S Wacke, T Gorecki, K Ksiazek - Relations between the cohesive energy molar volume bulk modulus and sound velocity in alkali halides - страница 1

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ВІСНИК ЛЬВІВ. УН-ТУ

Серія фіз. 2009. Bun. 43. С. 87-92

VISNYKLVIV UNIV. Ser. Physics. 2009. Is. 43. P. 87-92

PACS number(s): 61.41. +e, 61.25. Em

RELATIONS BETWEEN THE COHESIVE ENERGY, MOLAR VOLUME, BULK MODULUS, AND SOUND VELOCITY IN ALKALI HALIDES

S. Wacke1, T. Gorecki2, K. Ksiazek2

1 Department of Physics, Opole University of Technology 75 Ozimska Str., 45-370 Opole, POLAND e-mail: s.wacke@po.opole.pl 2 Institute of Physics, Opole University

48 Oleska Str., 45-052 Opole, POLAND

e-mail: kasiak@uni.opole.pl

Attention is drawn to the existence of hitherto unnoticed simple linear correlation between the cohesive energy density (internal pressure, Ec/V, where Ec is the molar bonding energy, and V - molar volume) and the bulk modulus B. By analyzing the experimental data available in the literature, it has been found that for alkali halides:

E

B = kE^

V

were k is a proportionality factor, the same for all the alkali halides.

The existence of this correlation leads to others, hitherto unrevealed relations between the cohesive energy, Ec, and different physical properties of condensed phases. For example, as the sound velocity u in liquids is related to the bulk modulus B and the mass density, p, by

2 B

u  = —

p

the two relations above lead to:

2 Ec

u  = c-

1

where li is the molar mass, and c - proportionality factor. Experimental data available in the literature confirm the validity of this relation.

Other consequencies of the revealed relations are also discussed.

Key words: cohesive energy density, bulk modulus, sound velocity, alkali halides.

The cohesive energy as well as the compressibility, or its reciprocal - bulk modulus, are important quantities determining the stability and other physical properties of condensed phases of substances. In the past several authors attempted to develope, both theoretically and semiempirically, formulas relating the cohesive energy and bulk

© Wacke S., Gorecki T., Ksiazek K., 2009

modulus to various physical properties of solids, such as the melting temperature, atomic volume, lattice constants, vacancy formation energy, Debye temperature, valence of constituent atoms, etc. Attempts to find correlation between the cohesive energy and bulk modulus have been made, too [1-3]. All of the above mentioned relationships are, as a rule, restricted to some groups of substances of similar structure, bond type or other physicochemical property. A closer examination of these relationships suggests, that instead of searching for a direct relationships between the bulk modulus and the cohesive energy one should look for the relation between the bulk modulus and the volume density of the cohesive energy.

Cohesive energy density, defined as the ratio of the molar energy of cohesion Ec, to the molar volume V, may be expressed in units J/m3, or, equivalently, in the pressure units - Pa. It has the meaning of some internal pressure preventing the atomization of the condensed phase.

The bulk modulus B (reciprocal of compressibility) of condensed phases defines their resistance against hydrostatic compression. At fixed temperature T, it is defined by the relation

were VT is the volume at temperature T. Its value may be expressed in units of pressure (N/m2 = Pa) or in J/m3. So, the bulk modulus defines the volumetric energy density connected with the compression work. The available literature data, give, as a rule, the values of B determined at room temperature and under atmospheric pressure. In principle, when comparing the values of B for different substances, these values should be reduced to the temperature corresponding to the thermodynamically equivalent states, e.g. to the melting point, or to the absolute zero. Unfortunately, such reduction is often impossible because of the lack of appropriate data.

By changing the interionic or intermolecular distances, the compression work changes both the energy of their interaction (cohesion) as well as the volume of compressed material, consequently changing the cohesive energy density. So, intuitively, the bulk modulus should be related to the cohesive energy density. In order to check this supposition in Table 1 we collected the available literature data on the cohesive energy (per mole) Ec [3-6] of 20 alkali halides. On this basis the cohesive energy density EJV has been calculated and plotted against the values of bulk modulus of corresponding salts (fig. 1). As it is seen, the experimental data points on figure 1 are only slightly scattered along the straight line described by equation

V

were k is a constant proportionality factor (k = 0,855±0,025).

E

B = kE^,

(1)

Table 1

Experimental data on the molar mass, molar volume, cohesive energy, bulk modulus and sound velocity (liquid state, melting point) of alkali halides.

Halide

Molar mass

V-

Г g 1

L mole J

Molar volume

V

Г    3 1 cm

mole

Cohesive energy

Ec

Г kJ 1 L mole J

V [GPa ]

Bulk modulus

B [GPa ]

10-6 Г m21

Г s2 J

Sound Velocity

u m

L s J

LiCl

42,39

20,48

831,2

40,59

32,88

19,61

2050

NaCl

58,44

26,93

762,9

28,33

23,23

13,05

1756

KCl

74,56

37,66

692,5

18,39

17,40

9,29

1590

RbCl

120,92

43,81

665,6

15,19

15,60

5,50

1280

LiF

25,94

11,28

1012,3

89,76

69,80

39,02

2573

NaF

46,99

18,36

914,1

49,80

46,10

19,45

2082

KF

58,10

23,43

793,7

33,88

30,50

13,66

1835

RbF

104,47

26,99

758,0

28,08

26,20

7,26

1380

CsF

151,90

42,31

740,7

17,51

23,50

4,88

1200

LiBr

86,85

25,10

793,7

31,62

25,67

9,14

1477

NaBr

102,90

32,06

725,3

22,63

19,20

7,05

1340

KBr

119,01

43,28

662,7

15,31

14,80

5,57

1273

RbBr

165,38

49,37

637,6

12,92

13,00

3,86

1112

LiI

133,84

32,97

742,6

22,53

18,83

5,55

1232

NaI

149,89

40,84

681,9

16,70

15,10

4,55

1140

KI

166,01

53,21

626,1

11,77

11,70

3,77

1110

RbI

212,37

59,82

605,9

10,13

11,20

2,85

1007

CsCl

168,36

42,41

669,0

15,78

16,87

3,97

1160

CsBr

212,81

47,93

645,0

13,46

18,00

3,03

1118

CsI

259,81

57,61

612,0

10,62

12,40

2,36

923

u-t-1-1-■-1-■-1-■-1-■-1

0 20 40 60 80 100

£ /V [GPa]

Fig. 1. Linear correlation between the experimental data on the bulk modulus B and cohesive energy density EJV

0.0 г '—і—>—і1   і   1 і—1—і—1   t   1   і   1   і   1 і—1—г

0      2      4      6      8      10     12     14     16     18 20 1 О"6 -Л; и [mV]

Fig. 2. Linear correlation between the experimental data on the squared sound velocity u2 of liquid alkali halides at the melting point and their mass density of the cohesive energy at absolute zero, EJ-x.

The standard deviation of the experimental data points from values determined by eq. (1) is 3,54 GPa, and the correlation factor R depicting the quality of revealed relationship is very high - R = 0,9855. So, the experimental data available in the literature confirm the hypothesis on the existence of the proportionality between the bulk modulus and the cohesive energy density, and the agreement is surprisingly good.

The revealed relationship between the cohesive energy density and bulk modulus opens new possibilities for searching for other correlations between various, often very important both for pure science and applications, properties of condensed phases.

For example, as the sound velocity u in the liquid phase is related to the bulk modulus B and mass density p by equation

(2)

then, from relation (1) it simply follows that

u 2 = cEc-, (3)

were li is the molar mass, and c - proportionality factor, the same for all the alkali halides. The term E on the right side of equation (3) has dimensions m2/s2 and has the meaning of the mass density of cohesive energy.

In order to check the validity of equation (3), in Table 1 the experimental data on the sound velocity u of liquid alkali halides at their melting points [5, 7] are collected together with the values of their molar masses. On this basis the plot of the mass energy density E/li versus the squared value u2 of the sound velocity has been prepared (fig. 2).

The experimental data points on figure 2 are scattered along the straight line predicted by equation (3), with the proportionality factor c = 0,242 ± 0,008. The correlation factor amounting R = 0,9953 is very high. So, it may be concluded that the experimental data strongly corroborate the existence of the predicted linear correlation between the squared sound velocity in liquid alkali halides and the mass density of their cohesive energy.

The existence of the relationships presented in the present communication suggests that similar relationships should be valid for other classes of substances, with other nature of the interionic or intermolecular bonds (e.g. metals and condensed rare gases).

As the values of the bulk modulus are related with other elastic moduli (shear modulus, Young modulus) [8], the values of the two later moduli should be also simply related to the volumetric energy density Ec/V, and the squared values of the longitudinal and transversal sound velocities in solids should correlate with the mass density of the cohesive energy, E/u. The search for such correlations will be continued.

1. Plendl J. M., Gielisse J. M. Compressibility and Polymorphism of Solids // Office of Aerospace Research, USAF, Bedford MA 1969.

2. Srivastava G. P., Weaire D. The theory of the cohesive energies of solids // Advances in Physics 1987. Vol. 36. N 4. P. 463-517.

3. Magomedov M. N.   O temperature Debaya kubicheskogo binarnogo ionnogo kristalla // Zhurn. Fiz. Khimii 1993. Vol. 67. N 11. P. 2280-2286.

4. Ksiqzek K., Gorecki T. Vacancies and a generalized melting curve of alkali

halides // High Temperatures - High Pressures 2000. Vol. 32. N 2. P. 209-216.

5. Ksiqzek K. Vacancy model of melting of alkali halides // PhD Thesis, Pedagogical Univ. of Czestochowa 2003.

6. Kittel Ch. Wstep do fizyki ciala stalego // PWN, Warszawa 1999.

7. Minchenko W. І., SmirnovM. V.Skorost' zvuka v rasplavennykh halogenidach shshelochnykh metallov i ich idealnykh smesyakh // Rasplavy 1994. Vol. 2.

P. 42-48.

8. Gorecki T. The relations between the shear modulus, the bulk modulus and Young's modulus for polycrystalline metallic elements // Mat. Sci. and Engng. 1980.

Vol. 43. P. 225-230.

СПІВВІДНОШЕННЯ МІЖ ЕНЕРГІЄЮ ЗВ'ЯЗКУ, МОЛЯРНИМ ОБ'ЄМОМ, МОДУЛЕМ ОБ'ЄМНОЇ ДЕФОРМАЦІЇ І ШВИДКОСТІ ЗВУКУ В ЛУЖНИХ ГАЛОГЕНІДАХ

С. Вацке1, T. Турецький2, K. Ксьонжек2

'Кафедра фізики, Технічний університет Ополє вул. Озімска 75, 45-370 Ополє, Республіка Польща

2Інститут фізики, Університет Ополє вул. Олєска 48, 45-052 Ополє, Республіка Польща

Розглянуто лінійну кореляцію між енергією зв'язку та модулем об'ємної деформації. Наявність цієї кореляції спричинює невідомі досі лінійні співвідношення між енергією зв'язку та різними фізичними властивостями конденсованих фаз.

Ключові слова: енергія зв'язку, модуль об'ємної гнучкості, звукова швидкість, лужні галогеніди.

СООТНОШЕНИЕ МЕЖДУ ЭНЕРГИЕЙ СВЯЗИ, МОЛЯРНЫМ ОБЪЕМОМ, МОДУЛЕМ ОБЪЕМНОЙ ДЕФОРМАЦИИ И СКОРОСТЬЮ ЗВУКА В ЩЕЛОЧНЫХ ГАЛОГЕНИДАХ

С. Вацке1, T. Гурецкий2, К. Ксьонжек2

' Кафедра физики, Технический университет Ополе ул. Озимска 75, 45-370 Ополе, Республика Польша

2Институт физики, Университет Ополе ул. Олеска 48, 45-052 Ополе, Республика Польша

Рассмотрена линейная корреляция между энергией связи и модулем объемной деформации. Наличие этой корреляции вызывает неизвестные до сих пор линейные соотношения между энергией связи и разными физическими свойствами конденсируемых фаз.

Ключевые слова: энергия связи, модуль объемной гибкости, звуковая скорость, щелочные галогениды.

Стаття надійшла до редколегії 04.06.2008 Прийнята до друку 25.03.2009

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