1.1.5: Activity of Solvents

Classically, the colligative properties of non-ionic solutions were used to determine the molar mass of solutes. For example, the depression \(\Delta \mathrm{T}_{f}\) of the freezing point of water \(\mathrm{T}_{f}\) at a given molalilty \(\mathrm{m}_{j}\) of solute-\(j\) yields an estimate of the relative molar mass of the solute \(\mathrm{M}_{j}\). Key thermodynamic assumptions require that (a) on cooling only pure solvent separates out as the solid phase and (b) the thermodynamic properties of the solution are ideal. The key relationship emerges from the Schroder- van Laar equation [1]. The common assumption is that the thermodynamic properties of the solution are ideal. If the properties of a given aqueous solutions are determined to a significant extent by solute-solute interactions, a measured relative molar mass will be in error. Indeed McGlashan[2] was dismissive of the procedures based on Beckmann’s apparatus for the determination of the relative molar mass of solute using freezing point measurements.

The chemical potential of water in an aqueous solution, \(\mu_{1}(\mathrm{aq})\) at temperature \(\mathrm{T}\) and pressure \(p\) (assumed to be close to the standard pressure, \(p^{0}\)) is related to the molality of solute \(j\), \(\mathrm{m}_{j}\) using equation (a) where \(\mathrm{R}\) is the gas constant, \(\phi\) is the practical osmotic coefficient and \(\mathrm{M}_{1}\) is the molar mass of water, \(0.018015 \mathrm{ kg mol}^{-1}\) where \(\mu_{1}^{*}(\lambda)\) is the chemical potential of water(\(\lambda\)) at the same \(\mathrm{T}\) and \(p\).

\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]

Chemical potential \(\mu_{1}(\mathrm{aq})\) at temperature \(\mathrm{T}\) is also related to \(\mu_{1}^{*}(\lambda)\) using equation (b) where \(a_{1}\) is the activity of water in the aqueous solution.

\[\mu_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mu_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)\]

Comparison of equations (a) and (b) shows that \(\ln \left(a_{1}\right)\) is related to the molality of solute \(\mathrm{m}_{j}\) using equation (c).

\[\ln \left(a_{1}\right)=-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]

For a solution having thermodynamic properties which are ideal, the practical osmotic coefficient is unity.

\[\text { Then, } \ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]

Hence for a solution having thermodynamic properties which are ideal, \(\ln \left(a_{1}\right)\) is a linear function of molality \(m_j\), the plot having slope, \(-(\mathrm{M}_{1})\). Equation (d) forms a reference for a consideration of the properties of real solutions. For a solution having thermodynamic properties which are ideal, the solvent, water in an aqueous solution is at a lower chemical potential than the pure liquid. This observation is at the heart of the terms ‘depression of freezing point’ and ‘elevation of boiling point’. In the event that the thermodynamic properties of a given solution are not ideal then the form of the plot showing \(\ln \left(a_{1}\right)\) as a function of molality \(\mathrm{m}_{j}\) is determined by \(\phi\) which is, in turn, a function of \(\mathrm{m}_{j}\). The dependence of \(\phi\) on \(\mathrm{m}_{j}\) for a given solute in aqueous solutions (at fixed \(\mathrm{T}\) and \(p\)) is not defined ‘a priori’.

Bower and Robinson[3] report the dependence of osmotic coefficients for urea (aq) at 298 K over the range \(0 \leq \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1} \leq 20.0\); \(\phi\) decreases with increase in \(\mathrm{m}_{j}\). Similarly Stokes and Robinson [4] report the dependence of \(\phi\) on solute molality for sucrose(aq), glucose(aq) and glycerol(aq) over the range \(0 \leq \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1} \leq 7.5\).

For \(\mathrm{m}(\text { urea })=8 \mathrm{~mol} \mathrm{~kg}^{-1}\), \(\ln \left(a_{1}\right) \text { equals }-12 \times 10^{-2}\) whereas \(\ln \left(a_{1}\right)^{i d}\) equals approx. \(-15 \times 10^{-2}\). At this molality for urea(aq), \(\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq})^{\mathrm{id}}\) indicating that water in urea(aq) at this molality is at a higher chemical potential than would be the case for a solution where the thermodynamic properties are ideal. On the other hand for the hydrophilic solute sucrose where \(\mathrm{m}(\text { sucrose })=6 \mathrm{~mol} \mathrm{~kg}^{-1}, \ln \left(\mathrm{a}_{1}\right) \text { is }-15 \times 10^{-2}\) whereas \(\ln \left(a_{1}\right)^{i d}\) equals approx. \(-11 \times 10^{-2}\) indicating that adding sucrose at this molality to water lowers the chemical potential of water relative to that for a solution having ideal properties.

For a dilute solution of simple neutral solutes the difference between ideal and real properties can be understood [5,6] in terms of the dependence of pairwise Gibbs energy interaction parameters \(g_{jj}\) on molality using equation (e) where \(\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\); the units of \(g_{jj}\) are \(\mathrm{ J kg}^{-1}\).

\[1-\phi=-(1 / R \, T) \, g_{i j} \,\left(1 / m^{0}\right)^{2} \, m_{j}\]

Using equation (c),

\[\ln \left(a_{1}\right)=-M_{1} \, m_{j} \,\left[1+(R \, T)^{-1} \, g_{i j} \,\left(m^{0}\right)^{-2} \, m_{j}\right]\]

or

\[\ln \left(a_{1}\right)+M_{1} \, m_{j}=-M_{1} \,(R \, T)^{-1} \, g_{i j} \,\left(m^{0}\right)^{-2} \,\left(m_{j}\right)^{2}\]

Hence for dilute solutions \(\left[\ln \left(a_{1}\right)+M_{1} \, m_{j}\right]\) is a linear function of \(\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\), the gradient of the plot yielding the pairwise Gibbs energy interaction parameter \(g_{jj}\). If, for example, \(g_{jj}\) is positive indicating solute-solute repulsion, \(\left[\ln \left(\mathrm{a}_{1}\right)+\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\) decreases with increase in \(\mathrm{m}_{j}\) such that \(\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq} ; \mathrm{id})\). In the event that solute-solute interactions are attractive, \(g_{jj}\) is negative. Hence the difference between the properties of real and ideal solutions is highlighted by the contrast between equations (d) and (f).

The analysis described above is readily extended to aqueous solutions containing two solutes; e.g. urea(aq) + sucrose(aq), [7] and glucose(aq) + sucrose(aq).[8]

Salt Solutions

A given aqueous salt solution contains a single salt j; µ1(aq) and µj(aq) are the chemical potentials of water and salt respectively in the closed system. For water,

\[\text { For water, } \mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{v} \, \mathrm{m}_{\mathrm{j}}\]

\[\text { And } \mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)\]

Here \(ν\) is the stoichiometric parameter, the number of moles of ions produced by complete dissociation of one mole of salt \(j\); for a 1:1 salt, \(ν\) equals 2. According to equations (h) and (i),

\[\ln \left(a_{1}\right)=-\phi \, M_{1} \, v \, m_{j}\]

\[\ln \left(a_{1}\right)^{\text {id }}=-M_{1} \, v \, m_{j}\]

\[\text { If we confine attention to } 1: 1 \text { salts, } \ln \left(a_{1}\right)^{\text {id }}=-2 \, M_{1} \, m_{j}\]

With increase in \(\mathrm{m}_{j}\), \(\ln \left(a_{1}\right)^{i d}\) decreases linearly. With reference to equation (j), a ) \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\). With dilution of a salt solution, a plot of ln(a1) against mj approaches a linear dependence.

\[\text { For the salt } \mathrm{j}, \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]

Here \(ν_{+}\) and \(ν_{-}\) are the number of moles of cations and anions respectively produced by one mole of salt \(j\); \(v=v_{+}+v_{-} ; \gamma_{\pm}\) is the mean ionic activity γ coefficient of salt \(j\). By definition, \(\mathrm{Q}^{\mathrm{v}}=\mathrm{v}_{+}^{\mathrm{v}(+)} \, \mathrm{V}_{-}^{\mathrm{v}(-)}\). Also \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0\) at all \(\mathrm{T}\) and \(p\); \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) is the chemical potential of salt \(j\) in a solution where \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) and the thermodynamic properties of the solute are ideal; i.e. no ion-ion interactions.

For a 1:1 salt (e.g. \(\mathrm{KBr}\)),

\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]

For a 1:1 salt where the thermodynamic properties of the solution are ideal,

\[\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]

According to the Debye-Huckel Limiting Law, for very dilute solutions,

\[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}\]

where \(S_{\gamma}=f\left(T, p, \varepsilon_{r}\right)\) and \(\varepsilon_{\mathrm{r}}\) is the relative permittivity of the solvent at the same \(\mathrm{T}\) and \(p\).

\[\text { Further }[8,9], 1-\phi=\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]

At 298.15 K and ambient pressure [8], \(\mathrm{S}_{\gamma}=0.5115\). In other words,

\[\phi^{\mathrm{dh} l l}=1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]

Hence using equation (j), for a 1:1 salt,

\[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{dhll}}+2 \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\left[2 \,\left(\mathrm{S}_{\gamma} / 3\right) \, \mathrm{M}_{1} \,\left(\mathrm{m}^{0}\right)^{-1 / 2}\right] \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}\]

Then \(\ln \left(a_{1}\right)^{\text {dhll }}\) indicates that for a salt solution, molality \(\mathrm{m}_{j}\), \(\ln \left(a_{1}\right)\) exceeds that in the corresponding salt solution having ideal thermodynamic properties . In other words the activity of the solvent, water, is enhanced above that for water having ideal thermodynamic properties. For very dilute solutions \(\left[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{dhll}}+2 \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\) is a linear function of \(\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}\). However other than for very dilute solutions equation (q) is inadequate and so a more sophisticated equation is required relating \(\phi\) and \(\mathrm{m}_{j}\). Extensive compilations of \(\phi\) for salt solutions are given in references [8] and [10].

Footnotes

[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1954, equation 22.5.

[2] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 307.

[3] V. E. Bower and R. A. Robinson, J. Phys. Chem.,1963,67,1524.

[4] R. H. Stokes and R. A. Robinson, J. Phys. Chem.,1963,67,2126.

[5] J. J. Savage and R. H. Wood, J. Solution Chem., 1976,5,733.

[6] M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Annu. Rep. Prog. Chem., Sect C., Phys.Chem.,1990,87,45.

[7] H. Ellerton and P. J. Dunlop. J. Phys.Chem.,1966,70,1831.

[8] R. A.Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn., 1965, chapter 8.

[9] K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 3rd. edition, 1995.

[10] S. Lindenbaum and G. E. Boyd, J. Phys.Chem.,1964,68,911.