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doc/specs/chemistry.rst

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+Chemistry
+=============
+
+In chemistry, the most important parameter that is always needed for performing thermodynamic 
+computations is the activity coefficient :math:`\gamma`. The latter is the correction applied to 
+the concentration in order to compute the activity. The Debye-Huckel is extensively explained
+in Bockris.
+
+
+The Debye-Huckel (or ion-cloud) theory of ion-ion interaction
+------------------------------------------------------------------
+
+Activity coefficients and ion-ion interactions
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Evolution of the concept of an acitivty coefficients
+
+The existence of ions in solution, of interactions between these ions, and of a chemical-potential
+change :math:`\Delta \mu _{i-I}` arising from ion-ion interactions have all been taken to be self-evident
+in the treatment hitherto presented here. This, however, is a modern point of view. The thinking
+about electrolytic solutions actually developped along different path.
+
+Ionic solutions were at first treated in the same way as nonelectrolytic solutions, though the latter do not contain 
+charged species. The starting point was the classical thermodynamic formula for the chemical potential :math:`\mu _i`
+of a nonelectrolyte solute
+
+.. math:: 
+    :label: eq_chemical_potential
+
+    \mu _i = \mu ^{0}_{i} + RT \ln x_i
+
+In this expression, :math:`x_i` is the concentration of the solute in mole fraction units, and
+:math:`\mu ^0 _i` is its the chemical potential in the standard state, i.e., when :math:`x_i` a standard
+or a normalized value of unity
+
+.. math::
+    :label: eq_standard_chemical_potential
+
+    \mu _i = \mu _i ^{0} \text{ when } x_i = 1
+
+Since the solute particles in a solution of a nonlectrolyte are uncharged, they do not engage
+in long-range Coulombic interactions. The short-range interaction arising from dipole-dipole or dispersion forces
+become significant only when the mean distance between the solute particles is small, i.e., when the concentration
+of the solute is high. Thus, one can to a good approximation say that there are no between solute particles
+in dilute nonelectrolyte solutions. Hence, if :eq:`eq_chemical_potential` for the chemical potential of a solute
+in a nonelectrolyte solution (with noninteracting particles) is used fot the chemical potential of an ionic species :math:`i`
+in an electrolytic solution, then it is tantamount to ignoring the long-range Coulombic interactions between ions.
+In an actual electrolytic solution, however, ion-ion interactions operate whether one ignores them or not.
+It is obvious therefore that measurements of the chemical potential :math:`\mu _i` of an ionic species or, rather,
+measurements of any property that depends on the chemical potential would reveal the error in :eq:`eq_chemical_potential`,
+which is blind to ion-ion interactions. In other words, experiments show that even in dilute solutions,
+
+.. math::
+    :label: eq_chemical_potential_error
+
+    \mu _i - \mu _i^0 \neq RT \ln x_i
+
+In this context, a frankly empirical approach was adopted by earlier workers not
+yet blessed by Debye and Huckel's light. Solutions that obeyed :eq:`eq_chemical_potential` were
+characterized as *ideal* solutions since this equation applies to systems of noninteracting solute particles,
+i.e, ideal particles. Electrolytic solutions that do not obey the equation were said to be *nonideal*. In order to use
+an equation of the form of :eq:`eq_chemical_potential` to treat nonideal electrolytic solutions, an empirical
+correction factor :math:`f_i` was introduced by Lewis as a modifier of the concentration term.
+
+.. math::
+    :label: eq_chemical_potential_correction
+
+    \mu _i - \mu _i^0 = RT \ln x_i f_i 
+
+It was argued that, in nonideal solutions, it was not just the analytical concentration
+:math:`x_i` of species *i*, but its effective concentration :math:`x_i f_i` which determined the chemical-potential
+change :math:`\mu _i - \mu _i ^0`. This effective concentration :math:`x_i f_i` was also known as the *activity*
+:math:`a_i` of the species *i*, i.e.,
+
+.. math::
+    :label: eq_activity_definition
+
+    a_i = x_i f_i
+
+and the correction factor :math:`f_i`, as the *activity coefficient*. For ideal solutions, the activity coefficient
+is unity, and the activity :math:`a_i` becomes identical to the concentration :math:`x_i`, i.e.,
+
+
+.. math::
+    :label: eq_activity_ideal_solution
+
+    a_i = x_i \text{ when } f_i = 1
+
+Thus, the chemical-potential change in going from the standard state to the final state can be written as 
+
+.. math::
+    :label: eq_chemical_potential_activity
+
+    \mu _i - \mu _i ^0 = RT\ln x_i + RT \ln f_i
+
+:eq:`eq_chemical_potential_activity` summarizes the empirical or formal treatment of the behavior of electrolytic
+solutions. Such a treatment cannot furnish a theoretical expression for the acitivity coefficient :math:`f_i`.
+It merely recognizes that expressions such as :eq:`eq_chemical_potential` must be modified if significant forces exist
+between solute particules.
+
+The physical significance of activity coefficients
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+For a hypothetical system of ideal (noninteracting) particles, the chemical potential has been stated to be given by
+
+.. math::
+    :label: eq_chemical_potential_3_52
+
+    \mu _i (ideal) = \mu _i ^0 + RT \ln x_i 
+
+For a real system of interacting particles, the chemical potential has been expressed in
+the form 
+
+.. math:: 
+    :label: eq_chemical_potential_3_57
+    
+    \mu _i (real) = \mu _i ^0 + RT \ln x_i + RT \ln f_i
+
+Hence, to analyze the physical significance of the activity coefficient term in :eq:`eq_chemical_potential_3_57`
+, it is necessary to compare this equation with :eq:`eq_chemical_potential_3_52`. It is obvious that when 
+:eq:`eq_chemical_potential_3_52` is substracted from :eq:`eq_chemical_potential_3_57`, the difference is the 
+chemical-potential change :math:`\Delta \mu _{i-I}` arising from the interactions between the solute particles
+(ions in the case of electrolytic solutions). That is
+
+.. math::
+    :label: eq_chemical_potential_error_3_58
+
+    \mu _i (real) - \mu _i (ideal) = \Delta \mu_{i-I}
+
+and therefore,
+
+.. math::
+    :label: eq_chemical_potential_error_3_59
+
+    \Delta \mu _{i-I} = RT \ln f_i
+
+Thus, the activity coefficient is a measure of the chemical-potential change arising from ion-ion interactions.
+There are several well-established methods of experimentally determining activity coefficients, and these methods are
+treated in adequate details in standard treatises.
+
+Now, according to the Debye-Huckel theory, the chemical-potential change :math:`\Delta \mu _{i-I}` arising 
+from ion-ion interactions has been shown to be given by
+
+.. math::
+    :label: eq_chem_pot_change_3_51
+
+    \Delta \mu _{i-I} = - \frac{N_A(z_i e_0)^2}{2 \epsilon \kappa ^{-1}}
+
+Hence, combining :eq:`eq_chem_pot_change_3_51` and :eq:`eq_chemical_potential_error_3_58`, the result is
+
+
+.. math::
+    :label: eq_3_60
+
+    RT \ln f_i = - \frac{N_A(z_i e_0)^2}{2 \epsilon \kappa ^{-1}}
+
+Thus, the Debye-Huckel ionic-cloud model for ion-ion interactions has permitted a theoretical calculation
+of activity coefficients resulting in :eq:`eq_3_60`.
+
+The activity coefficient in :eq:`eq_chemical_potential_error_3_59` arises from the formula :eq:`eq_chemical_potential_3_57`
+for the chemical potential, in which the concentration of the species *i* is expressed in mole fraction units :math:`x_i`.
+One can also express the concentration in moles per liter of solution (molarity) or in moles per kilogram of solvent (molality).
+Thus, alternative formulas for the chemical potential of a species *i* in an ideal solution read
+
+.. math::
+    :label: eq_3_61
+
+    \mu _i = \mu _i^0 (c) + RT \ln c_i 
+
+and
+
+.. math::
+    :label: eq_3_62
+
+    \mu _i = \mu _i^0 (m) + RT \ln m_i 
+
+where :math:`c_i` and :math:`m_i` are the molarity and molality of the species *i*, respectively, :math:`\mu _i^0(c)`
+and :math:`\mu _i^0(m)` are the corresponding standard chemical potentials.
+
+When the concentration of the ionic species in a real solution is expressed as molarity :math:`c_i` and molality :math:`m_i`
+, there are corresponding activity coefficients :math:`\gamma _c` and :math:`\gamma _m` and corresponding expressions for :math:`\mu _i`
+
+.. math::
+    :label: eq_3_63
+
+    \mu _i = \mu _i^0 (c) + RT \ln c_i + RT \ln \gamma _c
+
+
+.. math::
+    :label: eq_3_64
+
+    \mu _i = \mu _i^0 (m) + RT \ln m_i + RT \ln \gamma _m
+
+
+The activity coefficient of a single ionic species cannot be measured
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^