A simple derivation procedure of equations about a mean activity, a mean activity coefficient, and a mean molar concentration was proposed based on the electrochemical potentials. The equations for the electrolytes, such as KCl, CaCl₂, and LaCl₃, were derived.

Keywords: mean activity; mean activity coefficient; mean molar concentration; electrochemical potentials; strong electrolytes; average potential

It is difficult to understand a mean activity ($a\pm$ ), a mean activity coefficient ($y\pm$ ), and a mean molar concentration ($C\pm$ ) described in several books.^1-3 In addition to this, many students seem to hardly derive their expressions and it is hard to do them for me too. As an idea, the use of electrochemical potentials (μ̅)^2-4 can be effective. We try here to introducing a brief derivation procedure about their equations.

Case (A): for 1: 1 strong electrolyte

We handle the aqueous solution of C mol/L KCl as this example. First, each component is expressed with the electrochemical potential μ̅ [4].

${\overline{\mu }}_{+}={\mu }^0{}_{+}+RT1\text{n}{a}_{+}+F{\varphi }_{+}$    (1)

${\overline{\mu }}_{-}={\mu }^0{}_{-}+RT1\text{n}{a}_{-}-F{\varphi }_{-}$    (2)

Here, the symbols, ${\mu }^0{}_{+},a_{+}$ and ${\varphi }_{+}$ , denote the standard chemical potential for K⁺, the activity of K⁺ in water, and the inner potential of the phase, respectively. Also the same is true of $C{l}^{-}$  in Eq. (2) and additionally R, T, and F show the usual meanings. Secondly, from the two equations, we calculate an average potential (or energy), $\left({\overline{\mu }}_{+}+{\overline{\mu }}_{-}\right)/2$ , for all the components in this KCl solution as follows.

$\left({\overline{\mu }}_{+}+{\overline{\mu }}_{-}\right)/2=\left({\mu }^0{}_{+}+{\mu }^0{}_{-}\right)/2+\left(RT/2\right)\ln a+a-=\left({\mu }^0{}_{+}+{\mu }^0{}_{-}\right)/2+RT\ln {\left(a+a-\right)}^{1/2}$    (3)

Here, the condition of the electroneutrality for the phase corresponds to $F{\varphi }_{+}-F{\varphi }_{-}=0$ . In Eq. (3), we can define (${\left(a+a-\right)}^{1/2}$  as the mean activity and accordingly do $a+a-$ ^1,2 as the activity (aKCl) of the electrolyte B, namely KCl. Moreover, using the relations,^1-3 $a+=y+C$  and $a-=y-C$ , for the individual ions, the $a_{\pm }$  is expressed as

$a_{\pm }=a_{KC1}{}^{1/2}={\left(a+a-\right)}^{1/2}={\left(y+C\cdot y-C\right)}^{1/2}={\left(y+y-\right)}^{1/2}C,$    (4)

where y+ and y- refer to the activity coefficients of the cation K⁺ and the anion $C{l}^{-}$ , respectively. Finally, from Eq. (4), we can define ${\left(y+y-\right)}^{1/2}$ as the mean activity coefficient $y\pm$  and C as the mean molar concentration $C\pm$  for aB at B = KCl.

Case (B): for 2: 1 electrolyte

Similarly, we handle the aqueous solution of C mol/L CaCl2 as this example. Expressing each component with μ̅, the following equations were obtained.

${\overline{\mu }}_{2+}={\mu }^0{}_{2+}+RT1\text{n}{a}_{2+}+2F{\varphi }_{+}$    (5)

${\overline{\mu }}_{-}={\mu }^0{}_{-}+RT1\text{n}{a}_{-}+F{\varphi }_{-}$     (2)

Next, from these equations, we estimate the average electrochemical potential, $\left({\overline{\mu }}_{2+}+2{\overline{\mu }}_{-}\right)/3$ , of this CaCl₂ solution as

$\left({\overline{\mu }}_{2+}+2{\overline{\mu }}_{-}\right)/3=\left({\mu }^0{}_{2+}+2{\mu }^0{}_{-}\right)/3+\left(RT/3\right)\ln a_{2+}{\left(a-\right)}^2+a\left(2F{\varphi }_{+}-2F\varphi -\right)/3$     (6)

Also, we can define $\{a_{2+}\left(a-\right){{}^2\}}^{1/3}$ as $a_{\pm }$  and do $a_{2+}{\left(a-\right)}^2$  as the activity(aCaCl₂) of the electrolyte CaCl₂.^1,2 Here, the electroneutral condition of $2F{\varphi }_{+}-2F{\varphi }_{-}$ basically holds.       So, the $a_{\pm }$  is expressed as

$a_{\pm }=\{a_{2+}\left(a-\right){{}^2\}}^{1/3}=\{y_{2+}C\cdot \left(y-2C\right){{}^2\}}^{1/3}={\left(y_{2+}y{-}^2\right)}^{1/3}\left(4^{1/3}C\right)$    (7)

From this equation, we can immediately define $\left(y_{2+}y{-}^2\right)$  as $y\pm$  and 4^1/3C as $C\pm$  for aB at B = CaCl₂.

Case (C): for 3: 1 electrolyte

Let’s handle C mol/L LaCl₃ solution. Expressing each component with μ̅, the following equations were obtained in addition to Eq. (2).

${\overline{\mu }}_{3+}+{\overline{\mu }}_{3+}+RT\ln a_{3+}+3F{\varphi }_{+}$    (8)

From these equations, we calculate the average $\overline{\mu },\left({\overline{\mu }}_{3+}3\overline{\mu }-\right)/4$ , of this LaCl₃ solution as

$\left({\overline{\mu }}_{3+}+3\overline{\mu }-\right)/4=\left({\mu }^0{}_{3+}+3{\mu }^0-\right)/4+\left(RT/4\right)\ln a_{3+}{\left(a-\right)}^3+\left(3F{\varphi }_{+}-3F\varphi -\right)/4$     (9)

Also, we can define $\{a_{3+}\left(a-\right){{}^3\}}^{1/4}$ as $a_{\pm }$  and do $a_{3+}{\left(a-\right)}^3$ as the activity of the electrolyte LaCl₃,^1,2where the condition of $3F{\varphi }_{+}=3F\varphi -$ holds.   Hence, the $a_{\pm }$ is expressed as

$a_{\pm }=\{a_{3+}\left(a-\right){{}^3\}}^{1/4}=\{y_{3+}C\cdot \left(y-3C\right){{}^3\}}^{1/4}={\left(y_{3+}y{-}^3\right)}^{1/4}\left({27}^{1/4}C\right)$     (10)

From this equation, we can immediately define ${\left(y_{3+}y{-}^3\right)}^{1/4}$ as $y\pm$  and ${27}^{1/4}C$ as $C\pm$ for aB at B = LaCl₃.

A similar handling can be applied for other electrolytes, such as ZnSO₄, Na₂SO₄, and K₄[Fe(CN)₆]. In handling these electrolytes and those of the cases (B) and (C), it was assumed that all the electrolytes are strong ones. Also, its procedure is: (i) calculate the average electrochemical potential of the electrolyte, (ii) estimate  from its potential, and then (iii) obtain or define both $y\pm$  and $C\pm$ from rearranging $a_{\pm }$ . Table 1 summarizes such results, together with the above three cases.² At least, the results in Table 1 were the same as those in the book.² Thus, by estimating the average potentials of the electrolytes B, the mathematical styles about their activities aB were essentially derived. Except for the 1: 1 and 2 : 2 electrolytes, it is not still easy to understand physical and chemical meanings of their expressions. However, we can suppose that the $C\pm$  value is equivalent to a geometrical mean (Table 1) of the molar concentration which is based on the average μ̅ of the electrolyte.

Considering the average potential of the electrolyte B, we can easily derive the equations for the mean activity $a_{\pm }$ , the mean activity coefficient $y\pm$ , and the mean molar concentration $C\pm$ . The proposed handling must be applied for other concentration scales, such as the molal concentration and the mole fraction.

The author thanks Dr Hideaki Shirota (Chiba University) for his support about the book.³

The author declares there is no conflict of interest.

1. RA Robinson, RH Stokes. “Electrolyte Solutions. 2nd Revised Ed., Dover Publications, Inc., Mineola, New York, 2002, p. 24‒29.
2. R Tamamushi. “Denki Kagaku (Electrochemistry)”. 2nd Ed., Tokyo Kagaku-Dojin, Tokyo, 1991, p. 8‒12 & 26‒29.
3. PW Atkins. “Physical Chemistry”, Fourth Ed. Oxford University Press, Oxford, 1990, p. 249‒250 & 909‒911.
4. AJ Bard, LR Faulkner. “Electrochemical Methods: Fundamentals and Applications”, 2nd Ed., John Wiley & Sons, Inc., New York, 2001, p. 60‒62.