1.10.6: Gibbs Energies- Solutions- Pairwise Solute Interaction Parameters
A given solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) and \(\mathrm{m}_{\mathrm{j}}\) moles of solute \(j\). The chemical potential of the solvent water is related to \(\mathrm{m}_{\mathrm{j}}\) using equation (a) where pressure \(\mathrm{p}\) is close to the standard pressure, \(\mathrm{p}^{0}\)
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1} \nonumber \]
Here \(\mu_{1}^{*}(\ell)\) is the chemical potential of solvent water at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\phi\) is the practical osmotic coefficient which is unity for a solution having thermodynamic properties which are ideal; \(\mathrm{M}_{1}\) is the molar mass of water. The chemical potential of the solute \(\mathrm{j}\), \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{\mathrm{j}}\) using equation (b).
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
The chemical potentials of solute and solvent are linked by the Gibbs-Duhem equation which for aqueous solutions (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) containing \(1 \mathrm{~kg}\) of water takes the following form.
\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0 \nonumber \]
We draw equations (a) and (b) together in an equation for the Gibbs energy of a solution prepared using \(1 \mathrm{~kg}\) of solvent water. Then [1],
\[\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
Or,
\[\begin{aligned}
\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned} \nonumber \]
If the thermodynamic properties of the solution are ideal, both \(\phi\) and \(\gamma_{\mathrm{j}}\) are unity.
\[\begin{aligned}
\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \\
&+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned} \nonumber \]
The difference between \(\mathrm{G}(\mathrm{aq})\) and \(\mathrm{G}(\mathrm{aq} ; \mathrm{id})\) is the excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) [2];
\[\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
A related quantity is the excess molar Gibbs energy \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq}) \quad\left\{=\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right\} \nonumber \]
. Then [3]
\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
The dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) on \(\mathrm{m}_{\mathrm{j}}\) emerges from equation (g).
\[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
But, from the Gibbs-Duhem equation,
\[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=-\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right] \nonumber \]
Then [4]
\[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \nonumber \]
Equation (k) relates \((1-\phi)\) to the dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) on molality \(\mathrm{m}_{\mathrm{j}}\). The relationship between \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) and \(\gamma_{\mathrm{j}}\) is given by equation (l) [5].
\[\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{dm}_{\mathrm{j}}\right] \nonumber \]
At this point we make a key extrathermodynamic assumption. We assert that (at fixed temperature and pressure) the excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) is related to molality \(\mathrm{m}_{\mathrm{j}}\) of neutral solute \(\mathrm{j}\) using equation (m). Thus
\[\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{g}_{\mathrm{ij}}+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \ldots \ldots \nonumber \]
Here \(g_{j j}, g_{j j j} \ldots\) are the coefficients in a virial type of equation. Thus \(g_{j j}\) measures the contribution of pairwise solute-solute interactions to \(\mathrm{G}^{\mathrm{E}} (\mathrm{aq})\); \(g_{j j j}\) is a triplet interaction term. For quite dilute solutions the dependence of \(\mathrm{G}^{\mathrm{E}} (\mathrm{aq})\) on \(\mathrm{m}_{\mathrm{j}}\) is effectively described by the pairwise term, \(g_{j j}\).
\[\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
Here \(g_{j j}\) is expressed in \(\left[\mathrm{J kg}^{-1}\right]\), being the (Gibbs) energy of interaction in a solution containing \(1 \mathrm{~kg}\) of water. Pairwise solute-solute Gibbs energy interaction parameters are characteristic of solute \(j\), temperature and pressure.
At this stage we have not defined either the sign or magnitude of \(g_{j j}\). Clearly if pairwise solute-solute interactions are attractive/cohesive, both \(g_{j j}\) and \(\mathrm{G}^{\mathrm{E}}\) are negative. In the next stage of the analysis we use equation (l) to obtain an equation for \(\ln \left(\gamma_{j}\right)\) in terms of \(g_{j j}\) and molality \(\mathrm{m}_{\mathrm{j}}\). Thus [6]
\[\ln \left(\gamma_{\mathrm{j}}\right)=[2 / \mathrm{R} \, \mathrm{T}] \, \mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Hence equation (o) requires that (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) if \(g_{j j}\) is negative \(\ln \left(\gamma_{\mathrm{j}}\right)\) decreases with increase in \(\mathrm{m}_{\mathrm{j}}\) whereby \(\mu_{j}(\mathrm{aq})<\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\). We anticipate that the sign and magnitude of \(g_{j j}\) reflect the hydration characteristics of the two solute molecules because these characteristics determine the impact of cosphere overlap on the properties of the solution.
We turn to the properties of the solvent. Thus [7]
\[(1-\phi)=-(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Or,
\[\phi=1+(1 / R \, T) \, g_{j} \,\left(m^{0}\right)^{-2} \, m_{j} \nonumber \]
From the equation for \(\left[\mu_{1}(\mathrm{aq})-\mu_{1}(\mathrm{aq} ; \mathrm{id})\right]\), the difference in chemical potentials of solvent water in real and ideal solutions, it follows that negative \(g_{j j}\) requires that \(\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq} ; \mathrm{id})\), the solvent in the real solution being at a higher chemical potential than in the corresponding ideal solution. In other words the non-ideal properties of the solvent are also related to the pairwise interaction parameter \(g_{j j}\) and \(\mathrm{m}_{\mathrm{j}}\).
As a check on the procedures described above we draw the equations together to recover the original equation for \(\mathrm{G}^{\mathrm{E}} (\mathrm{aq})\).
\[\begin{aligned}
&\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \\
&=\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[-(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{jj}} \,\left(1 / \mathrm{m}^{0}\right)^{2} \, \mathrm{m}_{\mathrm{j}}+(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}} \,\left(1 / \mathrm{m}^{0}\right)^{2}\right] \\
&=\mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}
\end{aligned} \nonumber \]
Thus for dilute solutions both \(\phi\) and \(\operatorname{ln}\left(\gamma_{\mathrm{j}}\right)\) are linear functions of \(\mathrm{m}_{\mathrm{j}}\). Equation (n) forms the basis for understanding the properties of dilute aqueous solutions where the solutes are non-ionic [8]. The underlying theme is the idea that solute -solute interactions in these solutions can be understood in terms of cosphere- cosphere interactions [9,10]. Description of the properties of real solutions based on equation (a) is closely related to descriptions of dilute solutions developed for metallurgical systems [11,12]. Similarly procedures are discussed using site-site pair correlation functions for molecular interaction energies [13] and using quasi-chemical models [14].
Footnotes
[1] \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{W}_{1}=1 \mathrm{~kg}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\)
[2] \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,[\mathrm{I}]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\)
[3] \(\mathrm{G}_{\mathrm{m}}{ }^{\mathrm{E}}(\mathrm{aq})=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{I}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)
[4]
\[\begin{gathered}
(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)-\left(\mathrm{l} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right] \\
(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)-\mathrm{d}(1-\phi)-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \\
(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{j}\right]=-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}}
\end{gathered} \nonumber \]
[5] From equation (g), \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)\right]\)
\[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{dm}_{\mathrm{j}}=(1-\phi)-\mathrm{m}_{\mathrm{j}} \,\left(\mathrm{d \phi} / \mathrm{dm}_{\mathrm{j}}\right)+\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}} \nonumber \]
But from the Gibbs-Duhem equation, \(\mathrm{d}\left[\mathrm{m}_{\mathrm{j}} .(1-\phi)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)=0\)
Or,
\[-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+(1-\phi) \, \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{j}\right)=0 \nonumber \]
Or,
\[-\mathrm{m}_{\mathrm{j} \,} \,\left(\mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}\right)+(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}=0 \nonumber \]
[6]
\[\begin{aligned}
&\ell \mathrm{n}\left(\gamma_{j}\right)=(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] / \mathrm{dm}_{\mathrm{j}} \\
&\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)=(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}
\end{aligned} \nonumber \]
Or, \([1]=[1] \,\left[\mathrm { J } \mathrm { mol } ^ { - 1 } \mathrm { K } ^ { - 1 } \, \left[\mathrm{K}^{-1} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-2} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right.\right.\)
[7] From, \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\) But \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{g}_{i \mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\)
Then, \(\mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \mathrm{R} \, \mathrm{T}=\mathrm{m}_{\mathrm{j} \,}(1-\phi)+(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\)
Then \((1-\phi)=-(1 / R \, T) \, g_{i j} \,\left(m^{0}\right)^{-2} \, m_{j}\)
[8] M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Ann. Rep. Progr. Chem., Sect. C, Phys. Chem., C, 1990, 87 ,45.
[9] R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[10] See for example, W. G. McMillan Jr. and J. E. Mayer, J. Chem. Phys., 1945, 13 , 276.
[11] L. S. Darken, Trans.Metallurg. Soc.AIME, 1967, 239 , 80.
[12]
- R. Schuhmann, Metallurg. Trans.B, 1985, 16 , 807.
- A. D. Pelton and C. W. Bale, Metallurg. Trans.,A,1986, 17 ,1211.
- S. Srikanth, K. J. Jacob and K. P. Abraham,Steel Research, 89 ,6.
[13] R. P. Currier and J. P. O'Connell, Fluid Phase Equilib., 1987, 33 , 245.
[141 J. Abusleme and J. H. Vera, Can. J. Chem. Eng., 1985, 63 ,845.