Ionic equilibria form the foundation of physical chemistry, electrochemistry, analytical chemistry, and chemical thermodynamics. They describe the behaviour of ions in solution, including their interactions, chemical potentials, equilibrium constants, and participation in electrochemical processes.
At the undergraduate level, electrolyte solutions are often treated as ideal systems in which concentrations directly determine thermodynamic behaviour. However, real electrolyte solutions deviate from ideality because ions interact through electrostatic forces. These interactions become particularly important in concentrated solutions and in solutions containing highly charged ions.
To understand these deviations quantitatively, we introduce the concepts of chemical potential, activity, activity coefficient, ionic strength, mean ionic activity, and the Debye–Hückel theory.
The central idea is that concentration tells us how much of a species is present, whereas activity tells us how effectively that species participates in a thermodynamic process.
An electrolyte is a substance that produces mobile ions when dissolved in a suitable solvent or when present in the molten state. These ions carry electric current through the medium.
Electrolytes can be classified according to their degree of ionisation or dissociation.
For a strong electrolyte such as sodium chloride, the dissolution process is represented by:
\[\mathrm{NaCl(s)\rightarrow Na^+(aq)+Cl^-(aq)}\]
For a weak electrolyte such as acetic acid, ionisation is reversible:
\[\mathrm{CH_3COOH(aq)\rightleftharpoons H^+(aq)+CH_3COO^-(aq)}\]
The distinction between strong and weak electrolytes is important. Strong electrolytes may be almost completely dissociated but still exhibit substantial non-ideal behaviour because their ions interact with one another. Therefore, complete dissociation does not imply ideal behaviour.
In an electrolyte solution, each ion interacts electrostatically with surrounding ions. Oppositely charged ions attract one another, whereas ions of the same charge repel one another.
Consider a central positive ion. Its surroundings tend to contain an excess of negative charge. Similarly, a central negative ion tends to be surrounded by an excess of positive charge. This non-uniform distribution of charge is known as an ionic atmosphere.
The ionic atmosphere modifies the electrostatic environment of each ion and influences its chemical potential. Consequently, an ion in solution does not behave exactly as it would in an ideal solution of the same concentration.
The major consequences of ionic interactions include:
The Debye–Hückel theory provides a quantitative framework for understanding the electrostatic contribution to these effects in sufficiently dilute electrolyte solutions.
The chemical potential of a component is its partial molar Gibbs energy. It describes how the Gibbs energy of a system changes when an infinitesimal amount of that component is added while temperature, pressure, and the amounts of other components are appropriately specified.
For component \(i\), the chemical potential is defined as:
\[\mu_i=\left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\ne i}}\]
Here, \(\mu_i\) is the chemical potential, \(G\) is the Gibbs energy, \(n_i\) is the amount of component \(i\), \(T\) is temperature, and \(P\) is pressure.
The chemical potential is especially important because chemical equilibrium is governed by the equality of appropriate chemical potentials, not simply by equality of concentrations.
For a general chemical reaction:
\[\sum_i \nu_i A_i=0\]
where \(\nu_i\) is the stoichiometric coefficient, taken as positive for products and negative for reactants, the Gibbs energy change of reaction is:
\[\Delta_rG=\sum_i\nu_i\mu_i\]
At equilibrium:
\[\boxed{\Delta_rG=0}\]
This condition forms the thermodynamic basis of equilibrium constants, acid–base equilibria, solubility equilibria, and electrochemical reactions.
For an ideal mixture, the chemical potential of a species depends logarithmically on its concentration or another appropriate measure of composition. For a real solution, this expression is written in terms of activity:
\[\boxed{\mu_i=\mu_i^\circ+RT\ln a_i}\]
where:
This expression follows from the fundamental thermodynamic relation for a chemical potential. For an ideal solution, the composition-dependent contribution is proportional to the logarithm of the mole fraction. For a real solution, the mole fraction is replaced by an effective thermodynamic quantity, the activity.
The logarithm must act on a dimensionless quantity. Accordingly, activities are defined relative to a specified standard state.
For a solute described using the molality standard state:
\[a_i=\gamma_i\frac{m_i}{m^\circ}\]
where \(m_i\) is the molality of the solute, \(m^\circ=1\ \mathrm{mol\,kg^{-1}}\) is the standard molality, and \(\gamma_i\) is the molality-based activity coefficient.
Alternatively, using a concentration-based standard state:
\[a_i=\gamma_i^{(c)}\frac{c_i}{c^\circ}\]
Here, \(c_i\) is the molar concentration, \(c^\circ=1\ \mathrm{mol\,L^{-1}}\), and \(\gamma_i^{(c)}\) is the corresponding concentration-based activity coefficient.
Molality-based and concentration-based activity coefficients are not generally identical. Their numerical values can differ, particularly when solution density and solvent composition change significantly.
Activity is an effective thermodynamic measure of concentration. It accounts for deviations from ideal behaviour through the activity coefficient.
For a molality-based convention:
\[\boxed{a_i=\gamma_i\frac{m_i}{m^\circ}}\]
Three limiting situations are useful.
For example, suppose a solute has molality \(0.10\ \mathrm{mol\,kg^{-1}}\) and activity coefficient \(0.80\). Its activity is:
\[a_i=0.80\left(\frac{0.10}{1}\right)=0.080\]
Thus, its relative molality is 0.10, whereas its activity is 0.080. The difference indicates non-ideal thermodynamic behaviour.
Remember: activity is not an additional concentration unit. It is a dimensionless thermodynamic quantity defined relative to a standard state.
The activity coefficient, \(\gamma_i\), measures the deviation of a species from the ideal reference behaviour specified by the chosen standard state.
For an electrolyte reaction:
\[\mathrm{A^+ + B^- \rightleftharpoons AB}\]
the thermodynamic equilibrium constant is:
\[K^\circ=\frac{a_{AB}}{a_{A^+}a_{B^-}}\]
Using molality-based activities:
\[K^\circ=\frac{\gamma_{AB}(m_{AB}/m^\circ)}{\gamma_{A^+}(m_{A^+}/m^\circ)\gamma_{B^-}(m_{B^-}/m^\circ)}\]
If all activity coefficients are incorrectly assumed to be unity, the equilibrium constant calculated from measured concentrations or molalities may not agree with the thermodynamic equilibrium constant.
The thermodynamic equilibrium constant is constant at a specified temperature and set of standard-state conventions. However, a concentration-based conditional equilibrium constant can vary with the ionic medium because activity coefficients change.
A major difficulty in electrolyte thermodynamics is that the activity of an individual ion cannot generally be measured independently by a purely thermodynamic experiment. Measurable bulk properties of an electrically neutral solution involve combinations of ionic contributions.
For example, when sodium chloride dissolves:
\[\mathrm{NaCl\rightarrow Na^++Cl^-}\]
the solution contains both sodium and chloride ions. Thermodynamic measurements provide information about their combined effects rather than an independently measurable absolute activity for each ion.
Therefore, electrolyte thermodynamics commonly uses the mean ionic activity coefficient.
Consider an electrolyte that dissociates according to:
\[\mathrm{M_{\nu_+}X_{\nu_-}\rightarrow \nu_+M^{z_+}+\nu_-X^{z_-}}\]
Here, \(\nu_+\) and \(\nu_-\) are the stoichiometric numbers of cations and anions, respectively. For an electrically neutral electrolyte:
\[\nu_+z_++\nu_-z_-=0\]
The total number of ions produced per formula unit is:
\[\nu=\nu_++\nu_-\]
The mean ionic activity is defined as:
\[\boxed{a_\pm=(a_+^{\nu_+}a_-^{\nu_-})^{1/\nu}}\]
Substituting the individual ionic activities expressed through molalities gives:
\[a_\pm=\left[\left(\gamma_+\frac{m_+}{m^\circ}\right)^{\nu_+}\left(\gamma_-\frac{m_-}{m^\circ}\right)^{\nu_-}\right]^{1/\nu}\]
Separating the activity coefficients from the molalities:
\[a_\pm=\left(\gamma_+^{\nu_+}\gamma_-^{\nu_-}\right)^{1/\nu}\left[\left(\frac{m_+}{m^\circ}\right)^{\nu_+}\left(\frac{m_-}{m^\circ}\right)^{\nu_-}\right]^{1/\nu}\]
The mean ionic activity coefficient is therefore:
\[\boxed{\gamma_\pm=(\gamma_+^{\nu_+}\gamma_-^{\nu_-})^{1/\nu}}\]
Similarly, the mean ionic molality is:
\[m_\pm=(m_+^{\nu_+}m_-^{\nu_-})^{1/\nu}\]
Hence:
\[\boxed{a_\pm=\gamma_\pm\frac{m_\pm}{m^\circ}}\]
These expressions are particularly important in derivations involving electrolyte chemical potentials, ionic activity products, and the Debye–Hückel limiting law.
For NaCl:
\[\mathrm{NaCl\rightarrow Na^++Cl^-}\]
Both stoichiometric numbers are unity. Therefore:
\[\nu_+=1,\quad \nu_-=1,\quad \nu=2\]
The mean ionic activity coefficient is:
\[\boxed{\gamma_\pm=(\gamma_{\mathrm{Na^+}}\gamma_{\mathrm{Cl^-}})^{1/2}}\]
If the molalities of sodium and chloride are equal to \(m\), then:
\[m_\pm=(m\times m)^{1/2}=m\]
Consequently:
\[a_\pm=\gamma_\pm\frac{m}{m^\circ}\]
Calcium chloride dissociates as:
\[\mathrm{CaCl_2\rightarrow Ca^{2+}+2Cl^-}\]
For this electrolyte:
\[\nu_+=1,\quad \nu_-=2,\quad \nu=3\]
The mean ionic activity coefficient is:
\[\boxed{\gamma_\pm=(\gamma_{\mathrm{Ca^{2+}}}\gamma_{\mathrm{Cl^-}}^2)^{1/3}}\]
If the analytical molality of CaCl2 is \(m\), and dissociation is complete, then:
\[m_{\mathrm{Ca^{2+}}}=m\]
\[m_{\mathrm{Cl^-}}=2m\]
The mean ionic molality becomes:
\[m_\pm=(m(2m)^2)^{1/3}=4^{1/3}m\]
This illustrates why the mean ionic molality of an electrolyte is not necessarily equal to the analytical molality of its formula units.
Ionic strength is a quantitative measure of the overall electrostatic contribution of ions in solution. It accounts for both ionic concentrations and ionic charges.
The ionic strength is defined as:
\[\boxed{I=\frac12\sum_i m_i z_i^2}\]
where \(m_i\) is the molality of ion \(i\), and \(z_i\) is its signed charge number. The square of the charge means that cations and anions both make positive contributions.
For a concentration-based formulation, the corresponding expression is:
\[I_c=\frac12\sum_i c_i z_i^2\]
Here, \(I_c\) has concentration units, typically \(\mathrm{mol\,L^{-1}}\), whereas the molality-based ionic strength has units of \(\mathrm{mol\,kg^{-1}}\).
The factor of one-half prevents double counting in the conventional definition of ionic strength. More importantly, the squared charge term makes highly charged ions contribute disproportionately to the electrostatic environment.
Consider an electrolyte with formula \(\mathrm{M_{\nu_+}X_{\nu_-}}\), whose formula-unit molality is \(m\). Assuming complete dissociation:
\[m_+=\nu_+m,\qquad m_-=\nu_-m\]
The ionic strength is:
\[I=\frac12(m_+z_+^2+m_-z_-^2)\]
Substituting the ionic molalities:
\[I=\frac12(\nu_+mz_+^2+\nu_-mz_-^2)\]
Therefore:
\[\boxed{I=\frac{m}{2}(\nu_+z_+^2+\nu_-z_-^2)}\]
This expression assumes the stated electrolyte is the only contributor to ionic strength and that its dissociation is complete. In mixtures, contributions from all ions must be included.
Suppose an NaCl solution has molality \(0.10\ \mathrm{mol\,kg^{-1}}\).
Since NaCl dissociates into equal molalities of Na+ and Cl−:
\[m_{\mathrm{Na^+}}=0.10,\qquad m_{\mathrm{Cl^-}}=0.10\]
Both ions have charge magnitude one. Thus:
\[I=\frac12[(0.10)(1)^2+(0.10)(-1)^2]\]
\[I=\frac12(0.20)\]
\[\boxed{I=0.10\ \mathrm{mol\,kg^{-1}}}\]
For a 1:1 electrolyte with complete dissociation, the ionic strength equals its formula-unit molality.
Consider a CaCl2 solution of molality \(0.10\ \mathrm{mol\,kg^{-1}}\).
Complete dissociation gives:
\[m_{\mathrm{Ca^{2+}}}=0.10\]
\[m_{\mathrm{Cl^-}}=0.20\]
The ionic strength is:
\[I=\frac12[(0.10)(2)^2+(0.20)(1)^2]\]
\[I=\frac12(0.40+0.20)\]
\[\boxed{I=0.30\ \mathrm{mol\,kg^{-1}}}\]
Although the analytical molality of CaCl2 is only \(0.10\ \mathrm{mol\,kg^{-1}}\), its ionic strength is three times greater.
For aluminium sulphate:
\[\mathrm{Al_2(SO_4)_3\rightarrow2Al^{3+}+3SO_4^{2-}}\]
Let the formula-unit molality be \(m\). Then:
\[m_{\mathrm{Al^{3+}}}=2m,\qquad m_{\mathrm{SO_4^{2-}}}=3m\]
Substitution into the ionic strength equation gives:
\[I=\frac12[2m(3)^2+3m(2)^2]\]
\[I=\frac12(18m+12m)\]
\[\boxed{I=15m}\]
Thus, a hypothetical \(0.010\ \mathrm{mol\,kg^{-1}}\) solution of fully dissociated aluminium sulphate has:
\[I=15(0.010)=0.150\ \mathrm{mol\,kg^{-1}}\]
This example demonstrates the strong influence of multivalent ions on ionic strength.
For a solution containing several electrolytes, the ionic strength must be calculated from the concentrations or molalities of all ions present. It is not generally correct to calculate each electrolyte's ionic strength independently and then ignore the ionic species shared between them.
Consider a mixture containing \(0.10\ \mathrm{mol\,kg^{-1}}\) NaCl and \(0.020\ \mathrm{mol\,kg^{-1}}\) CaCl2, assuming complete dissociation.
The ionic molalities are:
\[m_{\mathrm{Na^+}}=0.10\]
\[m_{\mathrm{Ca^{2+}}}=0.020\]
\[m_{\mathrm{Cl^-}}=0.10+2(0.020)=0.14\]
Therefore:
\[I=\frac12[0.10(1)^2+0.020(2)^2+0.14(1)^2]\]
\[I=\frac12(0.10+0.080+0.14)\]
\[\boxed{I=0.160\ \mathrm{mol\,kg^{-1}}}\]
Notice that the chloride concentration includes contributions from both salts. This approach is essential for buffer solutions, supporting electrolytes, mixed salt systems, and electrochemical cells.
The term \(z_i^2\) in the ionic strength equation is a reflection of the electrostatic importance of ionic charge. An ion with charge magnitude two contributes four times as much per mole as a monovalent ion at the same molality. An ion with charge magnitude three contributes nine times as much.
For example, at equal molality:
Consequently, solutions containing Ca2+, Mg2+, Al3+, or other multivalent ions can show much stronger electrostatic non-ideality than solutions containing only monovalent ions at similar formula-unit concentrations.
The Debye–Hückel theory, developed by Peter Debye and Erich Hückel, explains the non-ideal thermodynamic behaviour of dilute electrolyte solutions using electrostatic interactions between ions.
Its central objective is to relate the mean ionic activity coefficient to the ionic strength of the solution.
The theory explains why activity coefficients depart from unity even when a strong electrolyte is fully dissociated. The deviation arises from interactions between ions, rather than necessarily from incomplete dissociation.
These assumptions are most appropriate at low ionic strength. They become less reliable as the solution becomes more concentrated or when specific ion–solvent and ion–ion interactions become important.
Imagine placing a positive ion in a dilute electrolyte solution. The central ion attracts surrounding negative ions and repels surrounding positive ions. Averaged over many microscopic configurations, its environment contains an excess of negative charge.
This diffuse distribution is called the ionic atmosphere. It does not represent a rigid shell of counterions; rather, it is a statistical distribution of charge around the central ion.
Similarly, a negative central ion is surrounded, on average, by an excess of positive charge.
The ionic atmosphere has two important consequences:
Because the ionic atmosphere depends on the surrounding ionic composition, the activity coefficient of an ion depends not only on its own concentration but also on the other ions present in solution.
Thermal motion tends to disperse ions throughout the solution, whereas electrostatic attraction and repulsion influence their local distributions.
The competition between these effects determines the ionic atmosphere. At higher temperatures, thermal energy becomes more significant relative to a fixed electrostatic interaction. At higher dielectric permittivity, electrostatic interactions between charges are weaker for a given separation.
For an electrolyte solution, the characteristic screening length depends on the ionic strength, temperature, and dielectric properties of the solvent. The relevant length scale is the Debye length.
The electrostatic potential \(\psi\) generated by a charge distribution is related to the local charge density through Poisson's equation:
\[\nabla^2\psi=-\frac{\rho}{\varepsilon}\]
Here, \(\rho\) is the local charge density and \(\varepsilon\) is the absolute permittivity of the medium.
For a solvent with relative permittivity \(\varepsilon_r\):
\[\varepsilon=\varepsilon_0\varepsilon_r\]
where \(\varepsilon_0\) is the permittivity of vacuum.
The local charge density is obtained by summing the charge contributions of all ionic species:
\[\rho=F\sum_i z_i c_i\]
when \(c_i\) is expressed in \(\mathrm{mol\,m^{-3}}\), \(F\) is Faraday's constant, and \(z_i\) is the signed charge number.
To derive the Debye–Hückel limiting law, the ion distribution around a central charge is approximated using the Boltzmann distribution.
For an ion of charge \(z_i e\) in an electrostatic potential \(\psi\), the electrostatic potential energy is:
\[U_i=z_i e\psi\]
where \(e\) is the elementary charge.
At thermal equilibrium, the local number density of that ion is:
\[n_i=n_i^\infty\exp\left(-\frac{z_i e\psi}{k_BT}\right)\]
Here, \(n_i^\infty\) is the bulk number density, \(k_B\) is Boltzmann's constant, and \(T\) is absolute temperature.
The exponential term indicates that ions of opposite charge are enriched near a central ion, while ions of the same charge are depleted.
When the electrostatic energy is small compared with thermal energy, the dimensionless quantity \(\lvert z_i e\psi/(k_BT)\rvert\) is much less than one. We may then use the first-order approximation:
\[\exp(-x)\approx1-x\]
Consequently:
\[n_i\approx n_i^\infty\left(1-\frac{z_i e\psi}{k_BT}\right)\]
This is the linearisation step that leads to the Debye–Hückel limiting description.
Substituting the linearised ionic distributions into Poisson's equation, and using bulk electroneutrality, gives the linearised Poisson–Boltzmann equation:
\[\boxed{\nabla^2\psi=\kappa^2\psi}\]
The parameter \(\kappa\) is the inverse Debye screening length. For ionic number densities \(n_i^\infty\), it is defined by:
\[\boxed{\kappa^2=\frac{e^2}{\varepsilon k_BT}\sum_i n_i^\infty z_i^2}\]
Here, the number density is expressed as the number of ions per unit volume. If molar concentrations \(c_i\) are used in \(\mathrm{mol\,m^{-3}}\), then \(n_i^\infty=N_Ac_i\), where \(N_A\) is Avogadro's constant.
Thus:
\[\kappa^2=\frac{F^2}{\varepsilon RT}\sum_i c_i z_i^2\]
where the concentrations are expressed in \(\mathrm{mol\,m^{-3}}\).
Since the concentration-based ionic strength in these units is:
\[I_c=\frac12\sum_i c_i z_i^2\]
the screening parameter becomes:
\[\boxed{\kappa^2=\frac{2F^2I_c}{\varepsilon RT}}\]
This equation shows that the screening parameter increases with ionic strength and decreases with increasing dielectric permittivity and temperature, with the latter two dependencies applying under the stated solvent model.
The Debye length, denoted by \(\lambda_D\), is the characteristic distance over which electrostatic potentials are screened by the ionic atmosphere:
\[\boxed{\lambda_D=\frac{1}{\kappa}}\]
Therefore:
\[\lambda_D=\sqrt{\frac{\varepsilon RT}{2F^2I_c}}\]
when \(I_c\) is expressed in \(\mathrm{mol\,m^{-3}}\).
The important proportionality is:
\[\boxed{\lambda_D\propto I_c^{-1/2}}\]
Thus, increasing ionic strength shortens the Debye length. The electrostatic influence of an individual ion is screened over a shorter distance because more ions contribute to the surrounding charge distribution.
At very low ionic strength, the Debye length is larger, and electrostatic correlations extend over a longer distance. This is one reason dilute electrolyte solutions require careful thermodynamic treatment even though their absolute concentrations are small.
For a spherically symmetric charge distribution around a central ion, the linearised Poisson–Boltzmann equation becomes:
\[\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\psi}{dr}\right)=\kappa^2\psi\]
The physically appropriate solution that decays at large distance has the form:
\[\psi(r)=\frac{A e^{-\kappa r}}{r}\]
The constant \(A\) is determined by the charge of the central ion. For a point charge \(z_ie\):
\[\boxed{\psi(r)=\frac{z_ie}{4\pi\varepsilon r}e^{-\kappa r}}\]
This expression is the screened Coulomb potential. Without an ionic atmosphere, the potential of an isolated point charge in a uniform dielectric medium would vary as \(1/r\). In an electrolyte solution, the additional exponential factor represents screening.
The ionic atmosphere modifies the electrostatic free energy of the central ion. In the Debye–Hückel treatment, the electrostatic contribution is obtained by accounting for the potential generated by the surrounding ionic atmosphere, excluding the singular self-potential of the central ion.
The resulting electrostatic contribution to the chemical potential is:
\[\mu_i^{\mathrm{ex}}=-\frac{z_i^2e^2\kappa}{8\pi\varepsilon}\]
Here, \(\mu_i^{\mathrm{ex}}\) is the excess chemical potential associated with electrostatic non-ideality in the limiting theory. The negative sign reflects the stabilising electrostatic contribution from the ionic atmosphere.
The thermodynamic relation between the excess chemical potential and the activity coefficient is:
\[\mu_i^{\mathrm{ex}}=RT\ln\gamma_i\]
Using this relationship gives:
\[RT\ln\gamma_i=-\frac{z_i^2e^2\kappa}{8\pi\varepsilon}\]
Therefore:
\[\boxed{\ln\gamma_i=-\frac{z_i^2e^2\kappa}{8\pi\varepsilon k_BT}}\]
This equation is written using a single-ion theoretical description. Experimentally, the directly accessible electrolyte quantity is the mean ionic activity coefficient.
Because the screening parameter obeys:
\[\kappa\propto\sqrt{I}\]
the electrostatic contribution to the logarithm of the activity coefficient is proportional to the square root of ionic strength.
For a dilute electrolyte solution, the mean ionic activity coefficient is described by the Debye–Hückel limiting law:
\[\boxed{\log_{10}\gamma_\pm=-A|z_+z_-|\sqrt{I}}\]
Here, \(A\) is a solvent- and temperature-dependent constant when the conventional molality scale is used, and \(I\) is the molality-based ionic strength expressed numerically relative to the standard molality. The symbols \(z_+\) and \(z_-\) are the signed cation and anion charge numbers; their absolute values are used in the coefficient.
For water at approximately \(25^\circ\mathrm{C}\), on the molality scale:
\[A\approx0.509\]
Accordingly:
\[\boxed{\log_{10}\gamma_\pm=-0.509|z_+z_-|\sqrt{I}}\]
This numerical form is intended for sufficiently dilute aqueous electrolyte solutions. It should not be applied indiscriminately to concentrated solutions or to solvents with substantially different dielectric properties.
For a 1:1 electrolyte such as NaCl, \(|z_+z_-|=1\). For a 2:1 electrolyte such as CaCl2, \(|z_+z_-|=2\). For a 3:2 electrolyte such as Al2(SO4)3, the charge product is six.
At the same sufficiently low ionic strength, the limiting law therefore predicts a larger deviation for an electrolyte with a larger charge product.
Suppose an aqueous NaCl solution at \(25^\circ\mathrm{C}\) has ionic strength \(I=0.0010\) on the molality scale.
For NaCl, both ions are monovalent:
\[|z_+z_-|=1\]
Using the limiting law:
\[\log_{10}\gamma_\pm=-0.509\sqrt{0.0010}\]
Since:
\[\sqrt{0.0010}=0.03162\]
we obtain:
\[\log_{10}\gamma_\pm\approx-0.0161\]
Taking the antilogarithm:
\[\gamma_\pm=10^{-0.0161}\]
\[\boxed{\gamma_\pm\approx0.964}\]
The mean ionic activity coefficient is slightly below unity, showing that even at low ionic strength the thermodynamic activity is not exactly equal to the relative molality.
Consider a solution with ionic strength \(I=0.0010\) on the molality scale. For CaCl2, the cation charge is \(+2\) and the anion charge is \(-1\).
\[|z_+z_-|=2\]
Therefore:
\[\log_{10}\gamma_\pm=-0.509(2)\sqrt{0.0010}\]
\[\log_{10}\gamma_\pm\approx-0.0322\]
Hence:
\[\boxed{\gamma_\pm\approx0.929}\]
The predicted mean ionic activity coefficient is smaller than that for NaCl at the same ionic strength. This follows from the larger charge product in the limiting law.
For a 1:1 electrolyte at \(25^\circ\mathrm{C}\), compare the limiting-law predictions at \(I=0.0010\) and \(I=0.0100\).
At \(I=0.0010\):
\[\log_{10}\gamma_\pm=-0.509\sqrt{0.0010}\approx-0.0161\]
At \(I=0.0100\):
\[\log_{10}\gamma_\pm=-0.509\sqrt{0.0100}=-0.0509\]
The second solution has a greater predicted deviation from ideality because its ionic strength is ten times higher. The square-root dependence means that the magnitude of \(\log_{10}\gamma_\pm\) increases by a factor of \(\sqrt{10}\), not by a factor of ten.
The limiting law is fundamental, but its assumptions restrict its quantitative range.
These limitations motivate extended Debye–Hückel equations, which introduce an empirical or effective ion-size parameter, and more advanced models such as the Davies equation and Pitzer models.
For a reaction:
\[\mathrm{A+B\rightleftharpoons C}\]
the thermodynamic equilibrium constant is expressed in terms of activities:
\[K^\circ=\frac{a_C}{a_Aa_B}\]
When ionic species are involved, activity coefficients are required to relate activities to concentrations or molalities. As a result, concentration-based equilibrium calculations can become inaccurate if non-ideality is ignored.
Consider the dissolution equilibrium of silver chloride:
\[\mathrm{AgCl(s)\rightleftharpoons Ag^+(aq)+Cl^-(aq)}\]
The thermodynamic solubility product is:
\[K_{\mathrm{sp}}^\circ=a_{\mathrm{Ag^+}}a_{\mathrm{Cl^-}}\]
For a pure solid, its activity is taken as unity in the standard formulation.
Using molality-based activities:
\[K_{\mathrm{sp}}^\circ=\gamma_{\mathrm{Ag^+}}\gamma_{\mathrm{Cl^-}}\left(\frac{m_{\mathrm{Ag^+}}}{m^\circ}\right)\left(\frac{m_{\mathrm{Cl^-}}}{m^\circ}\right)\]
This relationship shows that solubility cannot always be predicted from concentration products alone. The ionic medium can change the relationship between the measured concentrations and the thermodynamic equilibrium constant.
The Nernst equation is another direct application of chemical potential and activity. For a reduction half-reaction:
\[\mathrm{Ox}+ne^-\rightleftharpoons\mathrm{Red}\]
the electrode potential is:
\[E=E^\circ-\frac{RT}{nF}\ln Q\]
where \(Q\) is the reaction quotient constructed using activities and \(n\) is the number of electrons transferred.
At \(25^\circ\mathrm{C}\), the equation is commonly written as:
\[E=E^\circ-\frac{0.05916}{n}\log_{10}Q\]
For a silver electrode:
\[\mathrm{Ag^++e^-\rightleftharpoons Ag(s)}\]
the activity of the pure solid silver is unity, giving:
\[E=E^\circ+\frac{RT}{F}\ln a_{\mathrm{Ag^+}}\]
Using the molality-based activity expression:
\[E=E^\circ+\frac{RT}{F}\ln\left(\gamma_{\mathrm{Ag^+}}\frac{m_{\mathrm{Ag^+}}}{m^\circ}\right)\]
Therefore, changes in ionic strength can affect the electrode potential through changes in activity coefficients, even when the analytical concentration remains constant.
Answer: Complete dissociation describes the formation of ions, not their thermodynamic independence. Electrostatic attractions and repulsions between ions generate an ionic atmosphere, changing the excess chemical potential and causing activity coefficients to deviate from unity.
Answer: Ionic strength incorporates the square of the charge number. Consequently, a multivalent ion contributes more strongly to the electrostatic environment than a monovalent ion at the same molality.
Answer: The limiting law contains the factor \(|z_+z_-|\). A larger charge product produces a larger magnitude of the negative logarithm of the mean ionic activity coefficient at the same ionic strength.
Answer: Not generally by a purely thermodynamic experiment. Bulk thermodynamic measurements yield electrically neutral combinations of ionic contributions, so mean ionic activity coefficients are the standard experimentally accessible quantities. Individual-ion values require an adopted extrathermodynamic convention or a theoretical model.
Answer: At higher ionic strength, finite ion sizes, ion pairing, short-range interactions, specific solvation, and nonlinear electrostatic effects become increasingly important. The point-ion and linearisation assumptions are no longer sufficiently accurate.
The next part should develop the extended Debye–Hückel equation, the Davies equation, activity-coefficient calculations, thermodynamic equilibrium constants, ionic activity products, and the role of ionic strength in acid–base and solubility equilibria. It should also explain when each model is appropriate and how to solve CSIR-NET numerical problems systematically.
In Part 1, we established that electrolyte solutions exhibit non-ideal behaviour because of electrostatic interactions between ions. The Debye–Hückel limiting law provides a theoretical relationship between the mean ionic activity coefficient and ionic strength:
\[\log_{10}\gamma_\pm=-A|z_+z_-|\sqrt I\]
This equation is highly important for understanding dilute electrolyte solutions. However, it becomes progressively less accurate as ionic strength increases. One major reason is that real ions have finite dimensions and cannot be treated as mathematical point charges at all separations.
To address this limitation, several extensions and empirical modifications have been developed. The most important models at MSc Chemistry and CSIR-NET level are:
The limiting law explains the fundamental electrostatic origin of non-ideality. Extended models attempt to describe real solutions more accurately by incorporating additional physical effects.
The limiting law assumes that ions behave approximately as point charges. At very low ionic strength, the distance between ions is generally large enough that the point-ion approximation can be useful for describing long-range electrostatic interactions.
As ionic strength increases, however, the typical separation between ions decreases. The finite dimensions of hydrated ions, short-range interactions, and specific solvation effects become more important.
The extended Debye–Hückel equation introduces an effective ion-size parameter to account approximately for the finite distance of closest approach between ions.
A commonly used molality-scale form of the extended Debye–Hückel equation is:
\[\boxed{\log_{10}\gamma_i=-\frac{A z_i^2\sqrt I}{1+B a_i\sqrt I}}\]
Here:
The numerical values and units of \(A\), \(B\), and \(a_i\) must be consistent with the ionic-strength convention. In the common aqueous formulation at \(25^\circ\mathrm C\), when \(I\) is entered numerically in \(\mathrm{mol\,kg^{-1}}\), a frequently used parameterisation has approximately \(A=0.509\) and \(B=0.328\) when the effective ion-size parameter is expressed in ångströms.
For an electrolyte, an analogous mean ionic activity coefficient expression can be written using the stoichiometrically weighted contributions of its cations and anions. The simple ionic form above is commonly used in introductory calculations, but individual-ion coefficients are not independently measurable thermodynamic quantities.
Consider the extended equation:
\[\log_{10}\gamma_i=-\frac{A z_i^2\sqrt I}{1+B a_i\sqrt I}\]
At very low ionic strength:
\[I\rightarrow0\]
Therefore:
\[B a_i\sqrt I\rightarrow0\]
The denominator approaches unity:
\[\log_{10}\gamma_i\approx-Az_i^2\sqrt I\]
Hence, the extended equation approaches the Debye–Hückel limiting behaviour in the dilute-solution limit.
This is an important theoretical consistency condition: an extended model should reproduce the limiting law when its additional correction becomes negligible.
The denominator is:
\[1+B a_i\sqrt I\]
As ionic strength increases, this denominator increases for positive \(B\) and \(a_i\). Consequently, the magnitude of the negative logarithm of the activity coefficient is reduced relative to the unmodified limiting law.
This correction approximates the effect of finite ion size. However, it does not fully represent hydration shells, ion pairing, complex formation, or all short-range interactions. The effective ion-size parameter should not be interpreted as a universal, exact geometrical radius of an ion in every solution.
Suppose the effective ion-size parameter for a monovalent ion is \(a_i=4.0\ \text{Å}\), and the ionic strength is \(I=0.010\ \mathrm{mol\,kg^{-1}}\) at \(25^\circ\mathrm C\).
Use:
\[A=0.509,\qquad B=0.328\ \text{Å}^{-1}\]
For a monovalent ion:
\[z_i^2=1\]
The extended Debye–Hückel equation gives:
\[\log_{10}\gamma_i=-\frac{0.509\sqrt{0.010}}{1+(0.328)(4.0)\sqrt{0.010}}\]
Since:
\[\sqrt{0.010}=0.10\]
the denominator is:
\[1+(0.328)(4.0)(0.10)=1.1312\]
Therefore:
\[\log_{10}\gamma_i\approx-\frac{0.0509}{1.1312}\]
\[\log_{10}\gamma_i\approx-0.0450\]
Thus:
\[\boxed{\gamma_i\approx10^{-0.0450}\approx0.902}\]
This is an illustrative calculation using an assigned effective ion-size parameter. It should not be treated as an independently measurable single-ion activity coefficient.
The Davies equation is a widely used empirical modification of Debye–Hückel theory. It attempts to provide useful activity-coefficient estimates at ionic strengths for which the limiting law is no longer sufficiently accurate.
It is particularly useful in aqueous solution calculations involving buffers, acid–base equilibria, and analytical chemistry, provided its approximate range of applicability is respected.
A common form of the Davies equation is:
\[\boxed{\log_{10}\gamma_i=-Az_i^2\left(\frac{\sqrt I}{1+\sqrt I}-0.3I\right)}\]
For dilute aqueous solutions near \(25^\circ\mathrm C\), a commonly used value is:
\[A\approx0.509\]
Here, \(I\) is the numerical value of the ionic strength in the conventional molality scale, with the standard molality understood. The equation is an empirical approximation and should not be interpreted as an exact universal law.
The first term inside the parentheses is:
\[\frac{\sqrt I}{1+\sqrt I}\]
This term introduces a correction that reduces the magnitude of the limiting-law prediction as ionic strength increases.
The second term is:
\[0.3I\]
This empirical term further adjusts the prediction to improve agreement with experimental electrolyte behaviour over a wider range than the strict limiting law.
Because the expression is multiplied by \(z_i^2\), the predicted non-ideality remains strongly charge-dependent.
Consider a monovalent ion in an aqueous solution with ionic strength \(I=0.10\ \mathrm{mol\,kg^{-1}}\) at approximately \(25^\circ\mathrm C\).
For this ion:
\[z_i^2=1\]
First, calculate:
\[\sqrt I=\sqrt{0.10}=0.3162\]
Then:
\[\frac{\sqrt I}{1+\sqrt I}=\frac{0.3162}{1.3162}\approx0.2402\]
The correction term is:
\[0.3I=0.0300\]
Thus:
\[\log_{10}\gamma_i=-0.509(0.2402-0.0300)\]
\[\log_{10}\gamma_i\approx-0.1070\]
Therefore:
\[\boxed{\gamma_i\approx10^{-0.1070}\approx0.782}\]
This is an approximate model prediction. Its accuracy depends on the electrolyte system, solvent, temperature, and ionic-strength range.
| Feature | Limiting law | Extended equation | Davies equation |
|---|---|---|---|
| Basic basis | Electrostatic limiting theory | Electrostatic theory with effective ion-size correction | Empirical modification of Debye–Hückel behaviour |
| Ion-size parameter | Absent | Present | Not explicitly required |
| Very dilute solutions | Appropriate limiting model | Approaches the limiting law | Approaches the limiting-law form |
| Higher ionic strengths | Rapidly loses accuracy | Often improved, but still limited | Often convenient for moderate ionic strengths |
| Concentrated electrolytes | Generally unsuitable | Generally insufficient by itself | Not a reliable universal model |
Consider the general reaction:
\[\mathrm{aA+bB\rightleftharpoons cC+dD}\]
The thermodynamic equilibrium constant is defined in terms of dimensionless activities:
\[\boxed{K^\circ=\frac{a_C^c a_D^d}{a_A^a a_B^b}}\]
Here, the superscript degree denotes the equilibrium constant associated with the selected standard states. The exponents are the stoichiometric coefficients.
For ionic species, the activities must include the appropriate activity coefficients. For a species \(i\) on the molality scale:
\[a_i=\gamma_i\frac{m_i}{m^\circ}\]
Consequently, the thermodynamic equilibrium constant can be expressed as:
\[K^\circ=\frac{\gamma_C^c\gamma_D^d}{\gamma_A^a\gamma_B^b}\frac{(m_C/m^\circ)^c(m_D/m^\circ)^d}{(m_A/m^\circ)^a(m_B/m^\circ)^b}\]
This expression separates the effects of composition and non-ideality.
For the general reaction, the reaction Gibbs energy is:
\[\Delta_rG=\Delta_rG^\circ+RT\ln Q\]
where the reaction quotient is:
\[Q=\frac{a_C^c a_D^d}{a_A^a a_B^b}\]
At equilibrium:
\[\Delta_rG=0\]
and:
\[Q=K^\circ\]
Substitution gives:
\[0=\Delta_rG^\circ+RT\ln K^\circ\]
Therefore:
\[\boxed{\Delta_rG^\circ=-RT\ln K^\circ}\]
This is one of the fundamental equations of chemical thermodynamics. It relates the equilibrium constant to the standard Gibbs energy change of reaction.
If \(K^\circ\) is large, the standard Gibbs energy change is negative. If \(K^\circ\) is small, the standard Gibbs energy change is positive. The actual direction of a reaction at any instant, however, depends on the reaction quotient relative to the equilibrium constant.
At a fixed temperature, the thermodynamic equilibrium constant is determined by the standard Gibbs energy change and the chosen standard states. It does not change merely because the total concentration of the solution or its ionic strength changes.
However, the equilibrium concentrations required to satisfy that constant may change because the activity coefficients change.
This distinction is essential:
For the reaction:
\[\mathrm{A+B\rightleftharpoons C}\]
the thermodynamic equilibrium constant is:
\[K^\circ=\frac{a_C}{a_Aa_B}\]
Substituting the molality-based activity expressions:
\[K^\circ=\frac{\gamma_C(m_C/m^\circ)}{\gamma_A(m_A/m^\circ)\gamma_B(m_B/m^\circ)}\]
Rearranging:
\[\frac{m_C/m^\circ}{(m_A/m^\circ)(m_B/m^\circ)}=K^\circ\frac{\gamma_A\gamma_B}{\gamma_C}\]
The left-hand side is a concentration-based quotient. Its equilibrium value is therefore influenced by the activity coefficients.
For a reaction with general stoichiometric coefficients, the same reasoning gives:
\[\boxed{K_{\mathrm{conditional}}=K^\circ\frac{\prod_{\mathrm{reactants}}\gamma_i^{\nu_i}}{\prod_{\mathrm{products}}\gamma_i^{\nu_i}}}\]
This simplified expression assumes all species are represented consistently on the same concentration scale, with standard-state factors handled appropriately. It is useful for understanding why conditional equilibrium constants depend on the ionic medium.
Consider the formation of a complex ion:
\[\mathrm{Ag^++2NH_3\rightleftharpoons[Ag(NH_3)_2]^+}\]
The thermodynamic formation constant is:
\[K_f^\circ=\frac{a_{[\mathrm{Ag(NH_3)_2}]^+}}{a_{\mathrm{Ag^+}}a_{\mathrm{NH_3}}^2}\]
Since the complex ion and free silver ion are charged species, their activities depend on their respective activity coefficients.
Neutral ammonia may also exhibit non-ideal behaviour, although its activity coefficient is often treated differently from that of charged ions. If an approximation assumes its activity coefficient is unity, that is a modelling assumption rather than a universal fact.
The concentration-based formation constant can therefore vary with ionic strength even though the thermodynamic formation constant is fixed at a given temperature.
Consider the sparingly soluble salt silver chloride:
\[\mathrm{AgCl(s)\rightleftharpoons Ag^+(aq)+Cl^-(aq)}\]
The thermodynamic solubility product is:
\[\boxed{K_{\mathrm{sp}}^\circ=a_{\mathrm{Ag^+}}a_{\mathrm{Cl^-}}}\]
The activity of a pure solid in its standard state is taken as unity.
The ionic activity product, often denoted by \(Q_{\mathrm{sp}}\), is calculated using the actual activities of the dissolved ions:
\[Q_{\mathrm{sp}}=a_{\mathrm{Ag^+}}a_{\mathrm{Cl^-}}\]
Comparing \(Q_{\mathrm{sp}}\) with \(K_{\mathrm{sp}}^\circ\) determines the thermodynamic tendency toward precipitation or dissolution.
Assume AgCl dissolves in pure water and neglect other dissolved electrolytes. If its molar solubility is \(s\), then:
\[m_{\mathrm{Ag^+}}\approx s,\qquad m_{\mathrm{Cl^-}}\approx s\]
The ionic strength is approximately:
\[I=\frac12(s+s)=s\]
The thermodynamic solubility product on the molality scale is:
\[K_{\mathrm{sp}}^\circ=\gamma_{\mathrm{Ag^+}}\gamma_{\mathrm{Cl^-}}\left(\frac{s}{m^\circ}\right)^2\]
If the mean ionic activity coefficient is used:
\[K_{\mathrm{sp}}^\circ=\gamma_\pm^2\left(\frac{s}{m^\circ}\right)^2\]
Thus:
\[\boxed{\frac{s}{m^\circ}=\frac{\sqrt{K_{\mathrm{sp}}^\circ}}{\gamma_\pm}}\]
This simplified expression demonstrates that the solubility depends on the activity coefficient. The approximation \(I\approx s\) assumes no other significant ionic species are present and that AgCl is the only source of ions.
Suppose AgCl is dissolved in a solution that already contains NaCl. The chloride ion is a common ion. Its initial presence changes the equilibrium concentrations and generally suppresses the dissolution of AgCl.
The equilibrium condition remains:
\[K_{\mathrm{sp}}^\circ=a_{\mathrm{Ag^+}}a_{\mathrm{Cl^-}}\]
However, the chloride activity is no longer determined solely by the amount of AgCl that dissolves. The initial NaCl also contributes to the ionic strength, which can modify activity coefficients.
Therefore, two effects must be distinguished:
At higher ionic strengths, these effects can operate simultaneously. A correct quantitative treatment must include both when they are significant.
For a weak monoprotic acid:
\[\mathrm{HA\rightleftharpoons H^++A^-}\]
the thermodynamic acid dissociation constant is:
\[K_a^\circ=\frac{a_{\mathrm{H^+}}a_{\mathrm{A^-}}}{a_{\mathrm{HA}}}\]
Using molality-based activities:
\[K_a^\circ=\frac{\gamma_{\mathrm{H^+}}\gamma_{\mathrm{A^-}}}{\gamma_{\mathrm{HA}}}\frac{(m_{\mathrm{H^+}}/m^\circ)(m_{\mathrm{A^-}}/m^\circ)}{m_{\mathrm{HA}}/m^\circ}\]
The thermodynamic dissociation constant is fixed at a given temperature, but the ratio of equilibrium concentrations can change with ionic strength because the activity coefficients change.
The thermodynamic definition of pH is:
\[\boxed{\mathrm{pH}=-\log_{10}a_{\mathrm{H^+}}}\]
On a molality scale:
\[\mathrm{pH}=-\log_{10}\left(\gamma_{\mathrm{H^+}}\frac{m_{\mathrm{H^+}}}{m^\circ}\right)\]
Therefore, pH is not strictly equal to the negative logarithm of the hydrogen-ion molality or molar concentration.
In dilute solutions, the activity coefficient may be close to unity, making concentration-based approximations useful. In more concentrated solutions, neglecting the activity coefficient can introduce significant errors.
It is important to recognise that individual hydrogen-ion activity cannot be isolated by a purely thermodynamic measurement without convention. Operational pH measurements are established through standardisation procedures and measurement conventions.
For the equilibrium:
\[\mathrm{HA\rightleftharpoons H^++A^-}\]
the thermodynamic acid dissociation constant is:
\[K_a^\circ=\frac{a_{\mathrm{H^+}}a_{\mathrm{A^-}}}{a_{\mathrm{HA}}}\]
Rearranging:
\[a_{\mathrm{H^+}}=K_a^\circ\frac{a_{\mathrm{HA}}}{a_{\mathrm{A^-}}}\]
Taking the negative base-10 logarithm:
\[\mathrm{pH}=\mathrm{p}K_a^\circ+\log_{10}\frac{a_{\mathrm{A^-}}}{a_{\mathrm{HA}}}\]
Thus:
\[\boxed{\mathrm{pH}=\mathrm{p}K_a^\circ+\log_{10}\frac{a_{\mathrm{A^-}}}{a_{\mathrm{HA}}}}\]
Expressing activities through molalities gives:
\[\mathrm{pH}=\mathrm{p}K_a^\circ+\log_{10}\frac{\gamma_{\mathrm{A^-}}}{\gamma_{\mathrm{HA}}}+\log_{10}\frac{m_{\mathrm{A^-}}}{m_{\mathrm{HA}}}\]
When the neutral acid activity coefficient is approximated as unity:
\[\mathrm{pH}\approx\mathrm{p}K_a^\circ+\log_{10}\frac{\gamma_{\mathrm{A^-}}m_{\mathrm{A^-}}}{m_{\mathrm{HA}}}\]
The familiar concentration form is recovered when the activity-coefficient ratio is sufficiently close to unity or when its effect is absorbed into a conditional constant.
Buffer solutions commonly contain a weak acid and its conjugate base, often introduced as a salt. For example, an acetic acid–sodium acetate buffer contains neutral acetic acid and the ions Na+ and CH3COO−.
The ionic strength is determined by the concentrations of all ionic species, including added supporting electrolytes, buffer salts, and ions introduced through acid or base additions.
For an idealised mixture containing sodium acetate at molality \(m_s\), with complete dissociation and negligible additional ionic species:
\[m_{\mathrm{Na^+}}=m_s,\qquad m_{\mathrm{CH_3COO^-}}=m_s\]
Therefore:
\[I=\frac12[m_s(1)^2+m_s(-1)^2]\]
\[\boxed{I=m_s}\]
If a strong electrolyte is added to adjust ionic strength, the additional ions must be included in the ionic-strength calculation.
Buffer capacity measures the resistance of a buffer solution to pH change upon addition of acid or base. Activity coefficients, by contrast, describe thermodynamic deviations from ideal solution behaviour.
These concepts are related through chemical equilibria, but they are not interchangeable. A buffer may have substantial buffer capacity while also exhibiting non-ideal activity effects.
For accurate buffer preparation and pH prediction, the acid–base equilibrium, mass balance, charge balance, and relevant activity corrections should be considered together.
For an electrochemical reaction involving \(n\) electrons, the Gibbs energy change is related to the cell potential by:
\[\Delta_rG=-nFE\]
At standard conditions:
\[\Delta_rG^\circ=-nFE^\circ\]
The thermodynamic reaction relation is:
\[\Delta_rG=\Delta_rG^\circ+RT\ln Q\]
Substituting the electrochemical expressions:
\[-nFE=-nFE^\circ+RT\ln Q\]
Dividing by \(-nF\):
\[E=E^\circ-\frac{RT}{nF}\ln Q\]
Therefore:
\[\boxed{E=E^\circ-\frac{RT}{nF}\ln Q}\]
At \(25^\circ\mathrm C\):
\[\boxed{E=E^\circ-\frac{0.05916}{n}\log_{10}Q}\]
The reaction quotient \(Q\) must be constructed from activities, not blindly from raw concentrations.
A concentration cell consists of two electrodes of the same type exposed to solutions with different activities of the electroactive species.
For the half-reaction:
\[\mathrm{M^{n+}+ne^-\rightleftharpoons M(s)}\]
the electrode potential is:
\[E=E^\circ+\frac{RT}{nF}\ln a_{\mathrm{M^{n+}}}\]
For two solutions with activities \(a_1\) and \(a_2\), the potential difference is:
\[\boxed{E_{\mathrm{cell}}=\frac{RT}{nF}\ln\frac{a_2}{a_1}}\]
under the appropriate cell orientation and sign convention.
If the two solutions have different activity coefficients, the activity ratio is not necessarily equal to the concentration ratio. This is why concentration-cell calculations can require activity corrections when the solutions have different ionic strengths or compositions.
Why does the thermodynamic equilibrium constant remain constant when ionic strength changes, while a concentration-based equilibrium constant may change?
Answer: The thermodynamic equilibrium constant is defined using activities and is related to the standard Gibbs energy change. A concentration-based constant incorporates activity coefficients, which may vary with ionic strength.
Why does the extended Debye–Hückel equation approach the limiting law at low ionic strength?
Answer: As \(I\rightarrow0\), the term \(Ba_i\sqrt I\rightarrow0\), so the denominator approaches unity and the finite-size correction becomes negligible.
What is the difference between an ionic activity product and a solubility product?
Answer: The ionic activity product is calculated from the actual activities of dissolved ions in a particular solution. The thermodynamic solubility product is the equilibrium value of that product at saturation for a specified temperature and set of standard states.
Why should pH be defined using hydrogen-ion activity rather than simply hydrogen-ion concentration?
Answer: Chemical potential depends on activity. The activity coefficient accounts for non-ideal interactions, so concentration alone does not completely describe the thermodynamic behaviour of hydrogen ions.
Why might two solutions having identical analytical concentrations show different electrode potentials?
Answer: If their ionic compositions, solvent properties, or ionic strengths differ, their activity coefficients may differ. The Nernst equation depends on activities, so identical concentrations do not necessarily imply identical electrode potentials.
The next part will develop conductance of electrolyte solutions, specific and molar conductivity, Kohlrausch's law, ionic mobility, transport numbers, the Debye–Hückel–Onsager equation, and the relationship between ionic interactions and electrical conductivity, with derivations and numerical examples at MSc Chemistry and CSIR-NET level.
Electrical conductance is a fundamental property of electrolyte solutions and provides important information about ionic movement, dissociation, interionic interactions, and electrochemical processes. It connects the microscopic movement of ions with measurable macroscopic electrical properties.
When an electrolyte dissolves in water, it produces charged species that can move under an applied electric field. Cations migrate towards the cathode, whereas anions migrate towards the anode. Their movement generates an electrical current.
The magnitude of this current depends on several factors:
Electrical conductance measurements are used to determine limiting ionic conductivities, degree of dissociation of weak electrolytes, dissociation constants, solubility of sparingly soluble salts, ionic mobility, transport numbers, and properties of electrolyte solutions.
Conductance tells us how effectively ions carry electric current. The important distinction is that the number of ions and their ability to move are separate factors.
Electrical resistance measures the opposition offered by a material to the flow of electric current. According to Ohm’s law:
\[V=IR\]
where \(V\) is the potential difference, \(I\) is the electric current, and \(R\) is the resistance.
Therefore:
\[\boxed{R=\frac{V}{I}}\]
The SI unit of resistance is the ohm, \(\Omega\).
Conductance is the reciprocal of resistance:
\[\boxed{G=\frac{1}{R}}\]
Its SI unit is the siemens, S.
Substituting \(R=V/I\):
\[G=\frac{I}{V}\]
Conductance measures how easily current passes through a particular specimen or conductivity cell. Unlike conductivity, conductance depends on the dimensions and geometry of the specimen.
Consider a uniform conductor of length \(l\) and cross-sectional area \(A\). Its resistance is:
\[\boxed{R=\rho\frac{l}{A}}\]
where \(\rho\) is the resistivity of the material.
The resistivity is an intrinsic property under specified conditions of temperature, composition, and physical state.
Taking the reciprocal gives:
\[G=\frac{1}{R}=\frac{A}{\rho l}\]
Since conductivity \(\kappa\) is the reciprocal of resistivity:
\[\kappa=\frac{1}{\rho}\]
we obtain:
\[\boxed{G=\kappa\frac{A}{l}}\]
Therefore:
\[\boxed{\kappa=G\frac{l}{A}}\]
The ratio \(l/A\) is the geometrical factor of the cell, while \(A/l\) is the geometrical factor relating conductivity to conductance.
Specific conductivity, commonly called conductivity and denoted by \(\kappa\), is the conductance of a material having unit length and unit cross-sectional area under the idealised uniform-current geometry.
Its SI unit is:
\[\boxed{\mathrm{S\,m^{-1}}}\]
In laboratory chemistry, conductivity is also commonly expressed in \(\mathrm{S\,cm^{-1}}\) or \(\mathrm{mS\,cm^{-1}}\).
From the resistance equation:
\[\kappa=\frac{l}{RA}\]
Since \(G=1/R\):
\[\boxed{\kappa=G\frac{l}{A}}\]
The conductivity cell constant is defined as:
\[\boxed{K_{\mathrm{cell}}=\frac{l}{A}}\]
Its SI unit is \(\mathrm{m^{-1}}\), although it is often expressed in \(\mathrm{cm^{-1}}\) in laboratory measurements.
Therefore:
\[\boxed{\kappa=G K_{\mathrm{cell}}}\]
Since \(G=1/R\):
\[\boxed{\kappa=\frac{K_{\mathrm{cell}}}{R}}\]
The cell constant depends on the electrode geometry. It is usually determined experimentally using a standard electrolyte solution of known conductivity, such as a suitable potassium chloride solution.
A conductivity cell has a cell constant of \(1.20\ \mathrm{cm^{-1}}\). The measured resistance of an electrolyte solution is \(600\ \Omega\). Calculate its conductivity.
First, calculate conductance:
\[G=\frac{1}{600}=1.667\times10^{-3}\ \mathrm S\]
Then:
\[\kappa=G K_{\mathrm{cell}}\]
\[\kappa=(1.667\times10^{-3})(1.20)\]
Therefore:
\[\boxed{\kappa=2.00\times10^{-3}\ \mathrm{S\,cm^{-1}}}\]
To convert to SI units, use \(1\ \mathrm{S\,cm^{-1}}=100\ \mathrm{S\,m^{-1}}\). Thus:
\[\boxed{\kappa=0.200\ \mathrm{S\,m^{-1}}}\]
Conductivity generally increases when more ions are introduced into a solution. Therefore, a concentrated electrolyte solution may have a higher conductivity than a dilute solution simply because it contains more charge carriers per unit volume.
To compare electrolyte solutions at different concentrations, we introduce molar conductivity, which relates conductivity to the amount of electrolyte present.
Molar conductivity, \(\Lambda_m\), is the conductivity multiplied by the volume of solution containing one mole of electrolyte.
If the concentration is \(c\) in \(\mathrm{mol\,m^{-3}}\), then one mole occupies a volume:
\[V_m=\frac{1}{c}\]
Therefore:
\[\boxed{\Lambda_m=\frac{\kappa}{c}}\]
When conductivity is expressed in \(\mathrm{S\,m^{-1}}\) and concentration in \(\mathrm{mol\,m^{-3}}\), the unit of molar conductivity is:
\[\boxed{\mathrm{S\,m^2\,mol^{-1}}}\]
In conventional chemistry laboratory units, concentration is frequently expressed in \(\mathrm{mol\,L^{-1}}\) and conductivity in \(\mathrm{S\,cm^{-1}}\). The corresponding formula becomes:
\[\boxed{\Lambda_m=\frac{1000\kappa}{c}}\]
with \(\Lambda_m\) in \(\mathrm{S\,cm^2\,mol^{-1}}\), provided \(c\) is the numerical concentration in \(\mathrm{mol\,L^{-1}}\).
Suppose the concentration is \(c\ \mathrm{mol\,L^{-1}}\). One litre of solution contains \(c\) moles of electrolyte. Therefore, the volume containing one mole is:
\[V_m=\frac{1}{c}\ \mathrm L\]
Converting litres to cubic centimetres:
\[V_m=\frac{1000}{c}\ \mathrm{cm^3}\]
Molar conductivity is conductivity multiplied by this volume:
\[\Lambda_m=\kappa\frac{1000}{c}\]
Thus:
\[\boxed{\Lambda_m=\frac{1000\kappa}{c}}\]
An electrolyte solution has conductivity \(1.50\times10^{-3}\ \mathrm{S\,cm^{-1}}\) and concentration \(0.0100\ \mathrm{mol\,L^{-1}}\). Calculate its molar conductivity.
Using:
\[\Lambda_m=\frac{1000\kappa}{c}\]
\[\Lambda_m=\frac{1000(1.50\times10^{-3})}{0.0100}\]
Therefore:
\[\boxed{\Lambda_m=150\ \mathrm{S\,cm^2\,mol^{-1}}}\]
Equivalent conductivity is an older convention based on the number of chemical equivalents rather than moles of electrolyte.
It is defined as the conductivity multiplied by the volume of solution containing one equivalent of electrolyte.
If the normality is \(N\) in equivalents per litre:
\[\boxed{\Lambda_{\mathrm{eq}}=\frac{1000\kappa}{N}}\]
Its conventional unit is \(\mathrm{S\,cm^2\,equiv^{-1}}\).
Equivalent conductivity depends on the equivalent convention used for the reaction. Molar conductivity is generally more convenient for modern thermodynamic and physical chemistry treatments.
For a strong electrolyte, dilution reduces the number of ions per unit volume. Therefore, its conductivity usually decreases as the solution becomes more dilute.
However, the molar conductivity of a strong electrolyte generally increases upon dilution. This happens because interionic interactions become weaker and ions can move more independently.
At sufficiently low concentration, the molar conductivity approaches its limiting value, denoted by \(\Lambda_m^\circ\).
The superscript degree denotes the limiting infinite-dilution value, not necessarily the conductivity of a real solution at exactly zero concentration.
Weak electrolytes behave differently because dilution increases their degree of ionisation in addition to reducing interionic interactions.
For a weak electrolyte such as acetic acid:
\[\mathrm{CH_3COOH\rightleftharpoons H^++CH_3COO^-}\]
dilution shifts the dissociation equilibrium towards the ions under the usual idealised conditions.
Consequently, the molar conductivity of a weak electrolyte can increase substantially upon dilution. This increase reflects both improved ionic mobility and an increased fraction of ionised electrolyte.
The distinction is fundamental:
The limiting molar conductivity, \(\Lambda_m^\circ\), is the molar conductivity extrapolated to infinite dilution. At this limit, interionic interactions become negligible in the conventional electrolyte model.
For an electrolyte producing \(\nu_+\) cations and \(\nu_-\) anions per formula unit, the limiting molar conductivity is:
\[\boxed{\Lambda_m^\circ=\nu_+\lambda_+^\circ+\nu_-\lambda_-^\circ}\]
Here, \(\lambda_+^\circ\) and \(\lambda_-^\circ\) are the limiting ionic molar conductivities of the cation and anion, respectively.
The limiting ionic conductivities include the contributions of the respective ions under the specified solvent and temperature conditions.
At infinite dilution, each ion behaves approximately independently in its contribution to the total current. The conductivity contribution of a particular ion depends on its charge, concentration, and mobility.
The limiting molar conductivity of an electrolyte is therefore the sum of the limiting ionic contributions, weighted by the number of ions produced per formula unit.
For sodium chloride:
\[\mathrm{NaCl\rightarrow Na^++Cl^-}\]
the limiting molar conductivity is:
\[\boxed{\Lambda_m^\circ(\mathrm{NaCl})=\lambda_{\mathrm{Na^+}}^\circ+\lambda_{\mathrm{Cl^-}}^\circ}\]
For calcium chloride:
\[\mathrm{CaCl_2\rightarrow Ca^{2+}+2Cl^-}\]
the corresponding expression is:
\[\boxed{\Lambda_m^\circ(\mathrm{CaCl_2})=\lambda_{\mathrm{Ca^{2+}}}^\circ+2\lambda_{\mathrm{Cl^-}}^\circ}\]
Kohlrausch’s law of independent ionic migration states that at infinite dilution, each ion contributes independently to the total molar conductivity of an electrolyte, regardless of the identity of the counterion.
In mathematical form:
\[\boxed{\Lambda_m^\circ=\sum_i\nu_i\lambda_i^\circ}\]
For a binary electrolyte:
\[\boxed{\Lambda_m^\circ=\nu_+\lambda_+^\circ+\nu_-\lambda_-^\circ}\]
This law is extremely important because it allows limiting molar conductivities of weak electrolytes to be calculated from the limiting conductivities of strong electrolytes.
At finite concentration, ions interact electrostatically. These interactions influence ionic movement and make the conductivity contribution of an ion dependent on the surrounding ionic medium.
As the solution approaches infinite dilution, the electrostatic interactions between ions become negligible in the limiting description. Each ionic species then has a limiting conductivity contribution that can be added to those of the other ions.
This additivity does not mean that ions have identical mobilities. Different ions have different sizes, hydration characteristics, charges, and frictional interactions with the solvent.
Acetic acid is a weak electrolyte, so its limiting molar conductivity cannot usually be obtained by straightforward extrapolation of ordinary conductivity measurements using the simple strong-electrolyte relationship.
However, Kohlrausch’s law allows it to be calculated using suitable strong electrolytes.
The limiting molar conductivity of acetic acid is:
\[\Lambda_m^\circ(\mathrm{CH_3COOH})=\lambda_{\mathrm{H^+}}^\circ+\lambda_{\mathrm{CH_3COO^-}}^\circ\]
Consider the following three limiting molar conductivities:
\[\Lambda_m^\circ(\mathrm{HCl})=\lambda_{\mathrm{H^+}}^\circ+\lambda_{\mathrm{Cl^-}}^\circ\]
\[\Lambda_m^\circ(\mathrm{CH_3COONa})=\lambda_{\mathrm{Na^+}}^\circ+\lambda_{\mathrm{CH_3COO^-}}^\circ\]
\[\Lambda_m^\circ(\mathrm{NaCl})=\lambda_{\mathrm{Na^+}}^\circ+\lambda_{\mathrm{Cl^-}}^\circ\]
Add the first two equations and subtract the third:
\[\begin{aligned}\Lambda_m^\circ(\mathrm{HCl})&+\Lambda_m^\circ(\mathrm{CH_3COONa})\\&-\Lambda_m^\circ(\mathrm{NaCl})\end{aligned}\]
Substitution gives:
\[\lambda_{\mathrm{H^+}}^\circ+\lambda_{\mathrm{Cl^-}}^\circ+\lambda_{\mathrm{Na^+}}^\circ+\lambda_{\mathrm{CH_3COO^-}}^\circ-\lambda_{\mathrm{Na^+}}^\circ-\lambda_{\mathrm{Cl^-}}^\circ\]
The sodium and chloride contributions cancel, leaving:
\[\boxed{\Lambda_m^\circ(\mathrm{CH_3COOH})=\Lambda_m^\circ(\mathrm{HCl})+\Lambda_m^\circ(\mathrm{CH_3COONa})-\Lambda_m^\circ(\mathrm{NaCl})}\]
This is a standard application of Kohlrausch’s law.
Suppose the limiting molar conductivities at a specified temperature are:
Then:
\[\Lambda_m^\circ(\mathrm{CH_3COOH})=426.0+91.0-126.0\]
Therefore:
\[\boxed{\Lambda_m^\circ(\mathrm{CH_3COOH})=391.0\ \mathrm{S\,cm^2\,mol^{-1}}}\]
The numbers are illustrative. In an actual examination problem, use the supplied or tabulated values for the specified temperature.
The limiting conductivity of a weak electrolyte can be calculated by combining the limiting conductivities of strong electrolytes whose ions can be algebraically combined to reproduce the desired ionic contributions.
This is particularly useful for weak acids and weak bases, for which direct extrapolation is difficult because the degree of ionisation changes significantly with concentration.
For a sparingly soluble salt, the conductivity of a saturated solution can be used to estimate its solubility. The method is especially useful when the solubility is too small to measure accurately by ordinary concentration analysis.
The basic procedure is:
If the corrected conductivity is \(\kappa_{\mathrm{salt}}\) and the molar conductivity is approximated by its limiting value, then:
\[\boxed{s\approx\frac{1000\kappa_{\mathrm{salt}}}{\Lambda_m^\circ}}\]
Here, \(\kappa_{\mathrm{salt}}\) is in \(\mathrm{S\,cm^{-1}}\), \(\Lambda_m^\circ\) is in \(\mathrm{S\,cm^2\,mol^{-1}}\), and \(s\) is in \(\mathrm{mol\,L^{-1}}\).
The approximation is most suitable for extremely dilute saturated solutions, where the molar conductivity is close to its limiting value. The contribution of dissolved carbon dioxide or other impurities may need to be considered in practical measurements.
Combining conductivity measurements for several electrolytes enables the ionic contributions to be calculated. However, conductivity measurements alone do not independently provide every single-ion contribution without suitable combinations, reference information, or additional conventions.
The independent migration principle provides the mathematical basis for extracting limiting ionic conductivity contributions from electrolyte data.
The degree of dissociation, \(\alpha\), is the fraction of the original electrolyte molecules that dissociate into ions.
For a weak electrolyte undergoing dissociation:
\[\mathrm{AB\rightleftharpoons A^++B^-}\]
if the initial concentration is \(c\) and the fraction dissociated is \(\alpha\), the equilibrium concentrations under the simplest idealised model are:
\[[\mathrm{AB}]=c(1-\alpha)\]
\[[\mathrm{A^+}]=c\alpha\]
\[[\mathrm{B^-}]=c\alpha\]
These expressions assume the only source of these ions is the dissociation of AB and that side reactions are negligible.
For a weak electrolyte, the molar conductivity at a finite concentration reflects the fraction of electrolyte that has dissociated as well as the mobilities of the ions.
Under the classical approximation that the ionic conductivity contributions scale with the degree of dissociation:
\[\boxed{\alpha=\frac{\Lambda_m}{\Lambda_m^\circ}}\]
where \(\Lambda_m\) is the molar conductivity at the specified concentration and \(\Lambda_m^\circ\) is the limiting molar conductivity.
This relation is particularly useful for weak electrolytes. For strong electrolytes, it should not be interpreted as a literal degree of dissociation, because strong electrolytes are already almost completely dissociated and their molar conductivity changes substantially through interionic effects.
A weak electrolyte has molar conductivity \(40.0\ \mathrm{S\,cm^2\,mol^{-1}}\) at a particular concentration. Its limiting molar conductivity is \(400\ \mathrm{S\,cm^2\,mol^{-1}}\). Calculate the approximate degree of dissociation.
\[\alpha=\frac{\Lambda_m}{\Lambda_m^\circ}\]
\[\alpha=\frac{40.0}{400}=0.10\]
Therefore:
\[\boxed{\alpha=0.10=10\%}\]
The result indicates that approximately 10% of the electrolyte is dissociated under the stated idealised interpretation.
Consider a weak binary electrolyte:
\[\mathrm{AB\rightleftharpoons A^++B^-}\]
Let the initial concentration be \(c\), and let the degree of dissociation be \(\alpha\).
At equilibrium:
\[[\mathrm{AB}]=c(1-\alpha)\]
\[[\mathrm{A^+}]=c\alpha\]
\[[\mathrm{B^-}]=c\alpha\]
The concentration-based dissociation constant in the idealised treatment is:
\[K_c=\frac{[\mathrm{A^+}][\mathrm{B^-}]}{[\mathrm{AB}]}\]
Substituting the equilibrium concentrations:
\[K_c=\frac{(c\alpha)(c\alpha)}{c(1-\alpha)}\]
Cancel one factor of \(c\):
\[\boxed{K_c=\frac{c\alpha^2}{1-\alpha}}\]
This is the classical form of Ostwald’s dilution law for a weak binary electrolyte, assuming ideal solution behaviour.
For a very weak electrolyte, the degree of dissociation may be sufficiently small that:
\[\alpha\ll1\]
Then:
\[1-\alpha\approx1\]
Therefore:
\[K_c\approx c\alpha^2\]
Rearranging:
\[\boxed{\alpha\approx\sqrt{\frac{K_c}{c}}}\]
This equation predicts that the degree of dissociation increases as concentration decreases.
Since:
\[\alpha=\frac{\Lambda_m}{\Lambda_m^\circ}\]
substitution into Ostwald’s dilution law gives:
\[K_c=\frac{c(\Lambda_m/\Lambda_m^\circ)^2}{1-\Lambda_m/\Lambda_m^\circ}\]
Multiplying the numerator and denominator by \(\Lambda_m^\circ\):
\[\boxed{K_c=\frac{c\Lambda_m^2}{\Lambda_m^\circ(\Lambda_m^\circ-\Lambda_m)}}\]
This equation permits the dissociation constant of a weak electrolyte to be estimated from conductivity measurements.
Strictly, the expression is a concentration-based approximation. For accurate thermodynamic work, activity coefficients must be included, particularly when the ionic strength is not sufficiently small.
Ionic mobility, \(u_i\), is defined as the drift velocity of an ion per unit electric field strength.
\[\boxed{u_i=\frac{v_i}{E}}\]
where \(v_i\) is the magnitude of the steady drift velocity and \(E\) is the electric field strength.
The SI unit of mobility is:
\[\mathrm{m^2\,V^{-1}\,s^{-1}}\]
Mobility depends on the charge of the ion, its effective hydrodynamic size, the viscosity of the solvent, the temperature, and interactions with surrounding species.
Consider an ionic species with charge \(z_i e\), number density \(n_i\), and drift velocity \(v_i\). Its current density contribution is:
\[j_i=n_i z_i e v_i\]
The sign of \(z_i\) determines the current direction. Since drift is driven by the electric field, the conventional current contribution is in the direction of the field for both positive and negative charge carriers.
Using the magnitude of mobility, the ionic drift contribution gives a conductivity proportional to the square of the ionic charge:
\[\boxed{\kappa_i=n_i|z_i|e u_i}\]
This expression is written for the conductivity magnitude of a species, with \(u_i\) defined as a positive mobility magnitude and \(n_i\) expressed as number density.
If the molar concentration is \(c_i\) in \(\mathrm{mol\,m^{-3}}\), then \(n_i=N_Ac_i\). Since \(F=N_Ae\):
\[\boxed{\kappa_i=F c_i |z_i|u_i}\]
The total conductivity is the sum of the ionic contributions:
\[\boxed{\kappa=F\sum_i c_i|z_i|u_i}\]
This idealised expression highlights the dependence of conductivity on ionic concentration, charge, and mobility. At finite concentrations, interactions can influence the effective transport behaviour, so the measured conductivity need not equal the sum of independent infinite-dilution contributions.
At infinite dilution, the limiting ionic molar conductivity of ion \(i\) is related to its mobility by:
\[\boxed{\lambda_i^\circ=F|z_i|u_i^\circ}\]
where \(u_i^\circ\) is the limiting ionic mobility.
For a 1:1 electrolyte:
\[\Lambda_m^\circ=\lambda_+^\circ+\lambda_-^\circ\]
Therefore:
\[\boxed{\Lambda_m^\circ=F(u_+^\circ+u_-^\circ)}\]
For ions with different charge magnitudes, the charge factors must be retained in each ionic contribution.
The transport number, also called the transference number, is the fraction of the total electrical current carried by a particular ionic species under specified conditions.
For a binary electrolyte containing cations and anions:
\[\boxed{t_++t_-=1}\]
At infinite dilution, the transport numbers can be expressed in terms of limiting ionic conductivities:
\[\boxed{t_+^\circ=\frac{\nu_+\lambda_+^\circ}{\Lambda_m^\circ}}\]
\[\boxed{t_-^\circ=\frac{\nu_-\lambda_-^\circ}{\Lambda_m^\circ}}\]
Here, \(\nu_+\) and \(\nu_-\) are the stoichiometric numbers of cations and anions per formula unit.
For a 1:1 electrolyte, the total limiting molar conductivity is:
\[\Lambda_m^\circ=\lambda_+^\circ+\lambda_-^\circ\]
The cation contribution to the total current is proportional to its ionic conductivity, and the anion contribution is proportional to its ionic conductivity.
Thus:
\[t_+^\circ=\frac{\lambda_+^\circ}{\lambda_+^\circ+\lambda_-^\circ}\]
Similarly:
\[t_-^\circ=\frac{\lambda_-^\circ}{\lambda_+^\circ+\lambda_-^\circ}\]
Adding the expressions:
\[t_+^\circ+t_-^\circ=\frac{\lambda_+^\circ+\lambda_-^\circ}{\Lambda_m^\circ}\]
Therefore:
\[\boxed{t_+^\circ+t_-^\circ=1}\]
Suppose a 1:1 electrolyte has limiting ionic conductivities:
\[\lambda_+^\circ=50\ \mathrm{S\,cm^2\,mol^{-1}}\]
\[\lambda_-^\circ=150\ \mathrm{S\,cm^2\,mol^{-1}}\]
The limiting molar conductivity is:
\[\Lambda_m^\circ=50+150=200\ \mathrm{S\,cm^2\,mol^{-1}}\]
The cation transport number is:
\[t_+^\circ=\frac{50}{200}=0.25\]
The anion transport number is:
\[t_-^\circ=\frac{150}{200}=0.75\]
Hence:
\[\boxed{t_+^\circ=0.25,\qquad t_-^\circ=0.75}\]
The anion carries 75% of the total current in the infinite-dilution limiting description.
The Debye–Hückel limiting law describes activity coefficients, whereas the Debye–Hückel–Onsager equation describes the concentration dependence of the molar conductivity of strong electrolytes in sufficiently dilute solutions.
For strong electrolytes, molar conductivity decreases as concentration increases. The primary theoretical explanation involves the influence of the ionic atmosphere on ionic transport.
Two major effects are traditionally identified:
These effects reduce ionic mobility at finite concentration relative to its infinite-dilution limiting value.
For a sufficiently dilute strong electrolyte, the classical limiting form is:
\[\boxed{\Lambda_m=\Lambda_m^\circ-(A+B\Lambda_m^\circ)\sqrt c}\]
Here:
The numerical values of \(A\) and \(B\) are unit-dependent. Therefore, the constants must be taken from a consistent version of the equation rather than transferred between incompatible unit systems.
The essential theoretical result is that the leading concentration correction is proportional to \(\sqrt c\) in the dilute limit.
The equation can be rearranged into the linear form:
\[\Lambda_m=\Lambda_m^\circ-(A+B\Lambda_m^\circ)\sqrt c\]
Comparing with:
\[y=b+mx\]
we identify:
Therefore, a plot of \(\Lambda_m\) against \(\sqrt c\) is approximately linear at sufficiently low concentrations for a strong electrolyte. The intercept provides an estimate of the limiting molar conductivity.
This linear relationship is not generally applicable to weak electrolytes because their degree of dissociation changes substantially with concentration.
A conductivity cell usually contains two electrodes of known geometry immersed in the electrolyte solution. The measured resistance is converted into conductivity using the cell constant.
For accurate measurements:
When direct current is passed through an electrolyte cell, electrode reactions and accumulation of ions near the electrodes can alter the measured resistance. This phenomenon can introduce polarisation effects.
Alternating-current methods reduce some of these problems by periodically reversing the electric field. In modern conductivity instruments, the measurement frequency and electrode design are chosen to minimise systematic errors.
Conductivity generally increases with temperature in ordinary aqueous electrolyte solutions because ionic mobility increases as solvent viscosity decreases. Temperature also influences dissociation equilibria, dielectric properties, and other solvent characteristics.
Therefore, conductivity data and limiting ionic conductivities must be compared at the same temperature or corrected using an appropriate temperature model.
A \(0.0050\ \mathrm{mol\,L^{-1}}\) electrolyte solution has conductivity \(7.50\times10^{-4}\ \mathrm{S\,cm^{-1}}\). The limiting molar conductivity of the electrolyte is \(150\ \mathrm{S\,cm^2\,mol^{-1}}\). Calculate the molar conductivity and, under the weak-electrolyte approximation, the degree of dissociation.
Use:
\[\Lambda_m=\frac{1000\kappa}{c}\]
Substituting:
\[\Lambda_m=\frac{1000(7.50\times10^{-4})}{0.0050}\]
Therefore:
\[\boxed{\Lambda_m=150\ \mathrm{S\,cm^2\,mol^{-1}}}\]
For a weak electrolyte under the classical approximation:
\[\alpha=\frac{\Lambda_m}{\Lambda_m^\circ}\]
Hence:
\[\alpha=\frac{150}{150}=1.00\]
Therefore:
\[\boxed{\alpha=100\%}\]
This result indicates that the numerical data are consistent with complete dissociation under the simplified model. If the electrolyte were independently known to be weak, the equality would require scrutiny because experimental uncertainty, the assumed limiting conductivity, and activity or mobility effects could invalidate a literal interpretation.
Why does conductivity decrease on dilution while molar conductivity usually increases?
Answer: Conductivity depends on the number of charge carriers per unit volume, which decreases upon dilution. Molar conductivity is normalised to the amount of electrolyte and generally increases because ionic interactions weaken and, for weak electrolytes, the degree of dissociation also increases.
Why is Kohlrausch’s law particularly useful for weak electrolytes?
Answer: It allows the limiting molar conductivity of a weak electrolyte to be calculated from the limiting conductivities of suitable strong electrolytes, avoiding the difficulty of direct extrapolation when the degree of dissociation changes strongly with concentration.
Why is the plot of molar conductivity against the square root of concentration approximately linear for strong electrolytes at low concentration?
Answer: The Debye–Hückel–Onsager limiting treatment predicts a leading correction proportional to \(\sqrt c\), allowing the limiting molar conductivity to be estimated from the intercept.
What is the physical meaning of the relaxation effect?
Answer: The ionic atmosphere requires time to reorganise when an ion moves. This creates an asymmetric charge distribution that produces an opposing electrostatic contribution to the ion’s motion.
Can conductivity measurements directly determine the absolute mobility of every individual ion?
Answer: Conductivity measurements provide combined ionic contributions. Individual ionic mobilities can be determined using additional transport information or suitable reference conventions and experimental methods, but a single bulk conductivity measurement does not independently determine every ion’s mobility.
In standard conductometric measurements, the applied electric field is weak (typically less than \(5\ \mathrm{V\,cm^{-1}}\)) and alternating at audio frequencies (\(\sim 1\ \mathrm{kHz}\)). Under these conditions, the ionic atmosphere retards migration via relaxation and electrophoretic drags, yielding concentration-dependent conductivities described by the Debye–Hückel–Onsager equation.
When extreme field strengths or high frequencies are applied, these retarding forces break down, producing marked increases in electrical conductance.
The Wien effect refers to the increase in conductivity observed when an electrolyte solution is subjected to very high electrical potential gradients (on the order of \(10^4\) to \(10^5\ \mathrm{V\,cm^{-1}}\)).
The Debye–Falkenhagen effect describes the increase in molar conductivity observed when alternating electric fields of very high frequency (\(10^6\) to \(10^7\ \mathrm{Hz}\)) are applied.
At standard measurement frequencies, the central ion moves continuously in one direction long enough for the ionic atmosphere to lag behind, creating an opposing electrostatic force (relaxation drag). At radio frequencies, the direction of the field reverses more rapidly than the relaxation time \(\tau\) of the ionic atmosphere:
\[\tau = \frac{\varepsilon}{\kappa^2 \Lambda_m^\circ} \approx 10^{-10}\ \text{to}\ 10^{-8}\ \mathrm{s}\]
Because the central ion merely oscillates over minute sub-molecular amplitudes, the surrounding charge cloud does not have time to become asymmetric; it remains spherically symmetrical around the central position. The relaxation force vanishes, and molar conductivity increases toward a limiting value.
In a steady electric field \(E\), the electrostatic accelerating force acting on an ion of charge \(z_i e\) is balanced by the hydrodynamic frictional force exerted by the viscous solvent. Modeling the ion as a sphere of hydrodynamic radius \(r_h\) moving through a continuum of viscosity \(\eta_0\) gives:
\[z_i e E = 6\pi \eta_0 r_h v_i\]
Solving for the ionic mobility \(u_i = v_i / E\):
\[\boxed{u_i = \frac{|z_i|e}{6\pi \eta_0 r_h}}\]
Using the relation between mobility and limiting ionic conductivity \(\lambda_i^\circ = F |z_i| u_i^\circ = N_A e |z_i| u_i^\circ\):
\[\boxed{\lambda_i^\circ = \frac{z_i^2 F e}{6\pi \eta_0 r_h}}\]
This formulation explains why bare crystallographic radii do not determine ionic conductance in solution. For alkali metal ions, the crystal radii increase down the group:
\[r_{\mathrm{Li^+}} < r_{\mathrm{Na^+}} < r_{\mathrm{K^+}} < r_{\mathrm{Rb^+}} < r_{\mathrm{Cs^+}}\]
Because smaller ions have higher surface charge densities, they bind solvent dipoles more strongly. Consequently, the effective hydrated radii follow the reverse order:
\[r_{h,\mathrm{Li^+}} > r_{h,\mathrm{Na^+}} > r_{h,\mathrm{K^+}} > r_{h,\mathrm{Rb^+}} > r_{h,\mathrm{Cs^+}}\]
As a result, limiting ionic conductivities in water increase from lithium to caesium:
\[\lambda_{\mathrm{Li^+}}^\circ < \lambda_{\mathrm{Na^+}}^\circ < \lambda_{\mathrm{K^+}}^\circ < \lambda_{\mathrm{Rb^+}}^\circ < \lambda_{\mathrm{Cs^+}}^\circ\]
Multiplying the limiting molar conductivity of an ion by the dynamic viscosity of the solvent yields:
\[\boxed{\lambda_i^\circ \eta_0 = \frac{z_i^2 F e}{6\pi r_h} = \text{constant}}\]
Walden’s rule states that for large, poorly solvated ions whose hydrodynamic radius does not vary from solvent to solvent, the product \(\Lambda_m^\circ \eta_0\) is independent of the solvent at a given temperature. Deviations occur when strong specific ion–solvent interactions change the solvation sheath radius \(r_h\).
The limiting molar conductivities of \(\mathrm{H^+}\) (\(349.6\ \mathrm{S\,cm^2\,mol^{-1}}\)) and \(\mathrm{OH^-}\) (\(198.3\ \mathrm{S\,cm^2\,mol^{-1}}\)) in water at \(25^\circ\mathrm{C}\) are exceptionally larger than those of other monovalent ions (typically \(50\) to \(75\ \mathrm{S\,cm^2\,mol^{-1}}\)).
This anomaly occurs because protons and hydroxide ions do not transport solely via hydrodynamic bodily translation. Instead, charge migrates via the Grotthuss mechanism (proton jumping) across a hydrogen-bonded water network:
\[\mathrm{H_3O^+ + H_2O \rightleftharpoons H_2O + H_3O^+}\]
A proton transfers along a pre-existing hydrogen bond to an adjacent water molecule, converting it into a hydronium ion, which then passes a proton to the next neighbor. The rate-determining step in this pathway is not the proton transfer itself, but the dielectric rotation of neighboring water molecules required to orient the hydrogen-bonding network favorably.
Hittorf’s method evaluates transference numbers by measuring analytical concentration changes in isolated compartments (anode, cathode, and central) after passing a measured quantity of electricity through the cell.
Assuming non-attackable electrodes and migration of both ions:
Using a coulometer connected in series to record the total number of equivalents deposited (\(Q / F\)):
\[\boxed{t_+ = \frac{\text{Fall in equivalents in anodic compartment}}{\text{Total equivalents deposited on coulometer electrodes}}}\]
\[\boxed{t_- = \frac{\text{Fall in equivalents in cathodic compartment}}{\text{Total equivalents deposited on coulometer electrodes}}}\]
The central compartment must retain its initial analytical concentration throughout the run to verify that diffusion between the terminal compartments has not corrupted the data.
The moving boundary method tracks the physical displacement of a sharp boundary formed between two electrolyte solutions having a common ion (e.g., \(\mathrm{HCl}\) and \(\mathrm{CdCl_2}\)) housed in a calibrated vertical tube of cross-sectional area \(A\).
Let the leading electrolyte be \(M_1X\) (concentration \(c\) in equivalents per unit volume) and the indicator electrolyte be \(M_2X\). To maintain a sharp boundary during upward movement, the mobility of the leading ion must exceed that of the indicator ion (\(u_{M_1^+} > u_{M_2^+}\)), and the denser solution must remain below the less dense solution.
When a steady current \(I\) passes for time \(t\), the boundary moves a distance \(l\). The volume swept by the boundary is \(V = l A\), containing \(c l A\) equivalents of cation. The total charge passed is \(I t\), equivalent to \(I t / F\) chemical equivalents.
The transport number of the leading cation is therefore:
\[\boxed{t_+ = \frac{c l A F}{I t}}\]
The next part will cover electrochemical cells, electrode potentials, the Nernst equation in depth, reference electrodes, liquid-junction potentials, concentration cells, thermodynamic derivations of cell EMF, and the determination of thermodynamic quantities from electrochemical measurements. It will connect ionic activity and conductance with the electrochemical principles frequently tested in MSc Chemistry and CSIR-NET examinations.
Electrochemistry is the branch of physical chemistry that studies the relationship between chemical reactions and electrical energy. It explains how chemical energy can be converted into electrical energy and how electrical energy can drive chemical transformations.
Electrochemical principles are important in batteries, fuel cells, corrosion, electrolysis, sensors, electrochemical energy storage, industrial electroplating, and analytical chemistry.
The fundamental process in electrochemistry is the transfer of electrons between chemical species. Oxidation involves the loss of electrons, whereas reduction involves the gain of electrons.
For example, consider the reaction between zinc metal and copper(II) ions:
\[\mathrm{Zn(s)+Cu^{2+}(aq)\rightarrow Zn^{2+}(aq)+Cu(s)}\]
Zinc loses electrons and undergoes oxidation:
\[\mathrm{Zn(s)\rightarrow Zn^{2+}(aq)+2e^-}\]
Copper(II) ions gain electrons and undergo reduction:
\[\mathrm{Cu^{2+}(aq)+2e^-\rightarrow Cu(s)}\]
The overall reaction is obtained by adding the two half-reactions and cancelling the electrons.
Remember: oxidation and reduction always occur together. An electrochemical cell separates these processes spatially so that electron transfer can be used to produce electrical work.
A galvanic cell converts the Gibbs energy released by a spontaneous chemical reaction into electrical energy.
The reaction proceeds spontaneously in the direction that produces positive cell potential under the specified conditions and the chosen cell orientation.
For a conventional galvanic cell:
In a galvanic cell, the anode is negative relative to the cathode under the usual operating conditions.
An electrolytic cell uses an external electrical source to drive a non-spontaneous chemical reaction.
Oxidation still occurs at the anode, and reduction still occurs at the cathode. These definitions never change with the type of cell.
However, the electrode signs differ from those of a galvanic cell under ordinary operating conditions:
The sign of an electrode depends on the type of cell and its operating conditions, whereas the definitions of anode and cathode depend exclusively on oxidation and reduction.
A classical example of a galvanic cell is the Daniell cell, which uses zinc and copper electrodes immersed in solutions containing Zn2+ and Cu2+ ions, respectively.
The cell can be represented as:
\[\mathrm{Zn(s)|Zn^{2+}(aq)||Cu^{2+}(aq)|Cu(s)}\]
The single vertical line represents a phase boundary, while the double vertical line represents a liquid junction or salt bridge in the conventional cell notation.
The salt bridge provides an ionic pathway between the two half-cells and helps maintain electrical neutrality. Without it, charge accumulation would rapidly inhibit the sustained operation of the cell.
At the zinc electrode, oxidation occurs:
\[\mathrm{Zn(s)\rightarrow Zn^{2+}(aq)+2e^-}\]
At the copper electrode, reduction occurs:
\[\mathrm{Cu^{2+}(aq)+2e^-\rightarrow Cu(s)}\]
The overall reaction is:
\[\boxed{\mathrm{Zn(s)+Cu^{2+}(aq)\rightarrow Zn^{2+}(aq)+Cu(s)}}\]
Electrons flow through the external wire from zinc to copper when the cell operates spontaneously under ordinary conditions.
The salt bridge serves several purposes:
A salt bridge does not carry electrons through its bulk solution. Charge is transported through the electrolyte by ionic movement.
When a metal is placed in a solution containing its ions, a tendency exists for metal atoms to enter the solution as ions and for dissolved ions to deposit as metal.
For a metal M, these processes may be represented as:
\[\mathrm{M(s)\rightleftharpoons M^{n+}(aq)+ne^-}\]
Depending on the relative chemical potentials of the phases, charge separation can develop at the interface. This interfacial charge separation produces an electrical potential difference.
The resulting electrode potential reflects the equilibrium or operating conditions at the electrode–electrolyte interface.
A voltmeter measures a potential difference between two terminals. It does not directly measure the absolute potential of a single electrode in isolation.
Therefore, electrode potentials are measured relative to a reference electrode. The standard hydrogen electrode is conventionally assigned a standard electrode potential of zero volts.
This is a reference convention, not a statement that the hydrogen electrode has no interfacial electrical potential in an absolute physical sense.
The standard electrode potential, \(E^\circ\), is the electrode potential for a specified reduction half-reaction under standard-state conditions, measured relative to the standard hydrogen electrode.
For a reduction half-reaction:
\[\mathrm{Ox}+ne^-\rightarrow\mathrm{Red}\]
the standard potential refers to the specified standard states of the participating species. For dissolved species, activities are defined relative to their standard states; for pure solids and liquids in their standard states, the activity is conventionally unity.
Standard electrode potentials are commonly tabulated as reduction potentials.
A more positive standard reduction potential indicates a greater thermodynamic tendency for the specified reduction reaction relative to the reference electrode under standard-state conditions.
A more negative standard reduction potential indicates a less favourable reduction relative to the reference under those conditions.
However, standard reduction potentials are not the same as reaction rates. Kinetic barriers, electrode surfaces, mass transport, and activation overpotentials determine how rapidly an electrochemical process proceeds.
The maximum non-expansion electrical work obtainable from a reversible electrochemical cell is related to its Gibbs energy change by:
\[\boxed{\Delta_rG=-nFE}\]
Here:
The Faraday constant is approximately:
\[F=96485\ \mathrm{C\,mol^{-1}}\]
Under standard-state conditions:
\[\Delta_rG^\circ=-nFE^\circ\]
For any chemical reaction, the Gibbs energy change is related to the reaction quotient by:
\[\boxed{\Delta_rG=\Delta_rG^\circ+RT\ln Q}\]
For an electrochemical reaction:
\[-nFE=-nFE^\circ+RT\ln Q\]
Rearranging:
\[-nF(E-E^\circ)=RT\ln Q\]
Dividing by \(-nF\):
\[\boxed{E=E^\circ-\frac{RT}{nF}\ln Q}\]
This is the Nernst equation.
Using the relation:
\[\ln Q=2.303\log_{10}Q\]
the Nernst equation becomes:
\[E=E^\circ-\frac{2.303RT}{nF}\log_{10}Q\]
At \(25^\circ\mathrm C\), or approximately \(298.15\ \mathrm K\):
\[\boxed{E=E^\circ-\frac{0.05916}{n}\log_{10}Q}\]
The numerical coefficient \(0.05916\ \mathrm V\) is temperature-dependent. It should not be used unchanged at temperatures other than approximately \(25^\circ\mathrm C\).
The reaction quotient is constructed from the activities of the products divided by the activities of the reactants, each raised to its stoichiometric coefficient.
For the reaction:
\[\mathrm{aA+bB\rightleftharpoons cC+dD}\]
the quotient is:
\[Q=\frac{a_C^c a_D^d}{a_A^a a_B^b}\]
Pure solids and pure liquids in their standard states are omitted because their activities are unity. Dissolved species should be represented using activities when thermodynamic accuracy is required.
Consider the reduction reaction:
\[\mathrm{M^{n+}(aq)+ne^-\rightarrow M(s)}\]
The reaction quotient is:
\[Q=\frac{a_{\mathrm{M(s)}}}{a_{\mathrm{M^{n+}}}}\]
For a pure solid metal:
\[a_{\mathrm{M(s)}}=1\]
Therefore:
\[Q=\frac{1}{a_{\mathrm{M^{n+}}}}\]
Substituting into the Nernst equation:
\[E=E^\circ-\frac{RT}{nF}\ln\left(\frac{1}{a_{\mathrm{M^{n+}}}}\right)\]
Using the logarithm identity \(\ln(1/x)=-\ln x\):
\[\boxed{E=E^\circ+\frac{RT}{nF}\ln a_{\mathrm{M^{n+}}}}\]
At \(25^\circ\mathrm C\):
\[\boxed{E=E^\circ+\frac{0.05916}{n}\log_{10}a_{\mathrm{M^{n+}}}}\]
This equation shows that the electrode potential depends on the activity of the metal ion in solution.
Consider the half-reaction:
\[\mathrm{Ag^++e^-\rightarrow Ag(s)}\]
Suppose the standard reduction potential is \(E^\circ=+0.7996\ \mathrm V\), and the silver-ion activity is \(0.0100\).
At \(25^\circ\mathrm C\), the Nernst equation gives:
\[E=E^\circ+0.05916\log_{10}a_{\mathrm{Ag^+}}\]
Substituting:
\[E=0.7996+0.05916\log_{10}(0.0100)\]
Since:
\[\log_{10}(0.0100)=-2\]
we obtain:
\[E=0.7996-0.11832\]
Therefore:
\[\boxed{E=0.6813\ \mathrm V}\]
This result uses the specified activity directly. If \(0.0100\) were a concentration rather than an activity, the calculation would require the appropriate activity coefficient for a rigorous result.
Consider the general reduction half-reaction:
\[\mathrm{Ox}+ne^-\rightarrow\mathrm{Red}\]
The reaction quotient is:
\[Q=\frac{a_{\mathrm{Red}}}{a_{\mathrm{Ox}}}\]
Therefore:
\[E=E^\circ-\frac{RT}{nF}\ln\frac{a_{\mathrm{Red}}}{a_{\mathrm{Ox}}}\]
Equivalently:
\[\boxed{E=E^\circ+\frac{RT}{nF}\ln\frac{a_{\mathrm{Ox}}}{a_{\mathrm{Red}}}}\]
At \(25^\circ\mathrm C\):
\[\boxed{E=E^\circ+\frac{0.05916}{n}\log_{10}\frac{a_{\mathrm{Ox}}}{a_{\mathrm{Red}}}}\]
The potential increases when the activity ratio of the oxidised form to the reduced form increases, all other conditions being held constant.
Consider the reduction half-reaction:
\[\mathrm{Fe^{3+}+e^-\rightarrow Fe^{2+}}\]
Its electrode potential is:
\[E=E^\circ+\frac{RT}{F}\ln\frac{a_{\mathrm{Fe^{3+}}}}{a_{\mathrm{Fe^{2+}}}}\]
At \(25^\circ\mathrm C\):
\[E=E^\circ+0.05916\log_{10}\frac{a_{\mathrm{Fe^{3+}}}}{a_{\mathrm{Fe^{2+}}}}\]
If the two activities are equal, the logarithmic term is zero and:
\[\boxed{E=E^\circ}\]
This equality applies to the stated redox couple when the activities of the oxidised and reduced species are equal and the standard-state conventions are consistent.
The electromotive force, or EMF, of an electrochemical cell is the potential difference between its terminals under reversible open-circuit conditions.
When both electrode potentials are expressed as reduction potentials, the cell potential is:
\[\boxed{E_{\mathrm{cell}}=E_{\mathrm{cathode}}-E_{\mathrm{anode}}}\]
The cathode is the electrode where reduction occurs, and the anode is the electrode where oxidation occurs.
Under standard-state conditions:
\[\boxed{E_{\mathrm{cell}}^\circ=E_{\mathrm{cathode}}^\circ-E_{\mathrm{anode}}^\circ}\]
For the Daniell cell, using representative standard reduction potentials:
\[E^\circ_{\mathrm{Cu^{2+}/Cu}}=+0.340\ \mathrm V\]
\[E^\circ_{\mathrm{Zn^{2+}/Zn}}=-0.763\ \mathrm V\]
The standard cell potential is:
\[E_{\mathrm{cell}}^\circ=0.340-(-0.763)\]
Therefore:
\[\boxed{E_{\mathrm{cell}}^\circ=1.103\ \mathrm V}\]
The precise tabulated values depend on the reference data and conditions. The important point is that standard reduction potentials are subtracted according to cathode minus anode.
The Gibbs energy change is:
\[\Delta_rG=-nFE_{\mathrm{cell}}\]
Therefore:
These conclusions refer to the reversible thermodynamic cell potential. They do not determine how rapidly the reaction proceeds.
At equilibrium:
\[\Delta_rG=0\]
The general Gibbs energy relation is:
\[\Delta_rG=\Delta_rG^\circ+RT\ln Q\]
At equilibrium, \(Q=K^\circ\), so:
\[0=\Delta_rG^\circ+RT\ln K^\circ\]
Therefore:
\[\Delta_rG^\circ=-RT\ln K^\circ\]
For an electrochemical cell:
\[\Delta_rG^\circ=-nFE^\circ_{\mathrm{cell}}\]
Equating the two expressions:
\[-nFE^\circ_{\mathrm{cell}}=-RT\ln K^\circ\]
Thus:
\[\boxed{\ln K^\circ=\frac{nFE^\circ_{\mathrm{cell}}}{RT}}\]
In base-10 form:
\[\boxed{\log_{10}K^\circ=\frac{nFE^\circ_{\mathrm{cell}}}{2.303RT}}\]
At \(25^\circ\mathrm C\):
\[\boxed{\log_{10}K^\circ=\frac{nE^\circ_{\mathrm{cell}}}{0.05916}}\]
where \(E^\circ_{\mathrm{cell}}\) is expressed in volts.
A cell reaction transfers two electrons and has a standard cell potential of \(0.300\ \mathrm V\) at \(25^\circ\mathrm C\). Calculate the thermodynamic equilibrium constant.
Using:
\[\log_{10}K^\circ=\frac{nE^\circ_{\mathrm{cell}}}{0.05916}\]
\[\log_{10}K^\circ=\frac{2(0.300)}{0.05916}\]
\[\log_{10}K^\circ\approx10.14\]
Therefore:
\[\boxed{K^\circ\approx1.4\times10^{10}}\]
The large equilibrium constant indicates that products are strongly favoured at equilibrium for the reaction written in the specified direction, under the stated temperature and standard-state conventions.
A concentration cell is an electrochemical cell in which the EMF arises primarily from a difference in the activities of the same electroactive species in two half-cells.
Because the electrode materials and electrode reactions are identical, the standard electrode potentials cancel when the cell potential is calculated.
The driving force is the tendency of the system to reduce the difference in chemical potential between the two sides.
Consider two electrodes involving the same metal and metal ion, with activities \(a_1\) and \(a_2\).
The reduction potential at each electrode is:
\[E_1=E^\circ+\frac{RT}{nF}\ln a_1\]
\[E_2=E^\circ+\frac{RT}{nF}\ln a_2\]
Taking the second electrode as the cathode and the first as the anode:
\[E_{\mathrm{cell}}=E_2-E_1\]
Substitution gives:
\[E_{\mathrm{cell}}=\left(E^\circ+\frac{RT}{nF}\ln a_2\right)-\left(E^\circ+\frac{RT}{nF}\ln a_1\right)\]
The standard potentials cancel:
\[E_{\mathrm{cell}}=\frac{RT}{nF}(\ln a_2-\ln a_1)\]
Using logarithm rules:
\[\boxed{E_{\mathrm{cell}}=\frac{RT}{nF}\ln\frac{a_2}{a_1}}\]
At \(25^\circ\mathrm C\):
\[\boxed{E_{\mathrm{cell}}=\frac{0.05916}{n}\log_{10}\frac{a_2}{a_1}}\]
This expression assumes the stated cell orientation and neglects liquid-junction and other additional potential contributions.
Consider a concentration cell involving Ag/Ag+ electrodes with silver-ion activities \(a_1=0.0010\) and \(a_2=0.10\), at \(25^\circ\mathrm C\).
For silver, \(n=1\).
\[E_{\mathrm{cell}}=0.05916\log_{10}\frac{0.10}{0.0010}\]
\[E_{\mathrm{cell}}=0.05916\log_{10}(100)\]
Since \(\log_{10}(100)=2\):
\[\boxed{E_{\mathrm{cell}}=0.1183\ \mathrm V}\]
The more active silver-ion solution has the more positive reduction potential. Under the stated orientation, that half-cell acts as the cathode.
When two electrolyte solutions of different compositions or concentrations come into contact, their ions may diffuse at different rates because they have different mobilities. This can produce a transient or sustained separation of charge at the interface, creating a liquid-junction potential.
For example, if one solution contains a high concentration of HCl and the other contains a lower concentration, H+ and Cl− do not necessarily diffuse at identical rates. Their unequal mobilities can generate a potential difference across the junction.
A suitable salt bridge, often containing an electrolyte whose cation and anion have relatively similar mobilities, can reduce the junction potential. However, it does not guarantee that the junction potential is exactly zero.
The measured cell voltage may contain contributions from the electrode potentials and liquid-junction potentials. Consequently, accurate determination of thermodynamic quantities requires attention to junction effects and the experimental cell design.
In conventional electrochemical work, reference electrodes, appropriate salt bridges, and standardised measurement protocols are used to minimise or account for such contributions.
The standard hydrogen electrode, or SHE, is the primary reference convention for standard electrode potentials.
Its half-reaction is:
\[\mathrm{2H^+(aq)+2e^-\rightleftharpoons H_2(g)}\]
Under standard-state conditions, the standard electrode potential of the SHE is assigned:
\[\boxed{E^\circ_{\mathrm{SHE}}=0.000\ \mathrm V}\]
The electrode involves a suitable platinised platinum surface, hydrogen gas, and an aqueous solution with hydrogen-ion activity corresponding to the chosen standard state.
The saturated calomel electrode is a commonly used reference electrode based on the mercury/mercurous chloride system:
\[\mathrm{Hg_2Cl_2(s)+2e^-\rightleftharpoons2Hg(l)+2Cl^-(aq)}\]
Its potential depends on the chloride-ion activity. When the electrode is maintained in saturated potassium chloride solution, its potential is relatively stable at a fixed temperature.
Because mercury compounds present toxicity and disposal concerns, appropriate handling and institutional waste procedures are essential.
The silver/silver chloride reference electrode operates according to:
\[\mathrm{AgCl(s)+e^-\rightleftharpoons Ag(s)+Cl^-(aq)}\]
Its potential depends on the chloride-ion activity in the internal filling solution.
Silver/silver chloride electrodes are widely used because they are convenient and generally suitable for routine electrochemical measurements.
Potentials measured against different reference electrodes cannot be compared directly without accounting for the reference potential and the temperature and filling-solution conditions.
When reporting an electrode potential, the reference electrode should always be specified. For example, reporting a potential as “+0.250 V” is incomplete unless the reference scale and relevant conditions are known.
The reversible cell potential provides the Gibbs energy change of the overall cell reaction:
\[\boxed{\Delta_rG=-nFE}\]
At standard conditions:
\[\boxed{\Delta_rG^\circ=-nFE^\circ}\]
Therefore, EMF measurements can be used to determine the Gibbs energy change of an electrochemical reaction.
At constant pressure, the standard entropy change is related to the temperature dependence of the standard Gibbs energy:
\[\Delta_rS^\circ=-\left(\frac{\partial\Delta_rG^\circ}{\partial T}\right)_P\]
Using:
\[\Delta_rG^\circ=-nFE^\circ\]
and assuming \(n\) is constant:
\[\Delta_rS^\circ=nF\left(\frac{\partial E^\circ}{\partial T}\right)_P\]
Therefore:
\[\boxed{\Delta_rS^\circ=nF\left(\frac{\partial E^\circ}{\partial T}\right)_P}\]
The temperature coefficient of the standard cell potential is needed to determine the standard entropy change.
The Gibbs energy relation is:
\[\Delta_rG^\circ=\Delta_rH^\circ-T\Delta_rS^\circ\]
Rearranging:
\[\Delta_rH^\circ=\Delta_rG^\circ+T\Delta_rS^\circ\]
Substituting the electrochemical expressions:
\[\Delta_rH^\circ=-nFE^\circ+nFT\left(\frac{\partial E^\circ}{\partial T}\right)_P\]
Thus:
\[\boxed{\Delta_rH^\circ=-nF\left[E^\circ-T\left(\frac{\partial E^\circ}{\partial T}\right)_P\right]}\]
This relationship allows the standard enthalpy change to be obtained from the standard cell potential and its temperature coefficient, provided the measurements and thermodynamic conventions are consistent.
Faraday’s first law states that the amount of chemical change produced during electrolysis is proportional to the total electric charge passed through the electrolyte, provided the current efficiency is appropriately accounted for.
The total charge is:
\[Q=It\]
where \(I\) is current and \(t\) is time.
For a species requiring \(n\) electrons per mole of product:
\[\boxed{n_{\mathrm{product}}=\frac{It}{nF}}\]
Here, \(n_{\mathrm{product}}\) denotes the amount of product in moles, while \(n\) denotes the number of electrons transferred per formula unit in the electrode reaction.
If the molar mass of the deposited species is \(M\), the mass deposited is:
\[m=M n_{\mathrm{product}}\]
Substituting the Faraday relation:
\[m=M\frac{It}{nF}\]
Therefore:
\[\boxed{m=\frac{MIt}{nF}}\]
This is the ideal expression for the deposited mass when the current efficiency is 100% and the electrode reaction is known.
A current of \(2.00\ \mathrm A\) is passed through a solution containing Cu2+ for \(30.0\) minutes. Calculate the theoretical mass of copper deposited, assuming 100% current efficiency.
The electrode reaction is:
\[\mathrm{Cu^{2+}+2e^-\rightarrow Cu(s)}\]
Thus:
\[n=2,\qquad M=63.546\ \mathrm{g\,mol^{-1}}\]
Convert time into seconds:
\[t=30.0(60)=1800\ \mathrm s\]
The deposited mass is:
\[m=\frac{(63.546)(2.00)(1800)}{(2)(96485)}\]
Therefore:
\[\boxed{m\approx1.19\ \mathrm g}\]
The actual deposited mass may be lower if side reactions, incomplete current efficiency, or material losses occur.
A metal electrode follows the half-reaction \(\mathrm{M^{2+}+2e^-\rightarrow M(s)}\). Derive its electrode potential in terms of the activity of M2+.
Answer:
\[E=E^\circ+\frac{RT}{2F}\ln a_{\mathrm{M^{2+}}}\]
Derive the relationship between the standard cell potential and the thermodynamic equilibrium constant.
Answer:
\[\boxed{\log_{10}K^\circ=\frac{nE^\circ_{\mathrm{cell}}}{0.05916}}\]
This numerical form applies at \(25^\circ\mathrm C\), with the cell potential expressed in volts.
Why does a concentration cell produce an EMF even though both electrodes involve the same redox couple?
Answer: The two half-cells have different activities of the electroactive species. Their electrode potentials differ according to the Nernst equation, and the standard potentials cancel when the cell potential is calculated.
How can the entropy change of a cell reaction be determined experimentally?
Answer: Measure the standard cell potential at different temperatures, determine its temperature derivative at constant pressure, and apply \(\Delta_rS^\circ=nF(\partial E^\circ/\partial T)_P\).
What factors determine the mass deposited during electrolysis?
Answer: The mass depends on the total charge passed, the molar mass of the deposited species, the number of electrons required per formula unit, and the current efficiency.
The reversible reduction half-reaction is:
\[\mathrm{Q + 2H^+ + 2e^- \rightleftharpoons H_2Q}\]
The Nernst potential at \(25^\circ\mathrm{C}\) is:
\[E = E^\circ_{\mathrm{Q/H_2Q}} - \frac{0.05916}{2}\log_{10}\frac{a_{\mathrm{H_2Q}}}{a_{\mathrm{Q}} \cdot a_{\mathrm{H^+}}^2}\]
Because quinhydrone provides equal mole amounts of \(\mathrm{Q}\) and \(\mathrm{H_2Q}\), their activities are practically equal (\(a_{\mathrm{H_2Q}} \approx a_{\mathrm{Q}}\)). The expression simplifies to:
\[E = E^\circ_{\mathrm{Q/H_2Q}} + 0.05916\log_{10}a_{\mathrm{H^+}}\]
\[\boxed{E = E^\circ_{\mathrm{Q/H_2Q}} - 0.05916\,\mathrm{pH}}\]
At \(25^\circ\mathrm{C}\), \(E^\circ_{\mathrm{Q/H_2Q}} = +0.6996\ \mathrm{V}\). The quinhydrone electrode is limited to solutions with \(\mathrm{pH} < 8.5\); above this value, hydroquinone dissociates as a weak acid into \(\mathrm{HQ^-}\) and \(\mathrm{Q^{2-}}\), and atmospheric oxygen oxidizes the organic substrate, disrupting the 1:1 activity ratio.
The next part will cover electrochemical cells with multiple equilibria, electrode kinetics, overpotential, the Butler–Volmer equation, the Tafel equation, electrochemical polarisation, corrosion, batteries, and electrochemical energy storage. It will extend the thermodynamic foundation into kinetic and applied electrochemistry, including derivations and numerical examples suitable for MSc Chemistry and CSIR-NET preparation.
In Part 4, we established that the Nernst equation determines the reversible electrode potential under specified thermodynamic conditions. However, the Nernst equation alone does not explain how rapidly an electrochemical reaction proceeds when current flows.
This distinction introduces electrode kinetics, which investigates the rates of oxidation and reduction at electrode–electrolyte interfaces and the factors controlling electron transfer.
For example, a galvanic cell may have a positive reversible cell potential, but its operating voltage can be substantially lower because of electrode polarisation, internal resistance, and mass-transport limitations.
Electrode kinetics is essential for understanding:
Thermodynamics tells us whether a reaction is energetically favourable. Kinetics tells us how rapidly it occurs and what electrical driving force is required to achieve a particular rate.
The reversible electrode potential is the potential at which the electrode reaction is at equilibrium for the specified chemical activities, temperature, and pressure.
Consider the reduction half-reaction:
\[\mathrm{Ox}+ne^-\rightleftharpoons\mathrm{Red}\]
The equilibrium potential is given by the Nernst equation:
\[E_{\mathrm{eq}}=E^\circ-\frac{RT}{nF}\ln Q\]
At this potential, the forward and reverse interfacial reaction rates are equal. Consequently, there is no net Faradaic current, although oxidation and reduction events can continue microscopically.
Overpotential, denoted by \(\eta\), is the difference between the actual electrode potential and the reversible equilibrium potential for the specified reaction.
\[\boxed{\eta=E-E_{\mathrm{eq}}}\]
Here, the potential and overpotential must be referenced consistently.
For a cathodic reduction process, the overpotential is commonly negative under this convention. For an anodic oxidation process, it is commonly positive.
The magnitude of the overpotential indicates how far the electrode is driven away from equilibrium. A larger magnitude is often required to achieve a higher reaction rate, although the exact relationship depends on the reaction mechanism and transport conditions.
At equilibrium, oxidation and reduction proceed at equal rates. To produce a net current, the electrode potential must be displaced from its equilibrium value so that one direction
While chemical thermodynamics dictates the equilibrium position of a redox reaction, electrode kinetics governs the actual current that flows when an electrode is perturbed away from its equilibrium potential.
For an elementary single-electron exchange reaction:
\[\mathrm{Ox} + e^- \underset{k_a}{\overset{k_c}{\rightleftharpoons}} \mathrm{Red}\]
The net current density \(j\) (in \(\mathrm{A\,cm^{-2}}\)) passing across the electrode is the difference between the anodic current density \(j_a\) and the absolute cathodic current density \(|j_c|\):
\[j = j_a - |j_c|\]
At the reversible equilibrium potential \(E_{\mathrm{eq}}\) given by the Nernst equation, the forward and reverse rates are equal, resulting in zero net Faradaic current:
\[j_a = |j_c| = j_0\]
Here, \(j_0\) is the exchange current density. It reflects the intrinsic kinetic rate of continuous electron transfer occurring at equilibrium.
To produce a net current, the electrode potential \(E\) must depart from \(E_{\mathrm{eq}}\). The difference is defined as the overpotential \(\eta\):
\[\boxed{\eta = E - E_{\mathrm{eq}}}\]
The total operating overpotential across an electrode consists of three distinct contributions:
\[\boxed{\eta_{\mathrm{total}} = \eta_{\mathrm{act}} + \eta_{\mathrm{conc}} + \eta_{\mathrm{ohmic}}}\]
When an overpotential \(\eta\) is applied, the electrical energy barrier for reduction is modified by \(-\alpha F \eta\), and the activation barrier for oxidation is altered by \((1-\alpha) F \eta\), where \(\alpha\) is the transfer coefficient (or symmetry factor, typically \(\sim 0.5\)).
The forward and backward current densities are:
\[j_a = j_0 \exp\left[\frac{(1-\alpha)nF\eta}{RT}\right]\]
\[j_c = -j_0 \exp\left[-\frac{\alpha nF\eta}{RT}\right]\]
Combining these terms yields the Butler–Volmer equation:
\[\boxed{j = j_0 \left\{ \exp\left[\frac{(1-\alpha)nF\eta}{RT}\right] - \exp\left[-\frac{\alpha nF\eta}{RT}\right] \right\}}\]
This is the fundamental equation of phenomenological electrode kinetics for charge-transfer controlled processes.
When the electrode is subjected to very small perturbations such that \(|\eta| \ll \frac{RT}{nF}\) (typically \(|\eta| < 10\ \mathrm{mV}\)):
Using the linear series expansion \(\exp(x) \approx 1 + x\):
\[j \approx j_0 \left[ \left(1 + \frac{(1-\alpha)nF\eta}{RT}\right) - \left(1 - \frac{\alpha nF\eta}{RT}\right) \right]\]
\[\boxed{j = \frac{j_0 n F}{RT}\eta}\]
In this regime, the current density is directly proportional to the overpotential, resembling Ohm's law. The slope defines the charge-transfer resistance \(R_{\mathrm{ct}}\):
\[\boxed{R_{\mathrm{ct}} = \left(\frac{\partial \eta}{\partial j}\right)_{\eta \to 0} = \frac{RT}{n F j_0}}\]
When \(|\eta| > 100\ \mathrm{mV}\) at \(298\ \mathrm{K}\), one of the exponential terms in the Butler–Volmer equation becomes negligible compared to the other:
For a single-electron transfer (\(n=1\)) with \(\alpha = 0.5\) at \(298\ \mathrm{K}\), the standard Tafel slope is:
\[b = \frac{2.303 \times 8.314 \times 298.15}{0.5 \times 1 \times 96485} \approx 0.1183\ \mathrm{V\,dec^{-1}} \approx 120\ \mathrm{mV\,dec^{-1}}\]
At the boundary between an electrode and an electrolyte, charge separation occurs, establishing an electrical double layer that dictates interfacial capacitance and electron-transfer rates.
Helmholtz proposed that counter-ions align in a single, rigid sheet in the liquid phase at a distance of closest approach \(d_H\) (the Outer Helmholtz Plane, OHP), balancing the charge on the metal electrode.
The interface behaves as a parallel-plate capacitor with differential capacitance:
\[\boxed{C_H = \frac{\varepsilon_0 \varepsilon_r}{d_H}}\]
Limitation: The model assumes a fixed capacitance, failing to explain experimental observations showing that capacitance varies with both electrode potential and electrolyte concentration due to thermal dispersion.
Gouy and Chapman treated the counter-ions as a diffuse cloud governed by thermal agitation and electrostatic attraction, using a one-dimensional Poisson–Boltzmann equation.
The diffuse capacitance \(C_{\mathrm{diff}}\) is given by:
\[\boxed{C_{\mathrm{diff}} = \varepsilon \kappa \cosh\left(\frac{z F \phi_0}{2RT}\right)}\]
where \(\phi_0\) is the potential at the electrode surface relative to bulk solution.
Limitation: Because ions are treated as point charges without volume, the model predicts unrealistically large capacitances (\(C \to \infty\)) at high overpotentials.
Stern unified the two approaches by dividing the interfacial region into:
The overall capacitance \(C_{\mathrm{dl}}\) corresponds to the two capacitances in series:
\[\boxed{\frac{1}{C_{\mathrm{dl}}} = \frac{1}{C_H} + \frac{1}{C_{\mathrm{diff}}}}\]
Corrosion is an electrochemical process occurring spontaneously without an external power source. Anodic metal dissolution couples directly with a cathodic depolariser reduction (such as proton reduction in acid or oxygen reduction in aerated water) on the same conductive surface:
\[\mathrm{M \rightarrow M^{n+} + n e^-} \quad (\text{Anodic oxidation})\]
\[\mathrm{2H^+ + 2e^- \rightarrow H_2} \quad (\text{Cathodic reduction})\]
The metal establishes a steady-state mixed potential called the corrosion potential (\(E_{\mathrm{corr}}\)). At \(E_{\mathrm{corr}}\), the total rate of oxidation equals the total rate of reduction, defining the corrosion current (\(I_{\mathrm{corr}}\)):
\[\boxed{|I_a| = |I_c| = I_{\mathrm{corr}}}\]
In an Evans diagram (a plot of potential \(E\) versus \(\log |I|\)), the intersection of the anodic and cathodic Tafel polarization lines identifies the coordinates \((E_{\mathrm{corr}}, \log I_{\mathrm{corr}})\).
Near \(E_{\mathrm{corr}}\), the current–potential response is approximately linear. The slope defines the polarization resistance \(R_p\):
\[R_p = \left(\frac{\Delta E}{\Delta I}\right)_{\Delta E \to 0}\]
The Stern–Geary equation relates this resistance directly to the corrosion current:
\[\boxed{I_{\mathrm{corr}} = \frac{B}{R_p}}\]
where \(B\) is a constant determined from the empirical Tafel slopes:
\[\boxed{B = \frac{b_a b_c}{2.303(b_a + b_c)}}\]
Why does the molar conductivity of a weak acid increase dramatically at electric field strengths exceeding \(10^4\ \mathrm{V\,cm^{-1}}\)?
Answer: This is the second Wien effect (dissociation field effect). High potential gradients physically stretch and destabilize covalent and ion-pair bonds, shifting the dissociation equilibrium toward free ions, thereby increasing the degree of dissociation \(\alpha\).
For an electrode reaction with transfer coefficient \(\alpha = 0.50\) at \(298\ \mathrm{K}\), what is the expected cathodic Tafel slope for a single-electron transfer process?
Answer:
\[b_c = \frac{2.303 RT}{\alpha n F} = \frac{0.05916}{0.50 \times 1} \approx \mathbf{0.1183\ \mathrm{V\,dec^{-1}}\ (118.3\ \mathrm{mV\,dec^{-1}})}\]
Why does the differential capacitance curve of a dilute electrolyte show a minimum at the potential of zero charge (PZC), whereas in a concentrated electrolyte the curve remains nearly flat?
Answer: In the Stern model, the total capacitance follows \(1/C_{\mathrm{dl}} = 1/C_H + 1/C_{\mathrm{diff}}\). In dilute solutions near the PZC, the diffuse layer capacitance \(C_{\mathrm{diff}}\) is very small and dominates the series expression, producing a hyperbolic cosine minimum. In concentrated solutions, the Debye screening length is compressed, making \(C_{\mathrm{diff}} \gg C_H\), so the total capacitance is governed by the constant Helmholtz capacitance \(C_H\).
For the concentration cell with transference \(\mathrm{Ag \mid AgCl(s) \mid HCl(a_1)\ \vdots\ HCl(a_2) \mid AgCl(s) \mid Ag}\), write the expression for cell EMF in terms of transport numbers.
Answer: Because the electrodes are reversible to the anion (\(\mathrm{Cl^-}\)), the cell EMF depends on the transport number of the cation (\(\mathrm{H^+}\)):
\[\boxed{E = 2 t_+ \frac{RT}{F} \ln\frac{a_{\pm,2}}{a_{\pm,1}}}\]
Time Remaining: 900s