SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

CHAPTER 21 SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

In most circumstances of interest to chemists, the dominant experimental variables are temperature, pressure, and composition, and our attention has been concentrated on the dependence of a transformation on these factors. On some occasions, however,

a transformation takes place in a field: gravitational, electrical, or magnetic; chemists who work with macromolecules frequently use a centrifugal field in their work. It behooves us, therefore, to see how we can approach such problems. As a gravitational field is the most familiar in common experience, we shall focus initially on some representative problems in this area.

21.1 DEPENDENCE OF THE GIBBS FUNCTION ON EXTERNAL FIELD

In our exposition of the properties of the Gibbs function G (Chapter 7), we examined systems with constraints on them in addition to the ambient pressure. We found that changes in Gibbs function are related to the maximum work obtainable from an isothermal transformation. In particular, for a reversible transformation at constant pressure and temperature [Equation (7.79)],

dG T ,P ¼ DW net

Chemical Thermodynamics: Basic Concepts and Methods, Seventh Edition . By Irving M. Klotz and Robert M. Rosenberg Copyright # 2008 John Wiley & Sons, Inc.

SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

where DW is the net useful (non-PV) reversible work associated with the change in Gibbs function.

Of course, the equality in Equation (7.79) is symmetric; that is, the equation may be read in the mirror-image direction: If we perform reversible (non-PV) work DW net on a system at constant pressure and temperature, we increase its Gibbs function by the amount dG T,P . For example, if we reversibly change the position x of a body in the gravitational field of the earth [Fig. 21.1(a)], we perform an amount of work given by

(21 :1) where m is the mass of the body, g is the gravitational acceleration, and x is positive in

DW net ¼ mgdx

the upward direction. It follows then from Equation (7.79) that

@G ¼ mg ¼ force exerted on body that moves it against gravitational field @x T ,P

(21 :2) If we consider lowering a body down a shaft [Fig. 21.1(b)] it is convenient to change

our convention regarding the positive direction of x to downward. Hence,

Figure 21.1. Reversible processes in a gravitational field.

501 More generally, for constraints other than gravity, we can also state that

21.1 DEPENDENCE OF THE GIBBS FUNCTION ON EXTERNAL FIELD

(21 :4) Consequently, it follows that

DW net in field ¼ ( force exerted against field) dx ; Fdx

@G ¼ force exerted against field ; F

(21 :5) @x T ,P

For a system of constant composition in which fields are absent, we found in Chapter 7 that because the Gibbs function G is a function of pressure and temperature,

G ¼ f (T, P)

we can write for the total differential [Equation (7.42)]

Subsequently, when we examined systems in which composition, as well as T and P, can be varied (but fields are still absent or constant), we found [Equation (9.2)] that

G ¼ f (T, P, n 1 ,n 2 , ...,n i )

where n 1 ,n 2 , . . . are the moles of the respective components. So the total differential now becomes [Equation (9.3)]

@n i T ,P,n j

Now let us remove the constraint of a fixed field. To be concrete, let us move some unit of material from one position in the gravitational field of the earth to another. Under these circumstances, the Gibbs function G also depends on x, the position in the field, so we may write for the most general circumstances

(21 :6) Consequently it follows that the total differential should be expressed as @G

G ¼ f (T, P, n i ,n 2 , ...,n i , x)

@n i T ,P,n j ,x

@x T ,P,n j

SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

and, from Equations (9.4) – (9.6), and (21.2) dG Xm i dn i þ mgdx

21.2 SYSTEM IN A GRAVITATIONAL FIELD Let us now analyze some specific systems. First, we examine a column of pure fluid

perpendicular to the surface of the earth [Fig. 21.1(c)] and at equilibrium. In this case, it can be shown by the following argument that the pressure within the fluid varies with position in the gravitational field.

The column of pure fluid [Fig. 21.1(c)] is at a constant temperature, and the external pressure on it is constant. Thus, for the column of fluid

(21 :9) for any transfer of fluid from one level to another.

dG ¼0

Let us now analyze the contributions to dG if we take one mole of the pure fluid in the column at position x and move it to the position x þ dx. At each level within the column, the pressure is different (as the weight of fluid above it is different), although it remains fixed at each level. Hence, as the temperature and the composition remain fixed, when a unit of pure fluid is being moved from one position to another, Equation (21.8) can be written as

(21 :10) where G m and V m denote values of the respective properties per mole and M is the

dG ¼ dG m ¼V m dP þ Mgdx ¼ 0

molar mass. From this equation, we conclude that

where r is the density of the fluid. Thus, the pressure in the fluid is a function of x, and for the column rising from the surface of the earth [Fig. 21.1(c)], the pressure decreases as the distance x above the surface increases.

If the pure fluid were in the shaft extending below the surface [Fig. 21.1(b)], our analysis would correspond in every detail to that in the column above the surface except that the negative sign in Equation (21.11) would be replaced by a positive sign, because x is positive in the direction of the gravitational field. Thus, the pressure within the fluid would increase as the depth x down the shaft increases.

Let us examine now a column of a solution in the shaft of Figure 21.1(b). For simplicity, we shall assume only one dissolved solute is in a single-component solvent. If equilibrium has been attained, we find that the molality m of solute varies with the depth, and we can derive an analytic expression for this dependence of molality on depth.

503 Consider the transfer of one mole of solute from one position x in the column at

21.2 SYSTEM IN A GRAVITATIONAL FIELD

equilibrium to another position x þ dx. The transfer of the solute, in a column of very large cross section, does not change the molality at any position. The fluid in the shaft is at equilibrium, its temperature is invariant, and the external pressure on it is fixed; hence, dG ¼ 0. A mole of solute in solution has a molar Gibbs function G m2 . If the solute is moved from one position x to another, x þ dx, it could undergo a change in Gibbs function dG m2 resulting from the difference in pressure in the fluid, because of the change of position in the gravitational field, and as a result of any change in molality of solute at different levels. As, in our thought experiment, this transposition

is the only change being made in the system, we can write 1 in place of Equation (21.7), using ln m in place of n i ,

(21 :12) From Equation (15.11), we can obtain the following relation:

@G

m2 ¼ RT ln (m=m8) þ G8 m2 ¼ RT (21 :13) @ ln m P ,x

m2

From an equation analogous to Equation (9.25), it follows (see Exercise 4 of Chapter 9) that

@G m2

¼V m2 ¼M 2 v 2 (21 :14)

@P ln m,x

where V m2 is the partial molar volume of the solute, v 2 is the partial specific volume of the solute, and M 2 is its molar mass. The dependence of Gibbs function G on the gravitational field is expressed in Equation (21.3), which can be converted to

1 In the absence of a field, G m and m are identical. However, in the presence of a gravitational (or other) field, that identity no longer is valid because of historical reasons. As defined by Gibbs (J. W. Gibbs,

The Collected Works of J. Willard Gibbs , Vol. 1, Longmans, Green and Co., New York, 1928, pp. 144– 150), the chemical potential m is not a function of position x in a field. On the other hand, as used by G. N. Lewis (G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, New York, 1923, pp. 242– 244), the partial molar Gibbs function G m includes the energy associated with position x in a field. For this reason we have expressed our derivations in terms of G m . In a gravitational field (over dis- tances for which g is essentially constant), G m and m are related by the equation G m ¼ m þ Mgx. Some thermodynamicists define a “total chemical potential” m total or “gravitochemical potential” as the sum of

m þ Mgx; hence, they are essentially using G m .

SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

because M 2 and g are constants in the situation being analyzed. Recognizing that at equilibrium dG ¼ 0, and substituting Equations (21.13) to (21.15) into Equation

(21.12), we find [see Equation (21.11)]

2 v 2 rgdx If we integrate Equation (21.16) from x ¼ 0 to x ¼ d, the result is

¼M 2 gdx

m at depth d

m at surface

where d is the depth below the surface. Thus, whether an increase or decrease in molality occurs at depth d, in comparison with the surface, is determined by the factor (1 2 v 2 r). If v 2 r . 1, the molality of solute will decrease with increasing depth. On the other hand, if v 2 r , 1, the molality of solute will increase with increasing depth. Let us illustrate this phenomenon with a practical example, the variation of oxygen and of nitrogen equilibrium solubilities with depth in the ocean [1]. For seawater, the density r depends on temperature and salinity, and it could vary from 1.025

2 ¼ 0.97 cm g 21 in seawater at a water temperature near 258C. If d is expressed in meters, then at the lower limit of the water density, Equation (21.17) becomes

to 1.035 g cm 23 . For dissolved oxygen, v

log 10 d (21 :18)

m surface

Thus, for example, at a depth of 1000 m, and a density of 1.025 g cm 23 , the solubility of oxygen is 1.0007 times the solubility at the surface. If the density of the seawater is as high as 1.035, then the solubility of oxygen at a depth of 1000 m is 1 /1.007 times

the solubility at the surface, assuming that v 2 is 0.97 cm 3 g 21 . On the other hand, the situation with nitrogen is markedly different. Here, vari- ations in salinity and temperature have little effect on the factor (1 2 v 2 r) because the v

2 of nitrogen, 1.43 cm g 21 , is relatively so large. Thus, for nitrogen, Equation (21.17) becomes

RT ln

d (21 :19)

m surface

At a depth of 1000 m, the solubility of nitrogen decreases by 5 – 6%. In contrast to oxygen, the (equilibrium) solubility of nitrogen always decreases progressively with depth.

21.3 SYSTEM IN A CENTRIFUGAL FIELD

21.3 SYSTEM IN A CENTRIFUGAL FIELD Near the surface of the earth, the gravitational acceleration g is essentially constant.

For contrast, let us turn our attention next to a centrifugal field, where the acceleration is very sensitive to the distance from the center of rotation.

The centrifugal force F c at a distance r from the axis of rotation (Fig. 21.2) is

(21 :20) where m is the mass of the entity being centrifuged and v is the angular velocity.

F c ¼ mv 2 r

Thus, by an analysis similar to that presented for Equation (21.5), we find that

@G

@r P ,T,n i

because m and v are constant. This relation is analogous to that in the gravitational field, with the angular acceleration v 2 r replacing the gravitational acceleration g.

Figure 21.2. (a) Schematic diagram of a sedimentation apparatus for determination of mole- cular mass of a solute molecule. The sector-shaped container is actually mounted in a rotor that

spins about an axis of rotation A at an angular velocity v. The centrifugal field, F c at any point is directed along the axis r in the direction of increasing r. At any position r, the chord length is fr, which increases with increasing r. The position of the liquid meniscus is indicated by a. (b) Concentration distribution of solute in cell at sedimentation equilibrium.

SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

The negative sign on the right-hand side occurs because the Gibbs function decreases as r increases; that is, work would have to be performed on the sedimenting particle to bring it back from an axial position at large r to a position at small r.

If we now examine a cell (Fig. 21.2) containing a pure liquid (rather than a solution as shown), then again (as in the case of a gravitational field) we can show that the pressure in the pure fluid varies with position in the centrifugal field. Starting with Equation (21.8) we obtain

(21 :22) in place of Equation (21.10) for the gravitational field, where V m is the molar volume

V m dP

2 r dr ¼0

of the pure liquid. Thus, we conclude that

Therefore, the ambient pressure within the fluid in the cell in Figure 21.2 increases with increasing distance from the axis of rotation.

Finally, we consider the behavior of a solute in a solution in the cell subjected to the centrifugal field. At a suitable angular velocity, the tendency of the solute to sedi- ment toward the bottom of the cell is countered by its tendency to diffuse backward toward the meniscus, because the concentration increases with increasing r, as indicated in Figure 21.2(b). At some time, a sedimentation equilibrium is attained.

A typical equilibrium concentration distribution is depicted in Figure 21.2(b). Our aim is to find a quantitative analytical expression for this curve. We consider a transfer at constant temperature of an infinitesimal amount of any single solute i from a position r in the cell in the centrifugal field to a second position r þ dr. For this transfer at equilibrium, at constant T and external P, dG ¼ 0, so we write in place of Equation (21.12),

@c i P ,c k ,r (21 :24)

We distinguish between c i , the concentration of solute whose distribution we are focusing on, and the c k ’s, the concentrations of other solutes, because this type of multicomponent system is of frequent practical interest. Since the development by Svedberg [2], the ultracentrifuge has been used widely to determine the molecular weight of a macromolecule from its concentration distribution at equilibrium. The large molecule, natural or synthetic, which may be designated by i, is dissolved frequently in an aqueous solution containing other solute species k to buffer the solution or to provide an appropriate ionic strength.

507 For the individual terms and factors on the right-hand side of Equation (21.24), we

21.3 SYSTEM IN A CENTRIFUGAL FIELD

may insert the following substitutions:

where v i is the partial specific volume of the solute;

dP ¼ rv 2 rdr

(21 :27) @G mi

(21 :28) @c i

Although we shall carry along the term in Equation (21.28) for the variation of ln g i with c i —for in practice the macromolecule concentration may cover a wide range from meniscus to bottom of the cell (Fig. 21.2)—we shall assume that the change in ln g i of the macromolecule with change in concentration of other solutes c k in the solution is negligible to a good approximation. Within these specifications, Equation (21.24) can be reduced to

1 @ ln g

0 ¼M i i v i rv 2 rdr þ RT þ dc i

i v 2 rdr (21 :31)

@c i

which in turn can be converted into

An alternative form is

This equation suggests that a convenient graphical representation of the concentration distribution of species i would be one plotting ln c i versus r 2 . Three representative possible curves are illustrated in Figure 21.3.

SYSTEMS SUBJECT TO A GRAVITATIONAL OR A CENTRIFUGAL FIELD

Figure 21.3. Concentration distribution of solute in solution at sedimentation equilibrium. Curve A represents ideal behavior of a monodisperse solute; curve B represents nonideality; and curve C represents a polydisperse system.

If the solute i is monodisperse—that is, if no dissociation or aggregation of the (macro)molecules occurs and each one has exactly the same molecular weight at every position in the cell—then M i is the same for all macromolecular species in the solution. If, furthermore, these solute molecules do not interact with each other—that is, if they behave ideally—the term @ ln g i /@c i ¼ 0. Under these circum-

stances, ln c i varies linearly with r 2 , as shown in line A of Figure 21.3. If the molecu- lar weight of species i is unknown, it can be determined from the slope of line A, because Equation (21.33) becomes

Equilibrium ultracentrifugation has played a crucial role in establishing the molecular weights of protein molecules on an ab initio basis [3,4], that is, without requiring calibration with macromolecules of known molecular weight.

Should the macromolecules interact with each other, then @ ln g i /@c i does not vanish. In actual experience, its value is almost always positive, largely because of excluded volume effects. Then, c i [ @ ln g i /@c i ] will then increase in magnitude as

c i increases and r decreases. Thus, the downward curvature shown in curve B of Figure 21.3 is typical of nonideal behavior. It is also possible to observe upward curvature in a plot of ln c i versus r 2 , as in curve C of Figure 21.3. This curvature occurs when the macromolecules are polydis- perse, that is, when they possess a range of molecular weights. Common sense tells us, in this case correctly, that the heavier species in the class i will congregate toward the bottom of the cell. As the slope depends on M i , curve C will become steeper as we move toward the bottom of the cell, where r is greater.

EXERCISES

A mathematical analysis of equilibrium behavior in a polydisperse system leads to the conclusion that from the slope at any point on curve C, we can obtain a weight average molecular weight at that local concentration of solute i. Several software programs are available for carrying out the necessary calculations [5].