Activities

7.6 The solvent activity

General form of the chemical potential for a real or ideal solvent is given by modification of the equation for the chemical potential of a mixture.

For an ideal solution, Raoult’s law applies

The quantity a_A is the Activity of A, an “effective mole fraction” just as fugacity is an effective pressure. Since the chemical potential in terms of pressures is true for both real and ideal solutions we conclude that

A convenient way of expressing this convergence is to introduce the activity coefficient, g ,

7.7 The solute activity

The problem with defining activity coefficients and standard states for solutes is that they approach ideal dilute (Henry’s Law) behavior as

(a) ideal-dilute solutions

For a solute which obeys Henry’s Law, p_B = K_Bx_B

Both K_B and p_B^* are characteristic of the solvent, so the second term may be combined with the first to give a new standard chemical potential

(b) Real solutes

For real deviations from Henry’s law we introduce a_B into the equation above.

at all temperatures and pressures. Deviations of the solute from ideality disappear as zero concentration is approached.

Example 7.6 Measuring activity

Use the data in example 7.3 to calculate activity and activity coefficient of chloroform in acetone at 25 ⁰C, treating it first as a solvent then as a solute.

Method: For the activity of chloroform as a solvent, form a=p/p^* and g = a/x. For activity as a solute, form a = p/K and g = a/x.

Answer: Because p^* = 293 Torr and K = 165 Torr (previous problem), we construct the following table with x the mole fraction of chloroform.

Notice that g ® 1 as x ® 0 in the Raoult’s Law case;

g ® 0 as x ® 0 in the Henry’s Law case.

(c) Activities in terms of molalities

Another definition of activity. In dilute solutions, to a good approximation, x_B » n_B/n_A because n_B is proportional to molality, b, we can write

To simplify notation, we interpret b as a relative molality (replace b/b° by b; in practice, using for b its numerical value in moles per kg). Then we write

According to this definition, the chemical potential of solute has its standard value m ^° when molality of B is equal to b° (at 1 mol/kg). Note as b° goes to 0, m _B goes to ¥; that is the solution becomes diluted, so the solute becomes increasingly stabilized. The consequence of this result is that it is very difficult to remove the last traces of solute from a solution.
Now as before we incorporate deviations from ideality by introducing a dimensionless activity a_B and a dimensionless activity coefficient g_B.

Chapter 8 Phase Diagrams

We will review a systematic way to describe the physical changes mixtures undergo when they are heated or cooled and when their compositions are changed.

All phase diagrams can be discussed in terms of a relationship, the phase rule, derived by J. W. Gibbs.

8.1 Definitions

Phase: A state of matter that is uniform throughout in chemical and physical composition. (solid, liquid, gas, solution of two miscible liquids, etc.). The number of phases in a system is denoted P. A solution of water and acetone has one phase, P = 1, since they are uniformly mixed. A slurry of ice and water is a two-phase system, P = 2. A system in which calcium carbonate undergoes thermal decomposition consists of two solid phases (calcium carbonate and calcium oxide are not uniformly mixed) and one gas phase (carbon dioxide).

An alloy of two metals is a two-phase system (P = 2) if the metals are immiscible, but a single phase system (P = 1) if the metals are miscible.

A dispersion is uniform on a macroscopic scale but not on a microscopic scale, for it consists of grains or droplets of one substance in a matrix of the other. Such dispersions are important in materials where the microstructure is fixed to tailor important properties.

By a constituent of a system we mean a chemical species (ion or molecule) that is present. A mix of water and ethanol has two constituents. A Component is a chemically independent constituent of a system. The number of components, C, in a system is the minimum number of independent species necessary to define the composition of all phases present in the system.

When no reaction takes place, the number of components is equal to the number of constituents, chemical species. The mixture of ethanol and water is a two-component system (C = 2).

Consider the phase equilibrium

3 Phases - To specify the composition of the gas phase we need the species CO₂, and to specify phase two, we need the species CaO. We do not need an additional species to specify the composition of phase 3 because its identity can be expressed in terms of the other two constituents by making use of the stoichiometry of the reaction. Hence the system has 3 constituents but only two components (C = 2).

In general the number of constituents minus the number of chemical reactions equal the number of components.

Examples: How many components?

a) aqueous acetic acid – CH₃CO₂H and H₂O; C = 2

b) MgCO₃(s) in equilibrium with its decomposition products. MgCO₃(s) ® MgO(s) + CO₂(g)
3 constituents - 1 reaction – C = 2

c) water (including its ionization)
H₂O(l) ® H⁺(aq) + OH^-(aq) Looks like 3 constituents, 1 reaction. BUT H and OH are formed in equal molar amounts. Really only need the concentration of one and the other is the same.
2 constituents - 1 reaction – C = 1

d) NH₄Cl(s) ® NH₃(g) + HCl(g) in equilibrium
Again 2 constituents – 1 reaction – C=1

e) Reaction above with additional HCl(s) present.
Concentrations are not related by stoiciometry.
3 constituents - 1 reaction – C = 2

The Variance, F, of a system is the number of intensive variables that can be changed independently without disturbing the number of phases in equilibrium.

In a single component, single phase system (C = 1, P =1), the pressure and temperature may be changed independently without changing the number of phases, so F = 2. This system is bivariant and has two degrees of freedom.

If two phases are in equilibrium (ie - liq and vap) in a one component system such as H₂O(l) and H₂O(g) (C = 1, P = 2), the T or p can be changed at will, but the change in one demands an accompanying change in the other to preserve the number of phases in equilibrium. The system variance in this case is one (F = 1) one degree of freedom.

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8.2 The Phase Rule

In one of the most elegant calculations of the whole of thermodynamics, J. W. Gibbs deduced the phase rule, which is a general relation between the variance, F, the number of components, C, and the number of phases at equilibrium, P, for a system of any composition:

a) One-component system

b)

For a one-component system, such as pure water, F = 3 - P. When only one phase is present, F = 2 and both p and T can be varied independently without changing the number of phases. In other words, a single phase is represented by an area on a phase diagram. When two phases are in equilibrium, F = 1, which implies that pressure is not freely variable if temperature is set.

At a given temperature, a liquid has a characteristic vapor pressure (F = 1). Thus, the equilibrium of two phases is represented by a line on the phase diagram.

When the three phase are in equilibrium, F = 0 and the system is invariant. This special condition can be established only at a definite temperature and pressure that is characteristic of the substance and outside our control. The equilibrium of three phase is represented by a point (the triple point) on the phase diagram.

8.3 Vapor pressure diagrams

The vapor pressure of the components of an ideal solution of two volatile liquids are related to composition according to Raoult’s law:

The total pressure p_tot of the mixture is

The total pressure is a linear function of composition of the ideal solution.

a) The composition of vapor

The partial pressures of the components are given by Raoult’s Law. From Dalton’s law we write for the mole fractions of the vapor, y_A and y_B,

Provided the mixture is ideal, the total pressure can be expressed in terms of mole fractions in the liquid using Raoult’s Law for p_J and the equation above for total vapor pressure p, which gives

The compostion (mole fraction) in. the vapor, y_A, is related to the composition in the liquid, x_A. By solving the equation for x_A and substituting into the equation for p_tot we relate the total vapor pressure to composition of the vapor.