Dr. Cooper's comments: Good Work!

**(Red - Soo Hyeon Lee, sley@gmu.edu)**

**Ion Activities**

Many materials exist as ions in solution. Ions interact electrostatically over long distances and so they deviate strongly from ideal behavior even at low concentrations. Also, ions respond to electric potentials, and the presence of a favorable potential decreases their chemical activity. The thermodynamics of ions in solution depends on their chemical potential µ. Normally only electrically neutral solutions containing both cations and anions are encountered, and only the activity of an ion in the presence of ions of opposite charge will be presented in this section.

If the chemical potential of a univalent M^{+} is denoted µ_{+} and that of a univalent anion X^{-} is denoted µ_{-}, the total molar Gibbs energy of the ions in the electrically neutral solution is the sum of these partial molar quantities.

G_{m}^{ideal} = µ_{+}^{ideal}+ µ_{-}^{ideal } (Equation 5-61)

where G_{m}^{ideal} is molar Gibbs energy in a neutral solution, µ_{+}^{ideal}is the positive ion in a neutral solution, and µ_{-}^{ideal }is the negative ion in a neutral solution.

For a real solution of M^{+} and X^{-} of the same molality,

G_{m} = µ_{+}^{ideal}+ µ_{-}^{ideal} + RTlnγ_{+}+RTlnγ_{- } (Equation 5-62a)

G_{m} = G_{m}^{ideal} + RTlnγ_{+}γ_{- } (Equation 5-62b)

where G_{m} is molar Gibbs energy, RTlnγ_{+}+RTlnγ_{-} is the correction factor of ions, and γ_{+}γ_{-} is an activity coefficient. The electrostatic interactions leading to deviations from ideality are contained in Equation 5-62b

In reality, however, it is hard to separate into the two ions. That means, in case of a real solution, there is no way to separating part to the cations and part to the anions. Therefore, the best that can be done is to introduce the mean ionic activity coefficient, γ_{±.}

The mean ionic activity coefficient can be calculated as follows:

γ_{±} = (γ_{+}γ_{-})^{1/2} (Equation 5-63)

The individual chemical potentials of the ions can be written as follows so that the non-ideality term is distributed equally:

µ = µ_{+}+ µ_{-} (Equation 5-64a)

µ_{+} = µ_{+}^{ideal} + RTlnγ_{± } (Equation 5-64b)

µ_{-} = µ_{+}^{ideal} + RTlnγ_{± }(Equation 5-64c)

where µ_{+} and µ_{-} are the individual chemical potentials, and RTlnγ_{±} is the mean activity coefficient.

When the salt is M_{p}X_{q} which dissolves to give a solution containing the ions M^{+} and X^{-} in the ratio p:q, the mean activity coefficient is given by

γ _{±} = (γ_{+}^{p} γ_{-}^{q})^{1/S}, s = p + q (Equation 5-66)

and the chemical potential of either species is given by

µ_{i} = µ_{i}^{ideal} + RTlnγ_{±} (Equation 5-67)

and the molar Gibbs energy of the ions is the sum of their partial molar Gibbs energies:

G_{m} = pµ_{+}+ qµ_{-} (Equation 5-65a)

G_{m} = G_{m}^{ideal} +pRTlnγ_{+}+qRTlnγ_{-} (Equation 5-65b)

where p is the number of cation, q is the number of anion, G_{m} is molar Gibbs energy and µ_{i} and is potential. Once more the non-ideality term is distributed equally among the types of ion.

**Debye-Hückel Theory**

Ionic solutions depart sharply from ideality on account of the long range of their electrostatic interactions. This indicates not only that ions interact over long distances, but also that the departures from ideality are likely to be dominated by the direct electrostatic, Coulombic interaction. In other words, the departure from ideal behavior in ionic solutions can be mainly attributed to the Coulombic interaction between positively and negatively charged ions.

Oppositely charged ions attract each other. Cations and anions are not uniformly distributed, but anions tend to be found in the vicinity of cations, and vice versa. Overall the solution is neutral, but near any given ion there is a predominance of ions of opposite charge, which is called counter ions.

The situation is dynamic rather than static, and on the average more counter ions than same charged ions pass by any given ion, and they come and go in all directions. Averaged over time, counter ions are more likely to be found near any given ion. This time-averaged, spherical haze of opposite charge is called the *ionic atmosphere* of the ion. Figure 1 represents the Debye-Hückel theory.

*Figure 1. The fundamental picture of the Debye-Hückel theory that tendency for anions can be found near cations, and vice versa.*

The energy, and therefore the chemical potential, of the central ion is lowered by its favorable electrostatic interaction with its ionic atmosphere. This lowering of chemical potential is due to the activity of the solute and can be identified with correction factor, RTlnγ_{±}. At very low concentrations the activity coefficient can be calculated from the Debye-Hückel limiting law.

γ_{±} = -|z_{+}z_{-}|AI^{1/2} (Equation 5-69)

where A is a constant for an aqueous solution at 25℃, z_{+} is the charge number of cation, z_{-} is the charge number of anion, γ_{±}

is the mean activity coefficient, and I is the ionic strength of the solution.

All solutions are expected to conform to this expression in the limit of sufficiently low concentrations.

Figure 2 show how the observed activities of electrolytes of different valence type depend on the square root of the ionic strength, and compares them with the theoretical curves based on Equation 5-69.

*Figure 2. The dependence of the activity coefficient on concentration for various valence types, and the Debye-Hückel limiting law predictions.*

The ionic strength is a quantity, and it plays an important role because it takes into account the charges of the ions as well as their concentrations. The ionic strength can be defined as

I = 1/2*Σz_{i}^{2}(b_{i}/b^{θ}) (Equation 5-70)

where I is the ionic strength, b_{i} is the molality of ions, and b^{θ} is the standard molality of ions.

When the ionic strength is too high for the limiting law to be valid, the activity coefficient can be estimated from the Debye-Hückel extended law.

logγ_{±} = -(A|z_{+}z_{-}| I^{1/2} / 1+B I^{1/2}) + CI (Equation 5-72)

where I is the ionic strength, and A, B and C are the constant of each solution.

Figure 3 shows the how activity coefficient work depending on the ionic strength.

*Figure 3. The Debye-Hückel extended to take into account the size of the ions.*

__(Black Steven Ly Sly3@gmu.edu)__

__Ch. 6: Phase Diagrams__

**Phase Diagrams**

Let’s apply the knowledge of the thermodynamics of simple mixtures to discuss the physical changes of mixtures when they are heated or cooled and when their compositions are changed.

Phase diagrams can be used to judge whether or not two substances are mutually miscible.

Using these phase diagrams, it is possible to determine whether or no equilibrium can exist over a range of conditions or whether a system must be brought to a definite pressure, temperature and composition before equilibrium can be established.

There are many uses for phase diagrams. Some examples of which are industrial and commercial uses. Semiconductor, ceramics, steel and alloy industries rely heavily on phase diagrams to ensure uniformity of a product. Phase diagrams are also the basis for separation procedures in the petroleum industry and the formulation of foods and cosmetic preparations.

There are some definitions that are important to know to better understand phase diagrams.

First, a phase is a state of matter that is uniform throughout, not only in composition but also in physical state. The following examples are one phase systems:

- A pure gas
- A gaseous mixture
- Two totally miscible liquids
- A crystal
- A solution of sodium chloride
- Ice

Although ice set at zero *C will be a one phase system, a slurry of ice and water is considered a two phase system (one being the solid ice phase and the other, the liquid water phase.)

**Figure 6-1 shows the difference between an alloy of two metals. (A) shows a one phase system where the two metals are miscible. (B) shows a two phase system where the two metals are immiscible.**

An alloy of two metals is a two phase system if the metals are immiscible, but a single phase system if they are miscible as shown in *figure 6-1. Figure 6-1(A)* illustrates how a one phase system alloy of two metals should look like. There are no defined sections where one metal can be identified and separated from the other. As long as the two metals are uniformly mixed on a molecular level, it will be a one phase system. *Figure 6-1(B)* shows a two phase system where the two metals can be distinguished and depicted. A better way to understand this is if a scientist should observe an alloy of two metals on a molecular level, he/she should not be able to see where one metal ends and another begins.

In looking at *Figure 6-1(A)*, there is only one defined crystal structure. However in *Figure 6-1(B),* as you move along, you can see the two different crystal structures.

Dispersion can be uniform on a macroscopic level, but not on a microscopic scale. It is the dispersion on the microscopic level that determines whether a mixture will be a one phase system or two phase system. Dispersions are important in many advanced materials.

Heat treatment cycles are used to achieve the precipitation of a fine dispersion of particles of one phase within a matrix formed by a saturated solid solution phase.

The ability to control this microstructure resulting from phase equilibria makes it possible to tailor the mechanical properties of the materials.

The next thing to understand about a mixture is the constituents. A constituent of a system is a chemical species that is present, whether it be an ion, a molecule or atom. For example, a mixture of water and ethanol has two constituents. A solution of sodium chloride has three constituents: Na^{+}, Cl^{-}, H_{2}O.

Constituents are slightly different from components. **A component is a chemically independent constituent of a system.** To figure out the number of components in a system, you need to find the **minimum number of independent species necessary to define the composition of all the phases present in the system.**

When no reaction takes place and there are no other constraints, the number of components is the equal to the number of constituents. For instance, pure water is a one component system. A mixture of ethanol and water is two component system.

Although an aqueous solution of sodium chloride has three constituents, it is a two component system. This is because by the charge balance, the number of Na^{+} ions must be the same as the number of Cl^{-} ions.

A system that consists of hydrogen, oxygen and water at room temperature has three components. This is because all three are present at room temperature. Although hydrogen and oxygen does react to create water, at room temperature this reaction does not take place.

When a reaction can occur under the conditions prevailing in the system, we need to decide the minimum number of species that, after allowing for reactions in which one species is synthesized from others, can be used to specify the composition of all the phases.

Take for instance the following reaction of calcium carbonate going to calcium oxide and carbon dioxide:

**CaCO**_{3(s)} ßà CaO_{(s)} + CO_{2(g)}

Here, there are 3 phases: gas, one solid phase, and a second solid phase. Even though there are these are both solid phases, they are different chemicals and different crystal structures, giving two separate phases. There are also three constituents, or three different chemical species involved in the reaction.

To figure out how many components this reaction has, the minimum number of species involved to describe each phase.

To specify the composition of the gas phase, we need the species CO_{2}, and to specify the composition of the solid phase on the right, we need the species CaO.

No other additional species are needed to specify the composition of the phase on the left, because its identity (CaCO_{3}) can be expressed in terms of the other two constituents by making use of the stoichiometry of the reaction.

In doing this, it can be derived that this is a two component system.

Similarly, a reaction of:

** NH**_{4}Cl_{(s)} ßà NH3_{(g)} + HCl_{(g)}

has two phases. This is because the gases are uniformly mixed, giving only one phase. The reaction has 3 constituents, or three different chemical species involved in the reaction. In this reaction it is only a 1 component system.

This is different from the previous example because in the previous, there were 2 phases on the right hand side. In this example, there is only one phase.

Given the equation

**F = C – P + 2**

Where the number of phases = P.

The number of components =C.

The variance of the system = F is the number of intensive variables (e.g. p and T) that can be changed independently without disturbing the number of phases in equilibrium.

This is not an empirical rule based upon observations, it can be derived from chemical thermodynamics (Justification 6.1).

For a one component system,

** F = 3 – P**

When only one phase is present, F = 2 and both p and T can be varied without changing the number of phases

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