3.1: Mean Ionic Activity Coefficients and the Debye-Hückel Model
- Page ID
- 516013
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Introduction
Solutions of ions behave very non-ideally because of the charge-charge interactions between the ions and interactions with the solvent. Thus to model the behavior of ions in solution requires using the ion activities rather than concentrations as can be done reasonably well for molecular solutes. In terms of activities the chemical potential of NaCl(aq) would be:
\[\mu_{NaCl} = \mu^o_{Na^{+}} + RTlna_{Na^{+}} + \mu^o_{Cl^{-}} + RTlna_{Cl^{-}} = \mu^o_{Na^{+}} + \mu^o_{Cl^{-}}+ RTln\left(a_{Na^{+}}a_{Cl^{-}}\right) \]
If we write this in terms of activity coefficients we have:
\[\mu_{NaCl} = \mu^o_{Na^{+}} + \mu^o_{Cl^{-}}+ RTln\left(\gamma_{Na^{+}}[Na^{+}]\gamma_{Cl^{-}}[Cl^{-}]\right)\]
where the convention is that [] is the concentration divided by the standard reference concentration. The problem is that you cannot get the positive ions without also getting the negative ions that balance the charge. This means the separate activities or activity coefficients cannot be measured. The solution is to measure and use a combined mean activity coefficient.
Combined (mean) activity coefficients
Based on these Wikipedia Articles: Activity Coefficients; Debye-Hückel Theory
For the case of NaCl(aq) the mean activity coefficient is defined as:
\[\gamma_{\pm}(NaCl) = \sqrt{\gamma_{Na^{+}}\gamma_{Cl^{-}}}\]
Substituting this into the equation for the total chemical potential of NaCl(aq):
\[\mu_{NaCl} = \mu^o_{Na^{+}} + \mu^o_{Cl^{-}}+ RTln\left(\gamma_{\pm}^2[Na^{+}][Cl^{-}]\right)\]
More generally, the mean activity coefficient of an ionic compound of formula \(A_p B_q\) is given by
\[\gamma_\pm=\sqrt[p+q]{\gamma_\mathrm{A}^p\gamma_\mathrm{B}^q}\]
The behavior of this mean activity coefficient versus concentration can be understood theoretically. The Debye-Hückel theory and some extensions of it have proved to be useful for practical application.
Debye-Hückel Model
This theory explains the value of \(\gamma_{\pm}\) in terms of the shielding of like charges from each other by the surrounding opposite charges and the impact of the dielectric constant of the solvent's impact on the electric field strength felt by each ion. This requires integration over the distrubution of ions. Figure \(\PageIndex{1}\) is a 2-d image of ions in solution. Note that any charge you choose is mostly surrounded by opposite charges.
The combination of the higher probability that an ion is surrounded by oppositely charged ions and the dielectric constant of the solvent generally lowers the interaction strength between the ions. Thus like charges repel less and opposite charges are less attracted to each other. This leads to slightly lower energies of the ions than if they were completely randomly distributed. Thus the chemical potential is less than would be found in an ideal solution. A careful mathematical analysis of this model by Debye and Hückel lead the the model named after them. At very low concentrations their model takes the form of the Debye-Hückel limiting law:
\[log_{10}\gamma_\pm = -A\lvert z_{+}z_{-}\rvert\sqrt{I}\label{DebHucLim}\]
where \(z_{+}\) is the charge on the positive ions of the species, \(z_{-}\) the charge on the negative ions, A is a temperature dependent constant and I is the ionic strength (defined below). Beyond an ionic strength of about 10-3 a more complex version applies:
\[\log_{10}\gamma_\pm = -A \lvert z_{+}z_{-}\rvert \frac{\sqrt I}{1 +B\sqrt I}\label{DebHuc}\]
at room temperature in water for units of molarity or molality A ≈ 0.509 and B ≈ 3.2. Generally the values are arrived at empirically by fitting data in the units used for the experiment. At concentrations approaching 0.1 ionic strength and above extended versions of this and other models are used (see Extending Debye-Hückel).
Figure \(\PageIndex{2}\) shows how the first equation (limiting law) deviates from real data.
The most significant aspect of equations \ref{DebHucLim} and \ref{DebHuc} is the prediction that the mean activity coefficient is a function of ionic strength rather than the electrolyte concentration.
Ionic Strength
Ionic strength (\(I\)) is defined as:
\[I =\dfrac{1}{2} \sum b_iz_i^2 \nonumber \]
where \(b_i\) is the molality of ion \(i\) and \(z_i\) is its charge coefficient. Note that highly charged ions (e.g. \(z=3+\)) contribute strongly (nine times more than +1 ions), but the formula is linear in the molality.
When a solid is formed by a reaction from solution it is sometimes possible that it remains dispersed as very small particles in the solvent. The sizes typically range in the nanometers This is why it has become fashionable to call them nanoparticles, although they had been known as colloidal particles since the mid nineteenth century. They are smaller than the wavelength of the visible reason. This causes liquids that contain them to remain clear, although they can at times be beautifully colored. A good example is the reduction of AuCl4- with citrate to metallic gold. This produces clear wine red solutions, even at tiny gold concentrations.
\[\ce{2n AuCl4(aq)^{-} + n \,citrate^{3-}(aq) + 2n\, H_2O(l) \rightarrow} \\[4pt] \ce{2n\, Au(colloid) + 3n\, CH2O(aq) + 3n\, CO2(g) + 8n \,Cl^{-}(aq) + 3n\, H^{+}(aq)} \nonumber \]
The reason the gold does not precipitate completely is typically that the nanoparticle (AuNP) formed during the reaction are charged by the attachment of some of the ionic species in solution to its surface. This results in an charged particle with an atmosphere with a certain Debye length around it (Figure \(\PageIndex{3}\)). This charged cloud prevents the particle form coalescing with other particles by electrostatic repulsion.
Such a system is called a colloid. Of course these systems are metastable. Often they have a pretty small threshold to crashing to a real precipitate under influence of the strong van der Waals interactions that the particles experience once they manage to get in close contact. Under the right conditions colloids can survive for a long time. Some gold colloids prepared by Faraday in the 1850's are still stable today.
It will be clear from the above that addition of a salt -particularly containing highly charged ions like 3+ or 3-- may destabilize the colloid because the ionic strength will changed drastically and this will affect the Debye length.