## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 12. Mixing Liquids: Ionic Solutions

### 12.4. Strong and Weak Electrolytes

For strong electrolytes Bjerrum first made the proposition, now generally accepted, that complete dissociation is present [14]. NaCl in water provides an example. Generally, an electrolyte dissociates into *ν* = *ν*_{+} + *ν*_{−} ions according to

__(12.13)__

where *z*_{+} and *z*_{−} are the charges of M and X, respectively. Electroneutrality requires that

(12.14)

The chemical potential of the electrolyte with activities *a*_{+} and *a*_{−} is

__(12.15)__

with the activity *a*_{Q} defined by . As a reminder, the molality scale for component *i* is defined as *m _{i}*/

*m*

_{0}, with

*m*

_{0}= 1 kg mol

^{−1}. Frequently,

*m*

_{0}is omitted and we do so here (compare ln

*P*, which actually means ln (

*P*/

*P*°) with, say, the reference pressure

*P*° = 1 bar). The activity coefficient

__(on molality basis) is defined as__

^{5)}*γ*=

_{i}*a*/

_{i}*m*, and for the salt the activity becomes

_{i}(12.16)

using the mean activity coefficient , the mean stoichiometric coefficient and the fact that for an electrolyte of molality *m*_{Q} we have *m*_{+} = *ν*_{+}*m*_{Q} and *m*_{−} = *ν*_{−}*m*_{Q}. For a symmetric electrolyte, the above expression reduces to *a*_{Q} = (*γ*_{±}*m*_{Q})^{2}. For mixed electrolytes the full expression should be used. Similar equations can be written down using molarity *c* and mole fraction *x* instead of molality *m*. The mole fraction is, however, infrequently used in this connection.

For weak electrolytes, Arrhenius proposed in 1887 that dissociation is incomplete but that dissociation increases with increasing dilution. An example is a solution of acetic acid in water for which the dissociation equilibrium is

__(12.17)__

For the general electrolyte, __Eq. (12.13)__, the molarity of positive ions for a fully dissociated solution is *ν*_{+}*m*_{Q} with *m*_{Q} the molarity Q. With association this becomes *m*_{+} + *αν*_{+}*m*_{Q} with *α* the *degree of dissociation*. Because the total molarity of M^{+} ions, whether dissociated or not, is *ν*_{+}*m*_{Q}, we have

(12.18)

where *m*_{IP} denotes the molarity of the ion pairs. The total molarity of negative ions, whether dissociated or not, is *ν*_{−}*m*_{Q} so that

(12.19)

Since *γ*_{±} is given by and the chemical potential reads

(12.20)

we find, using __Eq. (12.15)__,

(12.21)

For *ν*_{+} = *ν*_{−} = 1, the activity coefficient reduces to (*αγ*_{±})* ^{ν}*. For the acetic acid example mentioned before, the dissociation constant

*K*is thus given by

__(12.22)__

For a solution of HA of molality *m* we have and

__(12.23)__

where the last step only can be made if we take *γ*_{±} = *γ*_{HA} = 1.

__Equation (12.23)__ is the *dilution law*, due to Ostwald in 1888, which expresses quantitatively that the lower the concentration, the larger the degree of dissociation. For acetic acid at 25 °C at infinite dilution (i.e., the limiting value), *K* ≅ 1.75 × 10^{−5} mol kg^{−1}, and does not deviate more than a few percent from this value up to 10^{−1} mol kg^{−1}. From the overall reaction, __Eq. (12.17)__, and the expression for *K*, __Eq. (12.22)__, the chemical potential becomes , where . An accurate empirical description for the dissociation constant *K* reads

(12.24)

In practice, one uses p*K*_{a} values, defined by p*K*_{a} ≡ −log*K* = *A*_{1}*/T* − *A*_{2} + *A*_{3}*T*. Values of the constants *A*_{1}, *A*_{2} and *A*_{3} for a few typical acids are listed in __Table 12.7__.

** Table 12.7** Constants for the p

*K*

_{a}expression values (mol kg

^{−1}) for some acids in H

_{2}O.

Van't Hoff found that the osmotic pressure of electrolyte solutions was always significantly higher than was predicted by colligative theory. For example, for the osmotic pressure equation Van't Hoff proposed *Π* = *icRT*, where *i* is now known as the *van't Hoff factor*. In principle this factor is equal to, but dependent on concentration, in practice usually less than the number of ions formed from one mole of compound. If, for a symmetric electrolyte 1 mole provides *ν* ions upon complete dissociation, the number of really dissociated molecules is *να* and the number of undissociated molecules is 1 − *α*. Hence, the total number of ions in solution *i* = 1 − *α* + *να* or *α* = (*i* − 1)/(*ν* − 1). The value for *α* obtained from colligative properties is only in approximate agreement with values determined via conductivity measurements.

Problem 12.10

Show that, if −*R* ln *K*_{a} = *A/T* − *C* + *DT*, the *K*_{a} value shows a maximum given by *R* ln *K*_{a} = *C* − 2(*AD*)^{1/2} at temperature *T*_{max} = (*A*/*D*)^{1/2}.

Problem 12.11

Calculate Δ*H*, Δ*S* and Δ*C _{P}* from Δ

*G*for the dissociation of a weak electrolyte.