Short communication
Internal Pressure, Energy of Vaporization and Solubility Parameter of Pb-Sn Molten Binary Mixture at Elevated Temperatures
Rajeev Kumar Shukla,1* Atul Kumar,1 Neetu Singh2 and Umakanti Tiwari2
'Department of Chemistry, V.S.S.D. College, Nawabganj, Kanpur(India) 208002 ^Department of Chemistry, A.N.D. College, Kanpur(India) * Corresponding author: E-mail: rajeevskukla47@rediffmail.com
Received: 15-02-2008
Abstract
Internal pressure, excess internal pressure, energy of vaporization, excess energy of vaporization and solubility parameter of Pb-Sn molten binary liquid mixture were calculated over a wide range of temperature and composition. A quantitative treatment has been carried out for these thermodynamic properties with the help of Hildebrand equation. Also, interaction study has been made in the light of excess thermodynamic functions.
Keywords: Molten, internal pressure, elevated temperature, binary mixture, solubility parameter, hildebrand equation.
1. Introduction
The role of internal pressure in liquid solution thermodynamics was recognized by Hildebrand.1-2 The use of this property for a long time was qualitative study of inter-molecular forces. Pioneer attempts have been made by several workers3-14 to show the significance and its correlation with other properties. A liquid under a small isothermal volume expansion does work against the cohesive force which causes the change in the internal energy (E). The function (dE/dV)T is called internal pressure. Hildebrand1-2 showed that for non polar liquids, (dE/dV)T = n AEvap/V, where AEvap represents the energy of vaporization of the liquid and V its molar volume. The quantity n approaches energy density. For polar liquids n ranges from 0.32-1.624. Internal pressure and cohesive energy density (c.e.d.), evidently, do not reflect the same physical property of these liquids. Our one of the aim is to analyze the physical significance of internal pressure and cohesive energy density and to demonstrate the usefulness of both properties.
Two liquids do not completely mix if one liquid has much greater cohesion than the other. Conversely, molecules in liquids of similar cohesion are just as likely to interact and mix with each other as with their own kind. Any
interaction between unlike molecules enhances the change of miscibility.
Hildebrand1 has referred to the square root of c.e.d. as the solubility parameter because of its frequent use in solubility problems. The internal pressure of Pb-Sn molten binary liquid mixtures were computed from the knowledge of thermal expansion coefficient (a) and isothermal compressibility (ßT) using Hildebrand equation. Extensive work has been carried out on excess thermodynamic functions like excess internal pressure15-16 and excess energy of vaporization.1 The present work deals with the theoretical prediction of internal pressure of Pb-Sn molten binary systems and its correlation with solubility parameter and energy of vaporization. Variations of these properties with the temperature and composition have been studied quantitatively. All the necessary parameters in the present work were taken from the literature.17 So for as our knowledge is concerned, nobody has made the temperature dependent study of these properties quantitatively.
2. Theoretical
The quantitative relation between solubility parameter and internal pressure has not been completely resol-
ved. The formal definitions of these quantities are;
(1)
(2)
where V, AE, Pi, 5, a & ßT are molar volume, energy of vaporization, internal pressure, solubility parameter, thermal expansion coefficient and isothermal compressibility respectively. To relate the two quantities, Hilderbrand proposed the empirical relation as;
P;= ti AE/V= nS^
(3)
For most of the liquids, the atmospheric pressure becomes negligible by compression.13 Hence,
T(ap/aT)v = (5E/av)-
(4)
The quantity (dP/dT)V is equal to a/ßT where a is the thermal expansion coefficient and ßT is the coefficient of compressibility at constant temperature.
If the phase happens to be an ideal gas, it has been shown that,
And substitutions of Eq. (5) to Eq. (2), we get, P = RT/V-(ÖE/6V)-r thus
Rendering (dE/dV)T = 0
For a real gas, say a vander Waal's gas the equation becomes;
Moelwyn-Hughes18 analysed Eq.(2) as that pressure of any system consists of two parts; the kinetic pressure, T(dP/dT)V, and static pressure, (dE/dV)T.
The former is due to intermolecular energy and may be positive and negative. Hence, Eq. (2) applies only to a
homogeneous phase. While applying this equation to liquids, many authors19-24 assume P as atmospheric pressure which is incorrect. In liquids, due to condensed phase, (dE/dV)T should be very high and the kinetic pressure should be low. Hence we can not assume P = 0. When we consider the effect of temperature, the kinetic pressure should increase with rise temperature and static pressure should decrease as intermolecular forces decrease.
Several workers23-27 discussed a suitable partition functions for liquid and as a result, free volume (Vf) was given by relation

bRT
MV
PV{dEidV)r where (dE/dV)T = Pj, hence equation becomes
(7)
(8)
All the notations used in the above equation have there usual significance.
Excess thermodynamic functions have been defined
as;
A^ = A^i, - and	(9)
Au, = X|A| + X2 A;
Where AE is the excess thermodynamic functions, Aide is the ideal function and Amix is value of liquid mixture respectively.
The internal molar latent heat of vaporization is a measure of the work done against the internal pressure in vaporizing one mole of liquid, occupying volume V, so that,
3. Results and Discussion
Parameters of pure components are listed in Table 1 where as parameters of Pb-Sn molten binary liquid mixtures are presented in Tables 2-3. Values of density, ther-
Table 1: Thermal Expansion Coefficient (a), isothermal compressibility (ßT), Molar Volume (V), Internal Pressure (Pi), Energy of Vaporization (AE) and Solubility Parameter (5) of Pb-Sn molten Liquids
Comp. X	T/K	a X 105	ßTX 109	V. X 103	Pi	AE	5
		K1	Pa1	dm3mol-1	kbar	K cal	J mole-1
Sn	400	8.8600	2.788	17.3097	21.451	8.8830	.0146
	500	8.9761	2.919	17.5016	23.770	9.9524	.0154
	600	8.9986	3.050	17.7079	25.757	10.9115	.0161
	700	9.0997	3.154	17.8910	28.072	12.0151	.0168
Pb	400	12.4533	3.650	19.7351	22.962	10.8410	.0152
	600	12.7670	4.195	20.1441	26.569	12.8038	.0163
	700	12.8993	4.490	20.4172	27.953	13.6536	.0167
Table 2: Thermal Expansion Coefficient (a), Isothermal Compressibility (ßT), Molar Volume (V), Internal Pressure (Pi), Ideal Internal Pressure Pijjjj,, Excess Internal Pressure (PiE) of Pb-Sn Molten Binary Mixtures
Comp X	T/K	a X 105	ßTx 109	Vm,xX 103	P i(mix)	P ^i(idl)	PiE
(Pb)		K-1	Pa-1	dm3 mol 1	kbar	kbar	kbar
10%	400	9.8008	2.909	17.9716	22.674	21.602	0.11
	600	10.1721	3.180	18.2601	27.925	25.838	0.21
	700	10.1861	3.310	18.4989	29.943	29.943	0.19
20%	400	10.2477	2.960	18.5657	23.299	21.753	0.15
	600	10.4071	3.243	18.9051	28.016	25.919	0.21
	700	10.4859	3.387	19.0913	30.124	28.049	0.21
30%	400	10.8838	3.110	19.0804	23.552	21.904	0.16
	600	11.0427	3.410	19.3646	28.270	26.000	0.21
	700	11.1502	3.610	19.7649	30.053	28.037	0.20
38%	400	10.8234	3.122	19.4623	23.332	22.026	1.30
	600	11.0136	3.460	19.8551	27.789	26.066	1.72
	700	11.0869	3.640	20.0838	29.636	28.027	1.60
45%	400	11.0988	3.130	19.6461	23.864	21.131	1.73
	600	11.2993	3.500	20.0899	28.184	26.122	2.06
	700	11.4169	3.610	20.3146	30.105	28.019	2.08
60%	400	12.7843	3.350	19.8644	25.683	22.357	3.32
	600	13.3438	3.850	20.3146	30.258	26.244	4.01
	700	13.6071	4.100	20.5791	32.292	28.001	4.29
80%	400	12.5712	3.425	19.8432	24.702	22.659	2.04
	600	12.9543	3.920	20.2769	28.849	26.406	2.44
	700	13.0263	4.181	20.5627	30.315	27.977	2.33
mal expansion coefficient and molar volume have been taken from the literature.17
A careful perusal of Table 2 reveals that internal pressure increases as temperature increases. Variation of
internal pressure with composition is not so much prevalent. Excess internal pressure of Pb-Sn liquid mixture also increases as temperature increases. At some places, the value of excess internal pressure decreases i.e. at 700 °C
Table 3: Thermal Expansion Coefficient (a), Isothermal Compressibility (ßT ), Molar Volume (V), Energy of Vaporization (AE^^^), Ideal Energy of Vaporization (AEjdj), Excess Energy of Vaporization (AEe) and Solubility Parameter (5) of Pb-Sn Molten Binary Mixtures
Comp	T/K	a X 105	ßT x 109 Pa-1	V X 103	AE . mix	AEidl	aee	5 J
X(Pb)		K-1	T Pa-1	dm3 mol-1	K cal	K cal	K cal	mole-1
10%	400	9.8008	2.909	17.5522	9.749	9.071	0.678	.0151
	600	10.1721	3.180	17.9515	12.199	11.992	0.207	.0167
	700	10.1861	3.310	18.1436	13.251	12.997	0.254	.0173
20%	400	10.2477	2.960	17.7948	10.348	9.261	1.087	.0153
	600	10.4071	3.243	18.1951	12.671	11.282	1.389	.0167
	700	10.4859	3.387	18.3962	13.759	12.344	1.415	.0174
30%	400	10.8838	3.110	18.0373	10.751	9.452	1.299	.0153
	600	11.0427	3.410	18.4387	13.097	11.469	1.628	.0168
	700	11.1502	3.610	18.6489	14.210	12.509	1.701	.0173
38.%	400	10.8234	3.122	18.2338	10.863	9.608	1.255	.0153
	600	11.0136	3.460	18.6361	13.120	11.621	1.499	.0167
	700	11.0869	3.640	18.8535	14.240	12.641	1.599	.0172
45%	400	11.0988	3.130	18.4011	11.216	9.742	1.474	.0154
	600	11.2993	3.500	18.8042	13.546	11.751	1.795	.0168
	700	11.4169	3.610	19.0278	14.631	12.755	1.876	.0174
60%	400	12.7843	3.350	18.7649	12.205	10.037	2.168	.0160
	600	13.3438	3.850	19.1696	14.758	12.036	2.722	.0174
	700	13.6071	4.100	19.4067	15.898	13.000	2.898	.0180
80%	400	12.5712	3.425	19.2500	11.726	10.435	1.291	.0157
	600	12.9543	3.920	19.6568	13.994	12.418	1.536	.0170
	700	13.0263	4.181	19.9119	14.913	13.327	1.586	.0174
Pb (10%, 30%, 38% & 80%).These discrepancies have been attributed that in these places interactions between the liquid mixture are much more prevalent. Excess ther-modynamic parameter is a measure of the extent of molecular interactions involved in the liquid mixture.
A careful observation of Table 3 reveals that energy of vaporization and solubility parameter increase as temperature increase and these values also increases as composition increases. This is due to the linear relationship of internal pressure with solubility parameter and energy of vaporization.
Thus, the effect of increasing temperature in Eq. 2 is that kinetic pressure always increases and static pressure decreases. In, real gases, the thermal pressure is predominant and potential energy is being too small, to be accounted for by a small value of internal pressure. However in liquid system internal pressure is too high to compare with the kinetic molecular motions and there is no method of calculating the pressure of the system to know its quantitative significance. Further, the temperature coefficient of kinetic pressure is most often positive whereas that of internal pressure is negative in liquid systems. Thus Eq. 2 has its roots in gas thermodynamics and is obviously not rigorously applicable to all liquid systems. This clearly explains the discrepancies observed in the computation of internal pressure of Pb-Sn liquid mixture. Variation of internal pressure with composition of Pb at various temperatures are presented in figures 1-3
XPb
Figure 1: Depedence of the internal pressure Pi on the composition of the Pb-Sn mixture at 400 K. XPb is the molar fracrion of Pb.
Figure 2: Depedence of the internal pressure Pi on the composition of the Pb-Sn mixture at 600 K. XPb is the molar fracrion of Pb.
XPb
Figure 3: Depedence of the internal pressure Pi on the composition of the Pb-Sn mixture at 700 K. XPb is the molar fracrion of Pb.
We can arrive at a conclusive juncture that Hilde-brand's equation is not sufficient to obtain the internal pressure and allied parameters of liquids and liquid mixtures untill its assumption is being modified up to some extent. However, equation is successfully applied to Pb-Sn molten liquid mixture and results obtained are much comparable. Variation of excess thermodynamic functions with temperature becomes powerful tool for predicting the intermolecular interactions in molten liquid mixture.
4. Acknowledgement
Authors are thankful to department of chemistry, V.S.S.D. College, Kanpur and VERSTHEN, a center for research and understanding for their help and support.
5. References
1.	J. H. Hildebrand, R. L. Scott (3rd ed.): The Solubility of Non Electrolytes New York, 1950, p104-136.
2.	J. H. Hildebrand, R. L. Scott: Regular Solutions, Prentice-Hall, Englewood Cliffs, New Jersey. USA, 1962 p204-216.
3.	V. Tiwari , J. D. Pandey, Z. Phys. Chemie. Leipzig 1981, 53, 262-267.
4.	J. D. Pandey, Alec. D. M. Devid, J .Chem. Phys. 1982, 72(2), 1064-1072.
5.	B. K. Sharma, Acustica 1982, 48, 121-129.
6.	J. A. R. Renunclo, G. J. F. Breedveld, J. M. Prausnitz, J. Phys. Chem. 1977, 81, 324-335.
7.	T. W. Richards, Chem. Rev. 1925, 2, 315-324.
8.	G. Tammann, Z. Phys. Chem. 1883, 11,676-681.
9.	P. Drude, W. Nernst, Z. Phys. Chem. 1894, 15, 79-87.
10.	J. E. Gordon, J. Amer. Chem. So.c 1965, 87, 4347-4357.
11.	E. D. Huges, C. K. Ingold (2nd ed.): Structure and Mekcha-nism in Organic Chemistry, Cornell University Press, Ithaca, 1953, p 117-128.
12.	M. R. Dack, J. Chem. Educ. 1974, 51, 231-235.
13.	C. V. Suryanarayana, Ind. J. Pure & Appl. Phys. 1909, 27, 751-756.
14.	M. L. McGlashan: Chemical Thermodynamics, Academic Press, London, 1979, p 47-66.
15.	R. P. Rastogi, J. Sci. Ind. Res 1980, 39, 480-487.
16.	S. V. Subrahmanyam, T. Ramanujappa, E. S. Rajgopal, Acustica 1983, 52, 125-134.
17.	G. V. Konyudhenb, High Temple 1972, 10, 272-281 Translation of Teplopic Vrs Tem., 1972, 10, 309-318.
18.	Moelwyn-Hughes: Physical Chemistry, Pergamon Press., London, 1957, p132-146.
19.	M. R. Dack, Aust. J. Chem. 1975, 28, 1643-1652.
20.	J. O. Hirschfelder, D. P. Sttevenson, H. Erying, J. Phys. Chem. 1937, 5, 897-908.
21.	J. H. Hildebrand, Science 1971, 174, 490-501.
22.	J. H. Hildebrand, R. H. Lamoreaux, Proc. Natl. Acad. Sci. 1972, 69, 3428-3437.
23.	H. Eyring, J. O. Hirshfelder, J. Phys. Chem. 1937, 41, 249258.
24.	J. O. Hirshfelder. D. P. Stevenson, H. Eyring, J. Phys. Chem. 1937, 5, 896-904.
25.	J. F. Kincaid, H. Eyring, J. Phys. Chem. 1937, 5, 587-596.
26.	J. F. Kincaid, H. Eyring, J. Phys. Chem. 1938, 6, 620-629.
27.	R. K. Shukla, S. K. Shukla, V. K. Panday, P. Awasthi, J. Mol. Liq. 2008, 137, 104-109.
Povzetek
V širokem obsegu sestave ter temperature smo izračunali notranji tlak, presežni notranji tlak, izparilno energija, presežno izparilno energijo ter parametre topnosti za binarne Pb-Sn taline. Navedene termodinamske količine smo obravnavali s pomočjo Hildebrandove enačbe. Iz vrednosti presežnih količin smo sklepali na interakcije v preiskovanem sistemu.