Activities and Activity Coefficients

The accuracy of our expression for cell potential can be increased by including the full activity corrections, rather than the approximations used earlier. The complexity of the calculations, however, increases significantly. The treatment below assumes that the reader has been exposed to the concepts of activity and fugacity. If these are new to you, you may want to learn more about them from a book on physical chemistry or chemical equilibrium (see Further Reading section at the end of the chapter). The fugacity is related to the chemical potential and is used as a surrogate for the chemical potential in phase-equilibrium calculations. The activity of a species is defined as the ratio of the fugacity of species i to the fugacity of pure i at the standard state:

(2.41)

Note that the activity is dimensionless.

For our analysis of electrochemical systems, we need the activity of solid, gas, solvent, and solute species. For solid species, the standard state is typically taken as the pure species. In this text, we will consider only pure solid species. Therefore, the fugacity is equal to the pure component fugacity, f_i, and

(2.42)

since the solid is in its standard state.

For the gas-phase species, the standard state fugacity is an ideal gas at a pressure of 1 bar. The fugacity of species i in a mixture is defined as

(2.43)

where  is the fugacity coefficient of species i in a mixture, y_i is the mole fraction of the component in the gas phase, and p is the total pressure. For the purposes of this book, we assume that , which is equivalent to assuming ideal gas behavior. With this assumption, the activity of the gas is

(2.44)

Remember,  is dimensionless, and the pressure units in the numerator cancel out with those of the standard state fugacity, p^o. However, since the numerical value of  is 1 bar, sometimes it is left off the standard state when writing the activity. Do so with great care.

Electrochemical systems include an electrolyte, which is a material in which current flows due to the movement of ions. A common liquid electrolyte consists of a solvent (e.g., water) into which one or more salts are dissolved to provide the ionic species. For electrolyte solutions, molality (m_i = moles solute i per kg solvent) is the most commonly used form of expressing the composition when dealing with nonideal solutions and activities, and hence will be used here. Molality is convenient from an experimental perspective because it depends only upon the masses of the components in the electrolyte solution, and does not require a separate determination of density. The temperature dependence of the density may also introduce error when dealing with concentration rather than molality. The relationship between molality and concentration is

(2.45)

where the subscript “0” refers to the solvent and M is the molecular weight. The summation in the denominator is simply the total mass of solute species per volume of solution.

We first consider the activity of the solvent. Since the concentration of the solvent is usually much higher than that of the dissolved solute species, the standard state fugacity is that of the pure solvent at the same pressure and phase of the system (e.g., liquid water).

(2.46)

In most cases, the vapor pressures are sufficiently low that the fugacity can be approximated by the pressure, as shown. An osmotic coefficient is usually used to express the activity of the solvent as follows:

(2.47)

where the summation is over ionic species and does not include the solvent. M_solvent is the molecular weight of the solvent and  is the osmotic coefficient. For a single salt (binary electrolyte), Equation 2.47 becomes

(2.48)

where m is the molality of the salt and ν is defined by Equation 2.54. When the ion concentration is zero, the right sides of Equations 2.47 and 2.48 both go to zero, and the solvent activity is equal to unity as expected. For our purposes in this text, we will assume unit activity for the solvent, unless otherwise specified.

We now turn our attention to the activity of the solute, which is the activity correction that is most frequently of concern in electrolyte solutions. The activity of a single ion is defined as

(2.49)

where  is the single-ion activity coefficient (unitless). Again, we have left off the standard state in the final expression, since it has a value of one. Remember, however, that the activity is dimensionless.

Where does the standard state come from, and why is it used? What is meant by an ideal solution in this context? A thought experiment is useful for answering these questions. Think about an ion in solution. That ion will interact with the solvent (e.g., water molecules) and with other ions of all types in the solution. However, as the ion concentration in the electrolyte approaches zero, only the ion–solvent interactions are important. Under such conditions, the fugacity of the ion is equal to . In other words, the fugacity of the ion depends only on the amount present. This is because the ions interact only with the solvent, and the nature of the interactions does not change with molality as long as ion–ion interactions remain insignificant (i.e., as long as changing interactions do not contribute to the fugacity). We define an ideal solution as a solution in which only ion–solvent interactions are important. Such behavior is approximated in real systems as the concentration approaches small values, and is seen as a linear asymptote in a plot of the activity as a function of composition.

The standard state defined in Equation 2.49 represents the fugacity of an ideal solution where only ion–solvent interactions are important at a concentration of one molal. This state is clearly hypothetical because ion-ion interactions are important in real electrolytes at this concentration. However, it is a convenient reference state and is widely used. The activity coefficient is used to account for deviations from this ideal state due to ion–ion interactions, including complex formation in solution. From the above, it follows that

(2.50)

There is, however, a practical problem with the single-ion activity and activity coefficient. Electrolyte solutions are electrically neutral and solutions containing just a single ion don’t exist. Therefore, single-ion activities cannot be measured, although they can be approximated analytically under some conditions (see Section 2.15).

To address this issue, we define a measureable activity that is related to the single-ion activities defined above. A single salt containing ν₊ positive ions and ν₋ negative ions dissociates as follows to form a binary electrolyte:

(2.51)

The activity of the salt in solution is

(2.52)

where the + and − subscripts are introduced for convenience to represent the positive and negative ions of the salt, respectively. We now define a mean ionic activity in terms of the single-ion activities:

(2.53)

where

(2.54)

Similarly, we define the mean ionic activity coefficient in terms of the single-ion activity coefficients

(2.55)

Assuming complete dissociation, the molality of the neutral salt in solution (moles of salt per kg solvent) is related to the molality of the individual ions according to

(2.56)

Combining Equations 2.52−2.56 with the definition of the single-ion activity coefficient (Equation 2.49) yields the following for the salt in solution:

(2.57)

Consistent with the above, the following limits are reached at low concentrations:

(2.58)

Equations 2.51–2.57 apply to a single salt in solution or binary electrolyte and permit activity corrections without requiring single-ion activity coefficients. Several systems of practical importance, such as Li-ion batteries and lead–acid batteries, have either binary electrolytes or electrolytes that can be approximated as binary. The mean ionic activity coefficient, , has been measured for a number of binary solutions, and the results have been correlated and can be found in the literature.

The activity relationships above provide a practical, measurable way to include activity corrections for binary electrolytes. Measurements for binary systems are typically fit to models in order to provide the needed activity coefficients in an accessible, usable way. Under some conditions, prediction of activity coefficients is possible. These issues are discussed briefly in the section that follows.