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International Journal of Pure and Applied Physics.
ISSN 0973-1776 Volume 13, Number 3 (2017), pp. 375-386
© Research India Publications
http://www.ripublication.com
Surface Thermodynamics of Liquid Binary
Copper-Bismuth Alloys at Elevated Temperatures
Gyan Prakash, Ashish K. Singh, Vikas.S.Gangwar and Sandeep K. Singh*
Department of Chemistry, VSSD College, Kanpur-208002(India).
Abstract
Theoretical results obtained from statistical mechanical concept of Flory for
Surface tension were investigated for liquid binary copper-bismuth alloys with
respect to their chemical composition at elevated temperatures and
atmospheric pressure over the entire concentration range 1-50% of bismuth,
from the experimental work of Oleksiak et al. Percent changes in the surface
tension (%σ) were evaluated and presented.Values of density and surface
tension for pure metals were evaluated from the relations detailed out in the
article. Conclusively, the model was compared and tested for copper-bismuth
alloys showing that theoretical results are consistent with the experimental
findings. The values of surface tension were investigated and correlated with
the composition and temperatures.
Key words: Surface tension, copper-bismuth alloy, Flory, binary mixture and
interactions
1. INTRODUCTION:
The surface tension of liquid metal on a liquid solid phase contact surface affect
many phenomenon at interface during hydrometallurgical process of metal production
and refinement as well as during casting process. In addition, an important factor is
presence of surface active substances in metals and alloys. Surface active substances
accumulate at interfaces and affect surface tension and contact angle values. For pyro-
metallurgical process, they can influence their rates by increasing at interfaces.
Surface tension highly affect refractory material corrosion as well as they are
important factors for processes of metal coating application, casting and composite
material production [1-10]. When metallurgical and casting modeling as well as
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376 Anitha T and Dr K Ramamorthy
material property modeling become increasingly popular, such parameters such as
surface tension, contact angle, density and viscosity of the liquid phase are of
particular importance [11-15]. The study involved theoretical treatment of surface
tension of liquid binary copper-bismuth alloys with respect to their chemical
composition temperatures as well as investigations of the liquid alloy.
The surface tension (σ) and density (ρ) of a liquid are important thermodynamic
properties in phenomena such as liquid-liquid extraction, gas absorption, distillation
and crystallization, also has been widely used to characterize surface of liquid in
chemistry and chemical engineering areas such as the manufacturing of plastics,
coatings, textiles and films. Values of gas–liquid interfacial tension are used in
studying of liquid–liquid and liquid–solid interfaces. They are also useful for
understanding and interpreting the nature of interactions between unlike molecules in
a mixture, control the growth of a material on a substrate as well as different
phenomena, such as melting, coalescence, evaporation, phase transition, growth of
nano particles etc. Surface tension is the physical property of liquids in which
exposed surface tendency to contract to the smallest possible area. A liquid that
molecules are strongly attracted to each other tends to have higher surface tension
and molecular density. Surface tension in dense systems has not been extensively
studied. The available experimental data for density and surface tension at some
subcritical temperatures are available for some ionic and normal fluids [16].
2.THEORETICAL:
In dealing with liquid state, Flory defined an element (or segment) as an arbitrary
chosen portion of the molecules. Considering such segments in a molecules and
following Prigogine treatment [17,18] of -mer chain molecules and representing the
number of external degree of freedom per segment by 3C, he was able to derive a
partition function of the form:
)kT/E(exp)vv(zz oncr33/13/1comb (1)
where k, n, T and Eo are respectively the Boltzmann constant,numberofparticles,
absolute temperature and excess inter molecular energy. Zcombis a combinatorialfactor
which takes into an account the number of ways inwhich rn elements intersperse
among one another, without taking into consideration the precise location of each
element relative to its chosen neighbour, is geometric factor which is given by the
expressi8ons
3/1)V( (1)
l)V( 3/1 (3)
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Surface Thermodynamics of Liquid Binary Copper-Bismuth Alloys at Elevated… 377
where, * is the length of such particles distributed within a space of length L and = L/N, is the space available per molecule. Using thesuggestion of Frank [19], Flory
[20-23] obtained the following expressions for the intermolecular free energ
V2
SrNEo
(4)
Here η is a constant characterising the energy of interaction for a pair ofneighbouring
sites, S is the number of inter molecular contact sites persegment and V is the volume
per segment. The reduced partition function, in the light of eqs 1 to 4 takes the form
)T~
V~
/rNC(.exp)1V()V(ZZ rNC33/1rNCcomb (5)
The reduced equation of state obtained from the resulting partition function is given
by, T~
V~1
)1V~
(
V~
T~V~
P~
3/1
3/1
(6)
Changing from the molecular to molar units per segment for V, V*, and η, one gets,
V
V
V
VV~
(7)
S
cRTV2
T
TT~
(8)
S
PV2
P
PP~
2
(9)
The reduced equation of state (6) at zero pressure becomes,
3/4
3/1
V~
1V~
T~ (10)
3
1T33(
TV~
(11)
where α is the coefficient of expansion at P=0. Thus the reduced volu V~
and reduced
temperature T~
can be obtained from the experimental volume of α. Knowing V~
and T~
, characteristic volume V* and characteristic temperature T* can be computed using
eqs (7) and (8). From the reduced equation of state it follows that:
2P2
T
V~
TV~
TP
(12)
where VP )T/P( is the thermal pressure coefficient at P= 0.Characteristic
Pressure P* is evaluated from this equation for molten binary liquid mixtures with the
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378 Anitha T and Dr K Ramamorthy
component subscript 1&2. Flory obtained the following expressions:
2211
1222
2211
222111
PP(
X1
PP
T~
PT~
P
T
TT~
(13)
and
)x(PPP 12212211
2
22
1
11
3/1
3/4
T
P
T
P
1V~
V~
T (14)
here 21 & are the segment fractions and 21 & are the site fractions which are
given by, 2211
22
1
222
1
111
1
2212
SS
S
)V~
VX(V~
VX(
V~
VX
(15)
)1( 21 (16)
where X1 and X2 are the mole fractions, X12 is the interaction parameter and V1 & V2
are the molar volumes of the component 1& 2 respectively. The ratio (S1/S2) = 3/1
21
3/1
21 )r/r()V/V( for spherical molecules.The interaction parameter X12 can
be expressed as,
26/1122/112112 )V/V()P/P(1PX (17) where X12 is the energy parameter and it is measured by the difference of interaction
energy between the unlike pairs and the mean of the like pairs.The surface tension of
binary liquid mixutes in terms of Flory's statistical theory can be described by the
expression,
)V~
(~ (18)
where V~
&~, are the characteristic and reduced parameters. Patterson & Rastogi
in their extension of the corresponding states theory dealt with the surface tension by
using as the reduction parameter
3/13/23/1 TPk (19)
called the characteristic surface tension of the liquid. Here k is the Boltzmann
constant and P* & T* are the characteristic pressure and temperature respectively.
Starting from the work of Prigogine and Saraga, they derived a reduced surface
tension equation which in the case of Vander Waals liquid can be written
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Surface Thermodynamics of Liquid Binary Copper-Bismuth Alloys at Elevated… 379
)1V~
(
)5V~
(In
V~
)1V~
(V~
M)V~
(~3/1
3/1
2
3/13/5 (20)
where M is the fraction of the nearest neighbours that a molecule loses on moving
from the bulk of the liquid to the surface. Its most suitable value ranges from 0.25 to
0.29 provided the liquid mixture is consist of hydrocarbons as taken by Flory et al. On
the basis of eq (20) the theory used for the evaluation of surface tension is
characterized as Flory-Patterson theory.
3.RESULTS AND DISCUSSION :
Parameters of the pure metal at various temperatures for Cu-Bi Molten mixtures have
been calculated by the following relations[24],
σ (T) = σᵧ⁰ + C ( T - Tm )
where σ(T), σ0,and Tm are the Surface tension of metal at temperature T, Surface
tension of metal at melting point and melting point respectively.
and C = dᵧ /dT
The experimental values of surface tension were taken from the work of Oleksiak et
al[25]. The values of reduced volume )V~
( , characteristic pressure )(* ,
characteristic temperature (T*), characteristic surface tension , calculated and
observed surface tension of Cu-Bi molten binary mixture and their percentage
deviations have been presented in Table over a wide range of temperature and
composition. The essential data required for the calculation of these values have been
taken from the literature. Surface tension of Cu-Bi mixtures have been computed with
the help of reduced surface tension equation (20). In this equation the most suitable
value of M varies indifferently from temperature to temperature. Prigogine and Sarga
suggested the most suitable value for hydrocarbons which ranges from 0.25 to 0.29.
Because this is directly related to the structural phenomenon of liquid molecules
involved in the liquid system. From hydrocarbons to Cu-Bi molten mixtures, structure
of the liquid molecules change abruptly i.e. their lattice structure becomes so
prominent that it changes the whole of the liquid geometry thus deviations in the most
suitable values have been observed. That is why, in the present context, we have taken
the most suitable value (M) for Cu-Bi system at 1373oK is 0.46, at 1423oK is 0.41,at
1473oK is 0.39,15230K is 0.38 and 15730K is 0.36 in the entire calculations.
Theoretical and experimental values are presented graphically in figures as function
of temperature for different mole fractions.
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380 Anitha T and Dr K Ramamorthy
1250
1260
1270
1280
1290
1300
1310
1320
1330
1300 1400 1500 1600
ᵧ, n
M.m
-1
T ,K
ᵧtheo , mN.m-1
ᵧexp , mN.m-1
Linear (ᵧtheo , mN.m-1)
Linear (ᵧexp , mN.m-1)
Bi % by mass =1
1100
1120
1140
1160
1180
1200
1220
1240
1260
1300 1400 1500 1600
ᵧm
N.m
-1
T,K
ᵧtheo , mN.m-1
ᵧexp , mN.m-1
Linear (ᵧtheo , mN.m-1)
Linear (ᵧexp , mN.m-1)
Bi % by mass =2
1040
1060
1080
1100
1120
1140
1160
1180
1300 1400 1500 1600
ᵧ, m
Nm
-1
T,K
ᵧtheo , mN.m-1
ᵧexp , mN.m-1
Linear (ᵧtheo , mN.m-1)
Linear (ᵧexp , mN.m-1)
Bi % by mass =3
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Surface Thermodynamics of Liquid Binary Copper-Bismuth Alloys at Elevated… 381
Figures Variation of Surface Tension with Temperatures at different composition of
Bismuth
920
940
960
980
1000
1020
1040
1060
1300 1400 1500 1600
ᵧ, m
Nm
-1
T,K
ᵧtheo , mN.m-1
ᵧexp , mN.m-1
Linear (ᵧtheo , mN.m-1)
Linear (ᵧexp , mN.m-1)
Bi % by mass =4
760
770
780
790
800
810
820
830
840
850
860
1300 1400 1500 1600
ᵧ, m
Nm
-1
T,K
ᵧtheo , mN.m-1
ᵧexp , mN.m-1
Linear (ᵧtheo , mN.m-1)
Linear (ᵧexp , mN.m-1)
Bi % by mass =5
700
720
740
760
780
800
820
1300 1400 1500 1600
ᵧ,m
Nm
-1
T,K
ᵧtheo , mN.m-1
ᵧexp , mN.m-1
Linear (ᵧtheo , mN.m-1)
Linear (ᵧexp , mN.m-1)
Bi % by mass =10
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382 Anitha T and Dr K Ramamorthy
A careful perusal of Table reveals that all the parameters i.e. reduced volume
characteristic pressure are increasing as the temperature and composition of Bi is
increasing whereas the values of surface tension of Cu-Bi system is decreasing.
Percentage deviations between experimental and theoretical values of surface tension
of Cu-Bi molten mixtures have been dipicted in the Table. Minimum percentage
deviation and maximum percentage deviation has been observed at 1373 oK for 1% Bi
the value 2.2 and at 1473oK for 50% Bi the value 18.3 respectively. At elevated
temperature, a high degree of coulombic interaction would be expected in the near
and next to near neighbours. The application of Prigogine corresponding state
principle to this problem for evaluating the surface tension of molten liquid mixture at
elevated temperature has been carried out in a pragmatic sprit as was that of the
original derivation of Patterson and Rastogi. The corresponding state treatment to Cu-
Bi system is somewhat unorthodox, since application of eq (20) to the molten liquid
mixture on the assumption that they are equivalent to single component liquids
effectively ignores differences in concentration occurring at the surface of the
mixture. Gibbs enrichment of a mixture surface by the component of lower surface
tension is well known. The normal results show a lowering of mixture surface tension,
which results in a negative deviation from a linear function of bulk mole fraction that
is why, there is a tendency of our theoretical values to be higher than the observed
values. However, it would not be proper to say that thls is the only reason for higher
discrepancies. A part of the discrepancy may be attributed to the approximations
made in the computation of the interaction parameter Xij.
Table Composition of Bi by mass, temperature, density(ρ),charecteristic
pressure(P*),molar volume(V),reduced volume( V~ ), theoretical surface tension(σtheo), experimental surface tension(σexp) and percent deviation for Cu-Bi
metal alloy at elevated temperatures
Bi% by
mass T/K ρ/ Kg.m-3 P*/ nM.m-1 V/ m-3 V~
m-3
σtheo/nM.m-1
σexp/ nM.m-1 %dev
1 1373 8004 1291.145 168.977 205.512 1323.8 1295 2.2
1423 7954 1318.831 170.039 202.289 1319.3 1285 2.7
1473 7904 1389.231 171.114 211.088 1309.2 1274 2.8
1523 7854 1460.952 172.204 220.001 1301.1 1268 2.6
1573 7803 1533.972 173.329 229.058 1294.2 1255 3.1
2 1373 8008 1284.290 168.447 25.301 1242.6 1193 4.2
1423 7956 1271.430 169.548 189.896 1236.6 1186 4.3
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Surface Thermodynamics of Liquid Binary Copper-Bismuth Alloys at Elevated… 383
1473 7906 1337.622 170.621 197.689 1227 1178 4.2
1523 7858 1404.911 171.663 205.497 1195.6 1133 5.5
1573 7808 1473.269 172.762 213.420 1197.4 1125 6.4
3 1373 8012 1277.416 167.912 25.090 1163.4 1107 5.1
1423 7962 1228.712 168.967 178.637 1153.1 1099 4.9
1473 7912 1291.321 170.035 185.574 1146.1 1086 5.5
1523 7862 1354.860 171.116 192.543 1133.3 1071 5.8
1573 7812 1419.301 172.211 199.542 1121.1 1059 5.9
4 1373 8016 1270.521 167.372 24.879 1053.8 994 6.0
1423 7956 1190.015 168.634 168.693 1037.8 967 7.3
1473 7916 1249.549 169.486 174.689 1027.7 961 6.9
1523 7866 1309.889 170.563 180.904 1013.2 953 6.3
1573 7816 1371.007 171.654 187.123 1003 936 7.2
5 1373 8021 1263.606 166.804 24.663 852.2 791 7.7
1423 7971 1154.799 167.850 159.220 842.3 786 7.2
1473 7920 1211.674 168.931 164.820 839.6 774 8.5
1523 7870 1269.262 170.004 170.388 825.4 769 7.3
1573 7820 1327.538 171.091 175.943 813.7 773 5.3
10 1373 8044 1228.731 163.913 23.586 794.4 738 7.6
1423 7993 1017.510 164.959 123.176 791.4 731 8.3
1473 7943 1065.283 165.997 126.632 786.3 722 8.9
1523 7893 1113.565 167.049 130.040 777.5 718 8.3
1573 7842 1162.341 168.135 133.417 761.2 713 6.8
20 1373 8095 1157.466 157.501 21.377 733.7 670 9.5
1423 8044 853.591 158.500 80.116 727.4 660 10.2
1473 7993 893.100 159.511 81.715 713.1 616 15.8
1523 7943 933.112 160.515 83.260 704.3 608 15.8
1573 7892 973.619 161.552 84.776 666.3 606 10.0
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384 Anitha T and Dr K Ramamorthy
30 1373 8156 1084.164 150.045 19.091 528.7 481 9.9
1423 8104 759.141 151.008 55.315 523.2 466 12.3
1473 8053 795.165 151.964 56.157 516 459 12.4
1523 8001 831.754 152.952 56.980 505 449 12.5
1573 7950 868.904 153.933 57.770 476.2 432 10.2
40 1373 8228 1008.824 141.308 16.730 503 454 10.8
1423 8176 697.743 142.207 39.213 499.3 440 13.5
1473 8124 732.002 143.117 39.695 483.2 444 8.8
1523 8072 766.877 144.039 40.160 472.1 432 9.3
1573 8019 802.365 144.991 40.615 473.8 430 10.2
50 1373 8315 931.467 130.916 14.295 516.4 453 14.0
1423 8263 654.642 131.740 27.944 517.1 437 18.3
1473 8210 687.897 132.590 28.229 481.4 420 14.6
1523 8157 721.807 133.452 28.505 476.7 420 13.5
1573 8104 756.368 134.324 28.772 462.5 415 11.4
4.CONCLUSION:
On the basis of the above discussion, it may be concluded that Flory's statistical
theory predict surface tension of Cu-Bi molten liquid mixture at elevated temperature
upto substantial extent. On the basis of our calculations, it can be inferred that the
Flory theory affords a useful estimation of the surface tension without considering
such concentration effects. The results of measurements of liquid Cu-Bi alloy surface
tension have revealed a significant effect of the bismuth additive on the parameter
values. In copper, bismuth acts as a surface-active substance like e.g. oxygen, sulphur
or lead. A slightly increased Bi fraction in the Cu-Bi alloy leads to a markedly
reduced surface tension, which is illustrated by the Cu-Bi surface tension values in
Table. When the bismuth fraction in the alloy is 40 %mass and higher, the surface
tension is near the value for pure bismuth. This may suggest accumulation of Bi at the
liquid alloy-gaseous phase interface as for other known surface-active substances. The
temperature rise causes a linear decrease in the surface tension of Cu-Bi alloys.
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Surface Thermodynamics of Liquid Binary Copper-Bismuth Alloys at Elevated… 385
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