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2. Thermodynamics of electrolyte solutions

2.3. Activity

2.3.1 Activity of non-electrolytes

In the expression of the chemical potential (2.6), activity ai is introduced. It describes the ‘effective’ concentration of a solute, i.e. its deviation from ideal behavior. Depending on the concentration scale, the chemical potential can be written in different ways:

\( \mu_i=\mu_i^{0,x}+RT\ln(f_ix_i) \)   mole fraction scale

(2.13a)

\( \mu_i=\mu_i^{0,m}+RT\ln(y_im_i) \) molality scale

(2.13b)

\( \mu_i=\mu_i^{0,c}+RT\ln( \gamma _ic_i) \) molarity scale

(2.13c)

The molality and molarity scales are identical in practice, except at very high concentrations. We will mainly use the latter one; γi is the activity coefficient and ci the concentration of the solute. It can be proven that the relation between the mole fraction and molarity scales is

\( \mu_i^{0,x}=\mu_i^{0,c}+RT\ln\overline{V} \),

(2.14)

where \( \overline{V} \) is molar volume of the solvent. Equation (2.14) is valid at relatively dilute solution where the ratio of activity coefficients fi :γi ≈ 1. The chemical potential of the solvent is usually expressed on the mole fraction scale because when the mole fraction of the solvent x0 → 1 also f0 → 1, and the chemical potential approaches its standard value (Raoult’s law). For the solute fi → 1 when xi → 0 (Henry’s law). The standard state of the molality scale is a hypothetical solution where the molality of the solute is m* = 1.0 mol/kg and yi = 1. The standard state of the molarity scale is also a hypothetical c* = 1.0 mol/dm3 and γi = 1. The chemical potentials should more likely be written

\( \mu_i=\mu_i^{0,m}+RT\ln(y_im_i/m^{ \ast} ) \) and \( \mu_i=\mu_i^{0,c}+RT\ln( \gamma _ic_i/c^{ \ast} ) \)

(2.15)

but for the sake of simplicity we leave the standard concentrations out and keep in mind that concentrations are written in mol/kg or mol/dm3 units.

 

2.3.2 Activity of electrolytes

Let’s consider a strong electrolyte \( \text{A}_{\upsilon+}\text{B}_{\upsilon-} \)that dissociates into ions Az+ and Bz-.  The electroneutrality  condition now becomes \( \upsilon_+z_++\upsilon_-z_-=0 \). The chemical potential of the electrolyte is the sum of the ionic ones:

\(\mu_{\pm}=\upsilon_+\tilde{\mu_+}+\upsilon_-\tilde{\mu_-}=\upsilon_+(\mu_+^0+RT\ln a_{+}+z_+F \phi)+\upsilon_-(\mu_-^0+RT\ln a_{-}+z_-F \phi)=\upsilon_+\mu_+^0+\upsilon_-\mu_-^0+RT\ln[(a_+)^{\upsilon_+}(a_-)\upsilon^{\upsilon_-}]= \mu_{\pm}^0+RT\ln(a_{\pm}^{\upsilon}) \)

(2.16)

Equation (2.16) defines the mean electrolyte activity a± and how it depends on the stoichiometric coefficients; \( \upsilon=\upsilon_++\upsilon_- \) . Since activity = concentration × activity coefficient,

\(a_{\pm}^{\upsilon}=(c_+^{\upsilon_+}\gamma_+^{\upsilon_+})(c_+^{\upsilon_-}\gamma_-^{\upsilon_-})=(\upsilon_+c\gamma_+)^{\upsilon_+}(\upsilon_-c\gamma_-)^{\upsilon_-}=c^{\upsilon}(\upsilon_+^{\upsilon_+}\upsilon_-^{\upsilon_-})(\gamma_+^{\upsilon_+}\gamma_-^{\upsilon_-})=c_{\pm}^{\upsilon}\gamma_{\pm}^{\upsilon} \) (2.17)

Equation (2.17) defines the mean electrolyte concentration c± and the mean activity coefficient γ±. It can be measured, unlike the ionic activity coefficients.

\( c_{\pm}=c(\upsilon_+^{\upsilon_+}\upsilon_-^{\upsilon_-})^{1/\upsilon}\)    ;    \( \gamma_{\pm}=(\gamma_+^{\upsilon_+}\gamma_-^{\upsilon_-})^{1/\upsilon} \)
 (2.18)

For the estimation of the ionic activity coefficients, the following convention can be used [1]

\( \displaystyle\sum\limits_{i}\frac{c_i}{z_i}\ln\gamma_i=0 \) (2.19)

For a 1:1 electrolyte (e.g. NaCl) it applies \( (\upsilon_+=\upsilon_-=z_+=-z_-=1) \)
\( \ln\gamma_+-\ln\gamma_-=0 \Rightarrow \gamma_+=\gamma_-=\gamma_{\pm} \) (2.20)

for 2:1 electrolyte (e.g. CaCl2, \( \upsilon_+=1, \upsilon_-=2, z_+=+2,z_-=-1 \)),

 \( \frac{1}{2}\ln\gamma_+=2\ln\gamma_- \Rightarrow \gamma_+=\gamma_{\pm}^2 \) and \( \gamma_-=\gamma_{\pm}^{1/2} \) (2.21)

and for a 1:2 electrolyte (e.g. Na2SO4, \( \upsilon_+=2, \upsilon_-=1,z_+=+1,z_-=-2 \)),
\( 2\ln\gamma_+=\frac{1}{2}\ln\gamma_- \Rightarrow \gamma_-=\gamma_{\pm}^2 \) and \( \gamma_+=\gamma_{\pm}^{1/2} \) (2.22)

Biosystems always contain polyelectrolytes such as proteins, polysaccharides, polypeptides or DNA. A polyelectrolyte is a large-sized molecule that can have the molar weight as high as millions and hundreds charged groups. Polyelectrolytes have counter-ions, typically H+, Na+ or Cl- that compensate its charge. Assume a polyelectrolyte Na100P in the concentration c, i.e. a polyanion P100- with 100 Na+ cations making it electroneutral. From Equations (2.17) and (2.18) it would be obtained that

\( a_{\pm}(100^{100} \cdot1^1)^{1/101}c\gamma_{\pm} \approx95.5c\gamma_{\pm} \)
 (2.24)

If the mean activity a± is measured, say, with the osmotic pressure or electrochemically, the mean activity coefficient \( \gamma_{\pm} \) can be calculated using the equation above. Without discussing this issue in greater detail, we are content to note that the activity of the counter-ion is usually much lower than its stoichiometric concentration because they do not exist in the solution as free ions due to extensive ion-binding to the polyanion. The mean activity coefficient of the polyelectrolyte can reach absurd values if Equation (2.18) is applied as such. Further discussion of this question is beyond the scope of this book.


[1] W.E.Morf, The principles of ion-selective electrodes and of membrane transport, Elsevier, Amsterdam 1981.


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