Liquid-Vapor Equilibrium in a Centrifugal Field: Binary

Russian Journal of Physical Chemistry, Vol. 78, No. 3, 2004, pp. 472-479. Original Russian Text Copyright © 2004 by Abakumov, Fedoseev. English Translation Copyright © 2004 by MAIK "Nauka/Interperiodica" =

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Translated from Zhurnal Fizicheskoi Khimii, Vol. 78, No. 3, 2004, pp.

563-570.

(Russia).

OTHER PROBLEMS OF PHYSICAL CHEMISTRY

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Liquid-Vapor Equilibrium in a Centrifugal Field: Binary Systems in Vessels with Common Gas Phase G. A. Abakumov and V. B. Fedoseev Institute of Organometallic Chemistry, Russian Academy of Sciences, ul. Tropinina 49, Nizhni Novgorod, 603600 Russia E-mail: [email protected] Received July 1,2003 Abstract—The effect of a centrifugal field on h e t e r o g e n e o u s equilibria in a l i q u i d - v a p o r s y s t e m in w h i c h t h e separated liquid parts of the system c o m m u n i c a t e t h r o u g h t h e c o m m o n g a s p h a s e w a s e x a m i n e d . It w a s e s t a b ­ lished that, d e p e n d i n g on the properties of the c o m p o n e n t s , the field m a y c a u s e substantial c h a n g e s in the c h e m ­ ical c o m p o s i t i o n , hydrostatic p r e s s u r e , a n d v o l u m e s of t h e p h a s e s in t h e different p a r t s of t h e s y s t e m . T h e s e effects w e r e d e m o n s t r a t e d to b e d e p e n d e n t not only on t h e strength of the field b u t also o n t h e configurations of the system a n d its parts.

While a centrifugal field does not affect the standard thermodynamic characteristics of phase transitions and individual chemical component, it can substantially change the physicochemical properties and the equilib­ rium chemical and phase compositions of multicomponent systems [1]. This effect is based on the redistribu­ tion of the components under the action of the field both within the individual phases and between the different phases.

tions in the system. The chemical composition and physicochemical properties of such systems depend on the distribution of the nonvolatile components. Indeed, when a liquid mixture occupies two or more vessels having a common gas phase, only the volatile compo­ nents can migrate from vessel to vessel. In this case, the redistribution of the components in a centrifugal field is accompanied by interesting, sometimes unexpected, results.

The criterion of thermodynamic equilibrium for a system in a centrifugal field is the constancy of the mechanochemical potential, i.e., the sum of the chemi­ cal potential and the potential energy of the compo­ nents at any point of the system [1, 2]. If the object is a liquid mixture of volatile components at equilibrium with its saturated vapor, the local pressures of the vapor components unambiguously determine the coordinates of the liquid-vapor interface and the local composition of the liquid phase. Given the shape and size of the ves­ sel, it is possible to calculate the overall composition of the liquid and gas phases. If one of the components is nonvolatile, the level of the liquid can be unequivocally determined from the amount of this component in the vessel.

Consider the simplest model: Let the vapor be an ideal gas and let all the liquid mixtures be ideal incompressible solutions obeying the Raoult law. All the processes are postulated to proceed under isother­ mal conditions. When more complicated models of solutions are used [6], the fundamental results remain the same.

In an external field, the thermodynamic parameters of a system depend not only on the strength of the mechanical field but also on the geometric configura­ tion of the system, the so-called vessel-shape effect [4]. To exclude the influence of the shape of the vessel, let us combine equivalent configurations under the term field-isomorphous. A vessel can be thought of as isomorphous to an external field if the field flux through an In [1, 3], an external field was interpreted as a per­ arbitrary cross section of the vessel formed by an equiturbation removing allosteric degeneracy in the system. potential surface is constant. For example, circular cyl­ inders, circular cylindrical rings, and segments thereof Any state in which the energy of each particle of the rotating about the principal axis are isomorphous to the system is independent of its position (coordinates) is corresponding centrifugal field, whereas cylinders ori­ termed allosteric degeneracy. An external field removes entated along a uniform gravitational field are isomor­ degeneracy, giving rise to the dependence of the ther­ phous to it (Fig. 1). If the thicknesses and the radii of modynamic state of the system on its geometric config­ the surfaces of identical-composition liquid layers uration. This effect is characteristic of gaseous [4] and rotating in isomorphous vessels are the same, the radial liquid [5] mixtures. For the heterogeneous systems con­ dependences of the concentration and pressure are also sidered in this work, the effect of shape exhibits specific the same. features associated with restrictions on the possibility of the individual components of the condensed phase For the sake of definiteness, we will model a vessel occupying the thermodynamically most favorable posi­ with a common gas phase by a cylindrical rotor divided 472

LIQUID-VAPOR EQUILIBRIUM IN A CENTRIFUGAL FIELD (a)

473

(b)

Fig. 1. Vessels isomorphous to (a) a centrifugal and (b) a uniform gravitational field.

into two sections with a diaphragm that has an orifice (at the center) whose radius is smaller that the radii of the rotating layers of the liquid. If the diaphragm is par­ allel to the bases of the cylinders, the vessels can be considered isomorphous (Fig. 2a). Non-isomorphous vessels will be modeled by a conical diaphragm (Fig. 2b). In the subsequent figures, we will depict only the axial cross sections of the rotors.

At the liquid-vapor interface of an equilibrium het­ erogeneous system, one more condition is fulfilled, namely ц°, = tf

g

+ RT\np°,

(a)

The conditions of constancy of the mechanochemical potentials of the components Ц of the liquid and gas phases at thermodynamic equilibrium in an external field read as 2

П,.(г) = U^ii^

+ RTlnp^-M^r /!, Щг) = П

= ц°, + RTliiXiir)

(1)

г

22 - Mfifrll

-

2

( )

+ ViP(r),

where x{r) is the mole fraction of component i for a radius r; and |U° are the standard chemical poten­ tials of component i in the liquid and gas phases, respectively; V,- and M are the molar volume and molecular mass of component i; со is the angular veloc­ ity of the centrifuge; p(j) is the partial pressure of com­ ponent i at a radius r; P{r) = j p(r) coVdr is the hydro­ static pressure at a radius r; p(r) is the local density of the liquid at a radius r; and r is the radius of the surface of the liquid. g

i

r

h

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Fig. 2. Two-section rotor with sections (a) isomorphous and (b) non-isomorphous to the centrifugal field. No. 3

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(3)

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ABAKUMOV,

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the pure component p(r ) = p . As a result, the vapor condenses, giving rise to the transfer of the component from vessel u to vessel d. According to Eq. (1), the equality p(r ) = p(r ) = p is fulfilled only if r = r ; therefore, the transfer will continue until the levels of the liquid in the vessels become equal. For a mixture obeying the Raoult law, the condition of equilibrium at the interface reads as h

Щ

(a)

hu

hd

Xi

!

I

0 1

:, 5 ,

1

l_

0

M

= |ij> + / г П п ( ° х , ( г ) ) ,

h

g

А

(4)

л

where x (r ) is the concentration of component i at the surface of the liquid. If the composition of the system is known at least at one point, Eqs. (1), (2), and (4) completely describe the radial distribution of the components and the radius of the interface. Using Eqs. (1) and (4), we obtained the following expression for the partial pressure of compo­ nent i over the surface t

1

0

hu

ц?, + RTln( (r ))

(c)

u

h

Pi(r) = p ^ ( r ) e x p h

I

2RT

r>r .

(5)

h

У

According to Eq. (2), the mole fraction of component i is given by 2

x (r) t

f

2

(1/2)М,ш (г -г0-У,Р(г)

= x (r )exp h

> (6)

RT r>r . h

Excluding the hydrostatic pressure from Eq. (6), we obtained the following relations for an arbitrary pair of components In

x (r) bin ~~x (r*)

*,(r) x,(r*)

2

coV*-r ) (M -bM ), 2RT

2

l

2

2

(7) b = M p IM p l

1.0 Ax, m o l e fractions Fig. 3. (a-c) Axial cross sections of rotors with sections isomorphous to the field (the parts occupied by the liquid, ele­ ments of the rotor, and gas phase are depicted in grey, black, and white, respectively), (d) The dependence of the radius of the surface of the pure volatile component at M - 150 g, 9 = 60000 rpm, and T = 298 K; the radius of the surface of the solution was r = 1 cm. h

where p° is the saturation vapor pressure over pure component p. Consider the case where vessels with a common gas phase (vessels u and d in Fig. 3a) contain a pure volatile component in such amounts that the rotating liquid has different levels (r > r ) . According to the barometric h d

h u

2

dependence p(r) = p°exp(McoV - r )/2RT), the vapor pressure exceeds the saturation vapor pressure of hu

2

2

{

=

Vi/V . 2

These relations make it possible to determine unequiv­ ocally the composition of the liquid mixture at any point of the system as a function of the composition at a level r*. In the subsequent calculations, we used r* = r . Note, however, that the limiting radius r* = max - o is more convenient reference level, because, in the general case, the level of the liquid depends on the compressibility of the components and the type of chemical transformations and, therefore, on the strength of the centrifugal field. Let us apply the equations obtained to a number of model systems. h

r

r

a

Vessels Isomorphous to the Field Let the rotor displayed in Fig. 2a contain a binary mixture of volatile components, and let the liquid be initially distributed between the sections in a random manner. The mixtures placed in the different sections can exchange components only through the gas phase.

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LIQUID-VAPOR EQUILIBRIUM IN A CENTRIFUGAL FIELD

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Let us calculate the difference 8 between the levels of the liquid in the sections in the equilibrium state of the system. The local partial pressures of the volatile compo­ nents on either side of the diaphragm are identical, as follows from Eq. (1) and the equality of the pressures at levels above the diaphragm. For the near-surface layers of the liquid in each of the sections, conditions (5) and identities

10 cm 0.5r

(a)

(8) (b)

are fulfilled. The quantities x (r + 8) are monotonic functions of 8, because Eq. (5) in combination with the Raoult law t

h

•explpj V 2RT Clearly, the sum of the monotonic functions x (r + 8) satisfies condition (8) only at 8 = 0; consequently, the levels of the liquid on either side of the diaphragm are identical and equal to r (Fig. 3a). yields x {r t

+ 8) =

h

t

h

h

Thus, the behavior of a liquid mixture of volatile components is similar to the phenomenon of communi­ cating vessels. Whatever the initial compositions and levels of the liquids in the vessels, at equilibrium, the levels of the liquids, the compositions at the surface, the radial dependences of the concentrations of the compo­ nents, and the hydrostatic pressures in the different sec­ tions of the rotor are identical. The equilibrium levels of the liquids (8 = 0) in the sections are also equal to one another when isomorphous vessels filled with a mixture of volatile compo­ nents have different maximum radii (Fig. 3b). Never­ theless, the mean concentrations of the components in the different sections will be different: in the section with larger radius, the liquid will be enriched in the heavy component. Indeed, in a wide (r ) and narrow (rj < r ) sections, the concentration profiles within the range from r (the radius of the surface) to r coincide, being, however, different (enriched in the heavy com­ ponent) over the range from r to r . 0

V V

x

x

0

If an equilibrium liquid mixture contains at least one nonvolatile component, the equalization of the levels does not occur. Let the nonvolatile component (component 2) of a mixture occupy the upper (u) section of the cylinder (Fig. 3c), while the volatile component (component 1) occupies the lower one (d). According to the Raoult law, component 1 should form with compo­ nent 2 a solution over which the pressure of the vol­ atile component equals p (r ) =x (r )p°. The pres­ sure over the surface (r + 8) of the pure volatile component is p (r + 8) = p°. Distribution (5) unequiv­ ocally relates p(r ) and p(r + 8). In this case, the levels u

h

l

h

h

d

h

h

h

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\®

I

\®

1

L

L

Fig. 4. Axial cross sections of rotors non-isomorphous to the field (the parts occupied by the liquid, elements of the rotor, and gas phase are depicted in grey, black, and white, respectively): (a) (x ) = (x ) p (r) > p ( r ) , (r) Ф x (r); (b) (x;> * (xi) p (r) = p (r), x ( r ) = xt ( r ) . t d

d

w

d

t w

d

u

u

u

d

d

it

u

u

of the pure component and solution should differ by 8, which, according to (5), reads as 8 = ±

-11/2

2RT

\n(x (r )) x

(9)

h

2

Mco

where x (r^) is the concentration of the volatile compo­ nent at the surface of the solution. Note that the nega­ tive and positive solutions are equivalent, because they describe radii symmetrical with respect to the axis of rotation. If a nonvolatile component (or different nonvola­ tile components) are present in either section, 8 is given by x

2

r

5= ± h

0

h

0

2RT, An Mco

1/2 *i(r) Y h

x (r x

h

+ b)J_

Ю ()

A particular case of (10) is Eq. (9). Formally, Eq. (9) or Eq. (10) are independent of the properties of the nonvolatile component. For a multicomponent mixture containing several volatile compo­ nents, Eq. (10) is fulfilled for each of the volatile com­ ponents simultaneously. Note that, for a system in a gravitational field, the expression for the difference of the levels, 8 = is similar to the van't Hoff forMg x {h) j mula for osmotic pressure [7]. These two phenomena are similar in many respects. The van't Hoff equation is derived based on the condition of the equality of the mechanochemical potentials on either side of the semi­ permeable membrane. Equations (9) and (10) were

Л

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ABAKUMOV,

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obtained from the condition of constancy of the mechanochemical potential of the volatile component over the entire volume of the vessel, including the interface. Along with the ordinary osmotic methods [8], measure­ ments of the difference between the levels of the liquid in vessels communicating through the gas phase can be used to determine the molecular masses of macromolecules. Consider a rotor rotating at 60000 rpm. Let the external radius r of the rotor and the radius r of the liq­ uid in the upper section (section u) of the rotor be 10 and 1 cm, respectively. Figure 3d shows how the radius of the surface of the pure component in section d should depend on the mole fraction of the volatile component in section u in order to keep the system in thermody­ namic equilibrium. Experiments in centrifugal fields make it possible to measure the quantity 8 for substances with moder­ ate molecular masses in a wide range of concentra­ tions. An accurate calculation of the levels of the liquid in vessels communicating through the gas phase can be conducted based on the total number of moles of the volatile component and the distribution of the non­ volatile component between the sections. For simplic­ ity, the amount of the volatile component in the gas phase can be disregarded. Therefore, for the rotor dis­ played in Fig. 3c, the conditions of conservation of the number of moles of the volatile n and nonvolatile n components, in the case where the nonvolatile compo­ nent is present in both sections (n = n + n ) , take the form 0

h

x

2

2

x

Lb

jCj

u

2

d

(r)2nrdr

x

2Vi Jbxi

2

u

Jr (r)2nrdr

d

J bx* Лг) + r +6 h

Lb r 2,u

"

(ID

x (r)2nrdr

" 2vJbx (r)

2fU

"'

r

u

2td

h

h

h

h

h

t

Vessels Non-isomorphous

to the Field

The shape of the vessel (rotor) was demonstrated to be a parameter that determines the state of the system in a field [4, 5]. In the systems under consideration, the vessel-shape effect manifests itself through the depen­ dence of the state of the system on the configuration of the diaphragm that separates the liquid layer. Consider this effect for a cylindrical rotor with a conical diaphragm (Fig. 4). The conical surface divides the internal cavity of the rotor into two sections non-isomorphous to centrifugal fields. To start with, let us consider a binary mixture of vol­ atile components placed into cavities isolated from one another by a continuous diaphragm (exchange through the gas phase is impossible) (Fig. 4a). Let the sections be filled with such amounts of the solution as to make the radii of the surfaces of the liquids identical during rotation. In this case, the generatrices for the upper (u) r-0.5r and lower (d) sections are given by h (r) = 2L 0

u

°

d

filled with an equimolar iodomethane-pentane mixture

x (r)2nrdr 2d

JK d O 0

" 2V,

2

and h (r) = 2L——- , respectively. Let both sections be

h

Lb

[u

r

r

+ x (rY

hu

d

tions of the components over the radius, which are needed to calculate the integrals in formulas (11), are described by expression (7). If the initial composition of the system is known, the system of equations (10), (11) in combination with identities (8) is sufficient to calculate all the concentrations and levels. For example, for a binary mixture, there are six unknowns (x (r ), *\,d( h + S), x , ( r ) , x (r + 5), r , and 8) equal to the number of equations. If the quantities r and 8 are mea­ sured in an experiment, it is possible to determine some of the parameters of the components, for example, molecular masses M or the ratio of the densities of the components, which enter into the coefficient b. The cor­ responding equations for real solutions [6] should addi­ tionally contain isothermal compressibility, activity coefficients, and other parameters of the equation of state for liquid components.

2yU

r

2

FEDOSEEV

« * 1 > = ). +

*2, (r)' d

r +5 h

In this case, the conditions of conservation of the composition, which make it possible to describe x (r) and P(r) in both sections, read as t

where b = V\/V = Mjp^A^pi, r is the external radius of the rotor, r is the radius of the surface of the solution in section u, r + 8 is the radius of the surface of the solution in section d, and L is the height of the cylindri­ cal rotor, which is divided by the diaphragm in halves. The concentration of the volatile component at the surface of the solutions in the sections, x (r ) and i, d( h 8) and the difference between the levels of the solutions are related by Eq. (10), whereas the distribu­ 2

0

h

h

lu

x

r

+

h

_ b r

x (r)2nrh (r) hu

' Vjbx (r) hu

b г

u

dr - n 2, u

+ x (r) 2tU

x (r)2nrh (r) 2tU

vjbx (r) hu

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY

u

+ x (r)

dr,

2)}i

Vol. 78

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LIQUID-VAPOR EQUILIBRIUM IN A CENTRIFUGAL FIELD P,

Ax, m o l e fractions 1.0

477

atm

2000

r

Fig. 5. Radial dependence of (a) the concentration of pentane (Ax) and (b) the hydrostatic pressure p for the upper (dashed line) and lower (solid line) isolated sections of a rotor containing an equimolar mixture of iodomethane and pentane.

b_ r

x (r)2nrh (r) ld

Vi J bx (r) ud

d

+

x (r)

dr - n 2,6

2d

r +5 h

x (r)2nrh (r) dr. J bx (r) + x (r) 2td

V

{

hd

d

2d

r +5 h

The calculation results for 60000 rpm and 300 К are displayed in Fig. 5. Under these conditions, the pres­ sure and concentration differentials across the mem­ brane are AP(r) = 820 atmosphere and Ax(r) = 0.62, respectively. Along with the pressure and concentra­ tions, there are differences in the chemical and mechanochemical potentials of the component in the isolated sections. What are reasons for the arising pressure and con­ centration differentials? The simplest way to under­ stand this effect is to perform the following thought experiment in a strong uniform gravitational field. Con­ sider a vessel that has a shape of rectangular prism (Fig. 6) and is equipped with an impenetrable movable diaphragm capable of separating the vessel along the diagonal plane into two equal sections. When the dia­ phragm is introduced, a vessel isomorphous to the field is replaced by two sections non-isomorphous to the field. The field can be switched on and off. Let us con­ duct two experiments: (1) The vessel with the diaphragm removed is filled with equal volumes of pentane and iodomethane. The field is switched on (operation field in Fig. 6) and allowed to operate until equilibrium is attained. Then, RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY

Vol. 78

in the presence of the field, the diaphragm is introduced (operation separation). Clearly, the diaphragm pro­ duces no effect on the equilibrium distribution of the components. Therefore, at any height, the concentra­ tions of the components in the left and right sections are identical. After switching off the field, we will find that the compositions in the left and right sections are differ­ ent: they are enriched in pentane and iodomethane, respectively. (2) The vessel is filled with the same mixture, the diaphragm is introduced, and the field is switched on. Then, in the presence of the field, the pressure is mea­ sured at various heights in the left and right sections. We will find that, at all levels except the uppermost, the hydrostatic pressure in the left section is higher than that in the right one. In addition, at any level, the local concentration of pentane in the right section is higher than that in the left one. The upper (narrow) part of the right section is too small to accommodate all the pen­ tane that tends to float, and therefore, a part of it larger than that in the left section remains at lower levels. Note, however, that, as before, the ratio of the concen­ trations of pentane at the surface and at the bottom of both of the sections is described by Eq. (7). Note also that the overall compositions of the solutions in the sec­ tions remain unchanged. Let us now, without switching off the field, make an orifice near the bottom of the ves­ sel (operation puncture) and observe how the pressure in system changes. Within a certain period of time, the system attains a new equilibrium, at which the pres­ sures on either side of the diaphragm are equal. Imme­ diately after switching off the field, when the state of No. 3

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ABAKUMOV, FEDOSEEV

while the equal volume of the heavy component is transferred in the opposite direction. As a result, the ini­ tially equimolar mixture ((x^ ) = (x ^ ) = 0.5) placed in a rotating rotor (9 = 60000 rpm; T = 300 K) with sec­ tions communicating through the gas phase becomes depleted of pentane in section u (*с н - 0.35) and enriched (*с н = 0.76) in section d. d

x

u

5

5

12

12

Consider a binary mixture composed of n moles of a volatile component and n moles of a nonvolatile component. The generatrices of the upper (u) and lower (d) sections have the same form as depicted in Fig. 4b. The concentrations of the volatile component at the sur­ faces of the solutions in the different sections, x (r ) and x (r + 8) and the difference 8 in the levels of these surfaces are related by Eq. (10) if the nonvolatile component is present in both sections (n = n + n ), or by Eq. (9) if the nonvolatile component is present only in one of the sections {x ( r + 8) = x (r) = 1). x

2

lu

ld

h

h

2

x

d

h

2

u

2

d

{d

The quantities x (r ) and x (r + 8) are deter­ mined from the conditions of conservation of the com­ ponents for the case of rotor sections of different shapes lu

h

ld

h

Г x (r)2nrh (r) Jbx (r) + x , (r) hu

u

hu

Г x (r)2nrh (r) J bx (r) + x (r)

+

hd

Г

2 u

^

d

hd

j

%d

r +5

'

h

R

Q

2 u

"'

_ b_ г vJbx (r) hu

x {r)2nrh (r) + x (r) 2n

n

2tU

Г

'

R

H

b_ г Fig. 6. State of a liquid mixture depends on whether it was exposed to a strong uniform field before the separation of the vessel into isolated sections or after.

x (r)2nrh (r) 2d

d

r +S h

The functions h^r) and h (r) determine the shapes of the upper and lower sections. In our case, h (r) + h (r) = L, with the shapes of both sections unambiguously deter­ mined by the containing of the diaphragm. It is the shape of the sections that determines the composition of the solutions in them, the levels of the surface; i.e., the configuration of the diaphragm is one of the most important parameters that determine the thermody­ namic state and properties of the system. d

u

equilibrium is not yet attained, the composition in the sections coincides with the final composition in the first experiment. Let us modify the experiment with the cylindrical rotor by removing a part of the diaphragm above the level of the surface of the liquid phase (Fig. 4b). Since both the components of the mechanism are assumed to be volatile, we will obtain the result described above: at equilibrium, the local concentration, hydrostatic pres­ sure, and the levels of the surface on either side of the diaphragm are identical, as if the diaphragm is absent. Note, however, that if we rapidly stop the rotor and ana­ lyze the compositions in the sections, we will find that they differ. Despite the fact that the levels of the liquid in the sections remain unchanged, a part of the light component is transferred from section u into section d,

d

Equilibrium is established by means of the transfer of the components through the gas phase. A criterion of equilibrium in the presence of a field is the constancy of the mechanochemical potential (but not the chemical potential) of the volatile component at any point of the system. In this case, at a given radius, the mecha­ nochemical and chemical potentials of the nonvolatile component and the hydrostatic pressures on either side of the diaphragm differ. Any change in the position of the separating diaphragm is accompanied by changes in

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY

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LIQUID-VAPOR EQUILIBRIUM IN A CENTRIFUGAL FIELD these parameters and the equilibrium difference of the levels 8. Thus, liquid mixtures in centrifugal field exhibit a variety of behaviors, a feature associated exclusively with the thermodynamic properties of the system. Experimental results can be used to determine the important parameters of the components of solutions, as discussed above. Models of nonideal solutions may reveal new aspects in the behavior of the liquids. Let us briefly describe the behavior of two volatile partially mixable liquids in a centrifuge. Irrespective of the shape of the rotor's sections communicating through the gas phase and the initial distribution of the compo­ nents, the radii of the surfaces of the liquids in the sec­ tions and the radii of the interfaces between the liquid phases should be identical. This conclusion follows from the condition of equality of the mechanochemical potentials of the components at the interfaces between the phases.

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1. 2. 3. 4. 5. 6. 7.

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REFERENCES G. A. Abakumov and V. B. Fedoseev, Ross. Khim. Zh. 42 (3), 36(1998). L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Nauka, Moscow, 1976; Pergamon, Oxford, 1980). G. A. Abakumov and V. B. Fedoseev, Chem. Rev. 24, 41 (2000). G. A. Abakumov and V. B. Fedoseev, Dokl. Ross. Akad. Nauk 365, 608 (1999) [Doklady Phys. 44 (4), 205 (1999)]. G. A. Abakumov and V. B. Fedoseev, in Testing Materi­ als and Constructions (Interservis, Nizhni Novgorod, 2000), Vol. 2, p. 164 [in Russian]. G. A. Abakumov and V. B. Fedoseev, Dokl. Ross. Akad. Nauk 383, 661 (2002) [Doklady Phys. Chem. 383, 89 (2002)]. H. A. Lorentz, Les theories statistiques en thermodynamique: conferences faites au College de France en novembre 1912 (B. G. Teubner, Leipzig, 1916; RKhD, Izhevsk, 2001). A. A. Tager, Physical Chemistry of Polymers (Khimiya, Moscow, 1978) [in Russian].

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