Surface Thermodynamics of Liquid Binary Copper-Bismuth ...ripublication.com/ijpap17/ijpapv13n3_12.pdf · binary liquid mixutes in terms of Flory's statistical theory can be described

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

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)

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


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


)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.


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



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



Bi % by mass =3


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



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



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



Bi % by mass =10


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


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


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 ...ripublication.com/ijpap17/ijpapv13n3_12.pdf · binary liquid mixutes in terms of Flory's statistical theory can be described