DEFINITIONS AND STANDARD STATES FOR DISSOLVED ELECTROLYTES

19.1 DEFINITIONS AND STANDARD STATES FOR DISSOLVED ELECTROLYTES

Uni-univalent Electrolytes As a plot of the activity of an electrolyte such as aqueous HCl against the first power

of m 2 /m8 gives a limiting slope of zero, we might examine graphs in which the activity is plotted against other powers of the molality ratio. Such a plot is shown in Figure 19.2, in which the activity of aqueous HCl is plotted against the square of the molality ratio. The curve has a finite, nonzero limiting slope.

This result suggests that the appropriate form of the limiting law for uni-univalent electrolytes (such as HCl) is

a HCI

m lim (m =m8) 2 ¼1

Figure 19.2. Activity as a function of the square of the molality ratio for aqueous HCl. The data are the same as for Figure 19.1.

19.1 DEFINITIONS AND STANDARD STATES FOR DISSOLVED ELECTROLYTES

Figure 19.3. A plot of the ratio of activity to the molality ratio to test for the validity of Henry’s law.

which is a modified form of Henry’s law. To test for agreement with this form of Henry’s law, we plot the ratio a

2 against (m /(m/m8) 2 /m8) , as in Figure 19.3. If followed in experimenrtally accessible dilute solutions, Henry’s law would be

manifested as a horizontal asymptote in a plot such as Figure 19.3 as the square of the molality ratio goes to zero. We do not observe such an asymptote. Thus, the modified form of Henry’s law is not followed over the concentration range that has been examined. However, the ratio of activity to the square of the molality ratio does extrapolate to 1, so that the data does satisfy the definition of activity [Equations (16.1) and (16.2)]. Thus, the activity clearly becomes equal to the square of the molality ratio in the limit of infinite dilution. Henry’s law is a limiting law, which is valid precisely at infinite dilution, as expressed in Equation (16.19). No reliable extrapolation of the curve in Figure 19.2 exists to a hypothetical unit molality

ratio standard state, but as we have a finite limiting slope at m 2 /m 2 8 ¼ 0, we can use that slope to guide us to the standard state; the appropriately modified Henry’s law line is shown in Figure 19.2. No real state of the system corresponds to the stan- dard state, but the properties of the standard state can be calculated from the Henry’s- law constant.

Thus far, we have not introduced any assumptions about the dissociation of electrolytes in order to describe their experimental behavior. As far as thermo- dynamics is concerned, such details need not be considered. We can take the limiting law in the form of Equation (19.1) as an experimental fact and derive thermodynamic relationships from it. Nevertheless, in view of the general applicability of the ionic theory, it is desirable to relate our results to that theory.

For example, the empirical relation between the activity and the molality ratio can be understood on the assumption that the chemical potential of the electrolyte is the sum of the chemical potentials of the constituent ions. That is, for HCl as the solute,

m HCI ¼m H þ þm CI (19 :2)

ACTIVITY, ACTIVITY COEFFICIENTS, AND OSMOTIC COEFFICIENTS

If we apply Equation (16.1), which is the definition of the activity, to Equation (19.2), the result is

m8 HCI þ RT ln a HCI ¼ m8 H þ þ RT ln a H þ þ m8 CI þ RT ln a CI (19 :3) We can also assume that

m8 HCI ¼ m8 H þ þ m8 CI (19 :4) so that

(19 :5) The individual ion activities should follow the limiting relations

HCI!0 m CI =m8

so the product of the limits is

(a H þ )(a CI )

m lim

¼1 HCI !0 (m H þ =m8)(m CI =m8) m HCI !0 (m HCI =m8) 2

¼ lim

HCI

which is the relation found empirically. No way exists within thermodynamics to determine the activity of a single ion because we cannot vary the concentration of a single ion while keeping the amounts of the other ions constant, because electroneutrality is required. As a þ

and a 2 approach m 2 at infinite dilution for a uni-univalent electrolyte, a þ must equal a 2 at infinite dilution. However, at any nonzero concentration, the difference between a þ and a 2 is unknown, although it may be negligibly small in dilute sol- ution. Nevertheless, in a solution of any concentration, the mean activity of the ions can be determined. By the mean activity a + , we refer to the geometric mean, which for a uni-univalent electrolyte is defined by the equation

a + ¼ (a þ a ) 1 =2 ¼a 1 2 =2 (19 :8) We can also define an activity coefficient g i for each ion in an electrolyte solution.

For each ion of a uni-univalent electrolyte,

g þ ¼ m þ =m8

443 and

19.1 DEFINITIONS AND STANDARD STATES FOR DISSOLVED ELECTROLYTES

m =m8

These individual-ion activity coefficients have the desired property of approaching 1 at infinite dilution, because each ratio a i /(m i /m8) approaches 1. However, individual- ion activity coefficients, like individual-ion activities, cannot be determined experimentally. Therefore, it is customary to deal with the mean activity coefficient

g + and the mean activity a + which for a uni-univalent electrolyte can be related to measurable quantities as follows:

where a + is given by Equation (19.8). From Equations, (19.1), (19.8), and (19.10), we can see that

Multivalent Electrolytes Symmetrical Salts. For salts in which anions and cations have the same valence,

activities and related quantities are defined in exactly the same way as for uni- univalent electrolytes. For example, for MgSO 4 , a finite limiting slope is obtained when the activity is plotted against the square of the molality ratio. Furthermore, m þ equals m 2 . Consequently, the treatment of symmetrical salts does not differ from that just described for uni-univalent electrolytes.

Unsymmetrical Salts. As an example of unsymmetrical salts, let us consider a salt such as BaCl 2 , which dissociates into one cation and two anions. By analogy with the case of a uni-univalent electrolyte, we can define the ion activities by the expression

a 2 ¼ (a þ )(a )(a )

(19 :12) In this case, the mean ionic activity a + also is the geometric mean of the individual-

¼ (a )(a ) 2 þ

ion activities:

a + ¼ [(a )(a ) 2 1 =3

] ¼a 2 (19 :13)

ACTIVITY, ACTIVITY COEFFICIENTS, AND OSMOTIC COEFFICIENTS

It is desirable that the individual-ion activities approach the molality ratio of the ions in the limit of infinite dilution. That is,

It follows from Equations (19.13) and (19.14) that

¼ [(m =m8)(2m

2 2 =m8) ]

(19 :15) It is also desirable that the mean ionic activity coefficient g + approach unity in

¼4 1 =3 (m 2 =m8)

the limit of infinite dilution. We can achieve this result if, as in Equation (19.9), we define

(a )(a ) 2 1 þ =3 ¼ (m 2 =m8)(2m 2 =m8) 2

(4 )(m 2 =m8)

Then, from Equations (19.15) and (19.16),

m lim g + 2!0 ¼1

445 For a uni-univalent electrolyte [Equation (19.10)]

19.1 DEFINITIONS AND STANDARD STATES FOR DISSOLVED ELECTROLYTES

g + ¼ m 2 =m8

To achieve a uniform definition of g + for all electrolytes, it is convenient to define a

mean molality m + (for BaCl 2 , for example) as m + ¼ [(m + )(m ) 2 ] 1 =3 ¼ [(m 2 )(2m 2 ] 1 2 =3 )

¼4 1 =3 m 2 (19 :17) With this definition, the relationship for the mean activity coefficient

m + =m8

holds for any electrolyte. It follows from Equation (19.13) and Equation (19.15) that

(4m 2 =m8)

Equation (19.19) is consistent with the empirical observation that a nonzero initial slope is obtained when the activity of a ternary electrolyte such as BaCl 2 is plotted against the cube of (m 2 /m8). As the activity in the standard state is equal to 1, by definition, the standard state of a ternary electrolyte is that hypothetical state of unit molality ratio with an activity one-fourth of the activity obtained by extrapolation

of dilute solution behavior to m 2 /m8 equal to 1, as shown in Figure 19.4.

Figure 19.4. Establishment of the standard state for a ternary electrolyte. Data from R. N. Goldberg and R. L. Nuttal, J. Phys. Chem. Ref. Data 7, 263 (1978).

ACTIVITY, ACTIVITY COEFFICIENTS, AND OSMOTIC COEFFICIENTS

General Case. If an electrolyte A v þ B v dissociates into v þ positive ions of charge Z þ and v 2 negative ions of charge Z 2 the general definitions for the activities and the activity coefficients are

2 ] þ (v m ) ] 1 2 =v (19 :22) The appropriate limiting law that is consistent with experimental observation is

and, as before,

m + =m8

Table 19.1 summarizes the empirical expression of the limiting law and the definitions of the ionic activities, molality ratios, and activity coefficients for a few substances and for the general case of any electrolyte.

Mixed Electrolytes In a solution of mixed electrolytes, the presence of common ions must be considered

when calculating the mean molality. For example, in a solution in which m NaCl ¼ 0.1 and m MgCl2 ¼ 0.2, the mean molality m + , for NaCl is

m + NaCl ¼ [(m

Na þ )(m Cl )]

¼ [(0:1)(0:5)] 1 =2 ¼ 0:244 mol kg

and for MgCl 2

m + MgCl

2 ¼ [(m Mg 2 þ )(m Cl ) ]

¼ [(0:2)(0:5) 2 ] 1 =3 ¼ 0:368 mol kg

TABLE 19.1. Thermodynamic Functions for Dissolved Solutes Sucrose

NaCl

Na 2 SO 4 AlCl 3 MgSO 4 A v þ B v a 2 1 1 4 27 1 v v þ þ v v 2

a 2 a sucrose

(a þ )(a 2 )

(a þ )(a ) 2

2 (a þ )(a 2 ) 3 (a þ )(a

(a þ ) þ (a 2 ) 2 m +

[(a þ )(a 2 )] 1 /2

[(a þ ) 2 (a 1 /3

[(a þ )(a 2 ) ]

[(a þ )(a 2 )]

1 /2 vv

m 2 4 m 2 27 1 /4 m 2 m 2 [(V þ ) v þ (V 2 ) v 2 ] 1 v m 2 g +

m + =m8

m + =m8

m + =m8

m + =m8

m + =m8

ACTIVITY, ACTIVITY COEFFICIENTS, AND OSMOTIC COEFFICIENTS

TABLE 19.2. Relationships between Ionic Strength and Molality Salt

NaCl Na 2 SO 4 AlCl 3 MgSO 4 A v þ B v Ionic

m 2 3m 2 6m 2 4m 2 1 v v

þ1 z 2 Strength m 2 v 2

Thus, when calculating the mean molality of an electrolyte in a mixture, we must use the total molality of each ion, regardless of the source of the ion.

Both on the basis of empirical data and on the grounds of electrostatic theory, it has been found convenient to introduce a quantity known as the ionic strength when considering the effects of several electrolytes on the activity of one of them.

The contribution of each ion to the ionic strength I is obtained by multiplying the molality of the ion by the square of its charge. One half the sum of these contributions for all ions present is defined as the ionic strength. That is,

The factor one half has been included so that the ionic strength will be equal to the molality for a uni-univalent electrolyte. Thus, for NaCl,

[m 2 (1) 2 þm 2 (1) 2 ]

2 ¼m 2

However, for BaCl 2 ,

because m þ ¼m 2 and m 2 ¼ 2m 2 , in which m 2 is the molality of the electrolyte. Several examples, together with a general formulation, are shown in Table 19.2.