Ionic Solutions

To express the chemical potential of an electrolyte in solution in terms of molality, let us use the example of a dissolved salt such as magnesium chloride, (MgCl_{2}).

[MgCl_{2}rightleftharpoons Mg^{2+}+2Cl^{-} label{1}]

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We can now write a more general equation for a dissociated salt:

[M_{nu+}X_{nu-}rightleftharpoonsnu_{+}M^{z+}+nu_{-}X^{z-} label{2} ]

where (nu_{pm}) represents the stoichiometric coefficient of the cation or anion and (z_pm) represents the charge, and M and X are the metal and halide, respectively.

The total chemical potential for these anion-cation pair would be the sum of their individual potentials multiplied by their stoichiometric coefficients:

[mu=nu_{+}mu_{+}+nu_{-}mu_{-} label{3} ]

The chemical potentials of the individual ions are:

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[mu_{+} = mu_+^{circ}+RTln m_+ label{4} ]

[mu_{-} = mu_-^{circ}+RTln m_- label{5} ]

And the molalities of the individual ions are related to the original molality of the salt m by their stoichiometric coefficients

[m_{+}=nu_{+}m]

Substituting Equations (ref{4}) and (ref{5}) into Equation (ref{3}),

[ mu=left( nu_+mu_+^{circ}+nu_- mu_-^{circ}right)+RTlnleft(m_+^{nu+}m_-^{nu-}right) label{6} ]

since the total number of moles (nu=nu_{+}+nu_{-}), we can define the mean ionic molality as the geometric average of the molarity of the two ions:

[ m_{pm}=(m_+^{nu+}m_-^{nu-})^{frac{1}{nu}}]

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then Equation (ref{6}) becomes

[mu=(nu_{+}mu_{+}^{circ}+nu_{-}mu_{-}^{circ})+nu RTln m_{pm} label{7} ]

We have derived this equation for a ideal solution, but ions in solution exert electrostatic forces on one another to deviate from ideal behavior, so instead of molarities we must use the activity a to represent how the ion is behaving in solution. Therefore the mean ionic activity is defined as

[a_{pm}=(a_{+}^{nu+}+a_{-}^{nu-})^{frac{1}{nu}}]

where

[a_{pm}=gamma m_{pm} label{mean}]

and (gamma_{pm}) is the mean ionic activity coefficient, which is dependent on the substance. Substituting the mean ionic activity of Equation (ref{mean}) into Equation (ref{7}),

[mu=(nu_{+}mu_{+}^{circ}+nu_{-}mu_{-}^{circ})+nu RTln a_{pm}=(nu_{+}mu_{+}^{circ}+nu_{-}mu_{-}^{circ})+RTln a_{pm}^{nu}=(nu_{+}mu_{+}^{circ}+nu_{-}mu_{-}^{circ})+RT ln a label{11}]

when (a=a_{pm}^{nu}). Equation (ref{11}) then represents the chemical potential of a nonideal electrolyte solutions. To calculate the mean ionic activity coefficient requires the use of the Debye-Hückel limiting law, part of the Debye-Hückel theory of electrolytes .

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