8066 2.0320 0.0284 0.7017 13.0347 2.0281 0.0326 0.5001 13.2541 1.9951 0.0356 0.2997 13.4632 1.9648 0.0289 0.1999 13.6786 1.9307 0.0186 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.22 (Online Publication: Dec., 2016) Table 3: Relevant data for Sgas/KB, Sη/KB and Sσ/KB used in the calculation. Alloys Temp (K) CRb Sgas/KB -Sη/KB Sσ/KB Na-Rb 400 0.2 13.649 3.036 0.04926 0.3 13.385 3.054 0.06952 0.5 12.841 3.099 0.09743 0.7 12.270 3.163 0.09944 0.8 11.971 3.206 0.08489 500 0.2 14.016 2.570 0.03864 0.3 13.751 2.589 0.0563 0.5 13.207 2.636 0.0789 0.7 12.635 2.702 0.0808 0.8 12.336 2.746 0.0690 600 0.2 14.322 2.229 0.0347 0.3 14.058 2.248 0.0471 0.5 13.512 2.294 0.0661 0.7 12.940 2.359 0.0676 0.8 12.640 2.404 0.0578 700 0.2 14.588 1.968 0.0271 0.3 14.323 1.987 0.0394 0.5 13.777 2.031 0.0566 0.7 13.204 2.059 0.0580 0.8 12.903 2.138 0.0496 800 0.2 14.824 1.762 0.0256 0.3 14.559 1.777 0.0342 0.5 14.012 1.815 0.0472 0.7 13.438 1.868 0.0519 0.8 13.137 1.904 0.0444 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.23 (Online Publication: Dec., 2016) Table 4: Relevant data for Sgas/KB, Sη/KB and Sσ/KB used in the calculation. Alloys Temp (K) Ck Sgas/KB Sη/KB Sσ/KB K-Rb 400 0.2022 13.8886 3.0419 0.00307 0.3998 13.6177 3.0711 0.00478 0.5004 13.4793 3.0864 0.00510 0.6005 13.3418 3.0980 0.00501 0.8514 12.9946 3.1353 0.00281 500 0.2022 14.2519 2.5910 0.00236 0.3998 13.9817 2.6177 0.00367 0.5004 13.8431 2.6342 0.00392 0.6005 13.7068 2.6415 0.01079 0.8514 13.3579 2.6883 0.00216 600 0.2022 14.5549 2.2586 0.00191 0.3998 14.2853 2.2817 0.00297 0.5004 14.1465 2.2979 0.00317 0.6005 14.0115 2.3011 0.00310 0.8514 13.6610 2.3516 0.00175 700 0.2022 14.8166 2.0044 0.00163 0.3998 14.5476 2.0229 0.00256 0.5004 14.4086 2.0380 0.00271 0.6005 14.2749 2.0369 0.00265 0.8514 13.9227 2.0873 0.00149 800 0.2022 15.0483 1.8047 0.00148 0.3998 14.7800 1.8177 0.00231 0.5004 14.6408 1.8309 0.00246 0.6005 14.5086 1.8254 0.00239 0.8514 14.1544 1.8728 0.00136 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.24 (Online Publication: Dec., 2016) Table 5: Relevant data for Sgas/KB, Sη/KB and Sσ/KB used in the calculation. Alloys Temp (K) Cna Sgas/KB Sη/KB Sσ/KB Na-Cs 400 0.1518 14.5366 2.8643 0.0503 0.3 14.0109 3.0122 0.0969 0.5023 13.2769 3.1965 0.1470 0.7014 12.5326 3.3140 0.1579 0.8 12.1507 3.3468 0.1378 500 0.1578 14.9034 2.3469 0.0380 0.3 14.3783 2.4667 0.0731 0.5023 13.6410 2.6420 0.1113 0.7014 12.8960 2.7679 0.1201 0.8 12.5133 2.8206 0.1053 600 0.1518 15.2099 2.0384 0.0391 0.3 14.6855 2.1377 0.0611 0.5023 13.9448 2.3007 0.0933 0.7014 13.1990 2.4167 0.1006 0.8 12.8115 2.4703 0.0884 700 0.1518 15.4754 1.8332 0.0285 0.3 14.9517 1.9164 0.0543 0.5023 14.2072 2.0680 0.0832 0.7014 13.4607 2.1692 0.0896 0.8 13.0763 2.2176 0.0787 800 0.1578 15.7112 1.6674 0.0263 0.3 15.1882 1.7355 0.0501 0.5023 14.4397 1.8732 0.0768 0.7014 13.6923 1.9547 0.0825 0.8 13.3070 1.9932 0.0724 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.25 (Online Publication: Dec., 2016) Table 6: Relevant data for Sgas/KB, Sη/KB and Sσ/KB used in the calculation. Alloys Temp (K) CRb Sgas/KB Sη/KB Sσ/KB Rb-Cs 400 0.2 14.888 2.775 0.0028 0.3 14.799 2.795 0.0038 0.5 14.621 2.848 0.0048 0.7 14.441 2.906 0.0043 0.8 14.350 2.936 0.0032 500 0.2 15.254 2.285 0.00177 0.3 15.166 2.311 0.00232 0.5 14.988 2.367 0.00303 0.7 14.808 2.429 0.00271 0.8 14.717 2.464 0.00213 600 0.2 15.560 1.987 0.0015 0.3 15.472 2.009 0.0021 0.5 15.294 2.056 0.0025 0.7 15.114 2.108 0.0023 0.8 15.023 2.137 0.0018 700 0.2 15.825 1.785 0.00149 0.3 15.737 1.802 0.00201 0.5 15.559 1.836 0.00253 0.7 15.379 1.874 0.00225 0.8 15.289 1.894 0.00169 800 0.2 16.061 1.626 0.00149 0.3 15.973 1.637 0.00201 0.5 15.795 1.662 0.00252 0.7 15.615 1.689 0.00225 0.8 15.525 1.705 0.00176 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.26 (Online Publication: Dec., 2016) Table 7. Relevant data for Sgas/KB, Sη/KB and Sσ/KB used in the calculation. Alloys Temp (K) Ck Sgas/KB Sη/KB Sσ/KB K-Cs 400 0.2 14.632 2.783 0.0106 0.3 14.400 2.815 0.0105 0.5 13.951 2.887 0.0191 0.7 13.493 2.975 0.0179 0.8 13.261 3.027 0.0145 500 0.2 14.988 2.298 0.00721 0.3 14.766 2.333 0.00993 0.5 14.315 2.413 0.01311 0.7 13.857 2.511 0.01234 0.8 13.625 2.568 0.01001 600 0.2 15.294 2.002 0.00598 0.3 15.071 2.032 0.00822 0.5 14.620 2.104 0.01085 0.7 14.162 2.191 0.01021 0.8 13.929 2.242 0.00827 700 0.2 15.558 1.801 0.00557 0.3 15.335 1.827 0.00767 0.5 14.884 1.884 0.01009 0.7 14.425 1.955 0.00950 0.8 14.193 1.997 0.00766 800 0.2 15.793 1.640 0.00536 0.3 15.570 1.660 0.00736 0.5 15.118 1.708 0.00967 0.7 14.659 1.765 0.01552 0.8 14.426 1.799 0.00732 B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.27 (Online Publication: Dec., 2016) Fig. 1: Entropy of mixing (SM/KB) of Na-K as a as a function of concentration and temperature. Fig. 2: Entropy of mixing of (SM/KB) Na-Rb alloys function of concentration and temperature. Fig. 3: Temperature dependence of the entropy of mixing (SM/KB) for liquid K-Rb alloys. Fig. 4: Temperature dependence of the entropy of mixing (SM/KB) for liquid Na-Cs alloys. corresponds to the experimental value at 380K of Neale & Caoack (1982). B.K. Singh et al./ BIBECHANA 14 (2017) 16-29 : RCOST p.28 (Online Publication: Dec., 2016) Fig. : Entropy of mixing (SM/KB) of (Rb-Cs) alloys as a function of Temperature. Fig. 6: Entropy of mixing (SM/KB) for liquid (K-Cs) alloys as a function of temperature & concentration. References [1] R. Hultgren, P.D. Desai, M. Glesier, K.K Kelley, D.D Wagman, Selected values of the thermodynamic properties of the elements and binary alloys. Vol. I & II (1973), American Society of Metals, Ohio. [2] I.H. Umar, A. Mayer, M. Watable , & W.H. Young, J. Phys. F : Metal Phys., 4 (1974) 1691. http://dx.doi.org/10.1088/0305-4608/4/10/016 [3] Hafner, J. Phys. 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