Chapter 8: Activity and the Systematic Treatment of Equilibrium

Chapter 8: Activity and the Systematic Treatment of Equilibrium

Overview

This chapter explores how ionic strength affects solubility and equilibrium, introduces the concept of activity and activity coefficients, revisits the definition of pH, and presents a systematic approach to solving equilibrium problems in analytical chemistry.

• The Effect of Ionic Strength on Solubility of Salts

• Activity Coefficients

• pH Revisited

• Systematic Treatment of Equilibrium

• Applying the Systematic Treatment of Equilibrium

Introduction

Ions, Hydration, and Activity

In aqueous solutions, ions and molecules are surrounded by an organized sheath of solvent molecules, a process known as hydration. The extent of hydration depends on the size and charge of the ion:

• Smaller, highly charged ions bind more water molecules and thus behave as larger hydrated species in solution.

• Water molecules bind to cations through the oxygen atom (which has a partial negative charge) and to anions through the hydrogen atoms (which have partial positive charges).

• The activity of aqueous ions is related to the size of the hydrated species, not just the bare ion.

Example: The hydrated radius of Na+ is much larger than its ionic radius due to strong hydration.

The Effect of Ionic Strength on Solubility of Salts

Ionic Strength and Solubility

Ionic strength (μ) is a measure of the total concentration of ions in solution, weighted by the square of their charges. It affects the solubility of sparingly soluble salts:

• Adding an inert salt (e.g., KNO3) increases the ionic strength of the solution.

• This increases the thickness of the ionic atmosphere around each ion, which reduces the electrostatic attraction between oppositely charged ions.

• As a result, the solubility of sparingly soluble salts increases with ionic strength.

Definition: The ionic atmosphere is a region of net positive or negative charge surrounding an ion in solution, composed of excess counter-ions.

Calculation of Ionic Strength

The ionic strength (μ) is calculated as:

$\mu = \frac{1}{2} \sum_{i} c_i z_i^2$

• $c_i$ = concentration of the i-th ion (mol/L)

• $z_i$ = charge of the i-th ion

Example: For 0.10 M NaNO3:

$\mu = \frac{1}{2}([\text{Na}^+] \times 1^2 + [\text{NO}_3^-] \times 1^2) = \frac{1}{2}(0.10 + 0.10) = 0.10$ M

Activity and Activity Coefficients

Definition of Activity

The activity (a) of a species is its effective concentration, accounting for non-ideal behavior due to ionic interactions:

• $a = \gamma [C]$ for solutes, where $\gamma$ is the activity coefficient.

• For gases: $a = \frac{P}{P^\circ}$, where $P^\circ$ is the standard pressure (1 bar).

• For pure solids and liquids: $a = 1$ by definition.

Activity coefficients ($\gamma$) measure the deviation from ideality. If $\gamma = 1$, the solution behaves ideally.

Thermodynamic Equilibrium Constant

Equilibrium constants should be written in terms of activities, not concentrations:

$K = \frac{a_{\text{products}}}{a_{\text{reactants}}}$

Example: For the dissolution of CaSO4:

$K_{sp} = a_{\text{Ca}^{2+}} a_{\text{SO}_4^{2-}} = [\text{Ca}^{2+}] \gamma_{\text{Ca}^{2+}} [\text{SO}_4^{2-}] \gamma_{\text{SO}_4^{2-}}$

Debye-Hückel Equation

The Debye-Hückel equation relates the activity coefficient to ionic strength:

$\log \gamma = -\frac{0.51 z^2 \sqrt{\mu}}{1 + a \sqrt{\mu}/305}$

• $z$ = charge of the ion

• $\mu$ = ionic strength (M)

• $a$ = effective hydrated radius (pm)

This equation is valid for $\mu < 0.1$ M. As ionic strength increases, $\gamma$ decreases (i.e., activity deviates more from concentration).

Factors Affecting Activity Coefficients

• Higher ionic strength ($\mu$) leads to lower $\gamma$ (greater deviation from ideality).

• Ions with higher charge ($|z|$) have lower $\gamma$.

• Smaller hydrated radius ($a$) increases the effect of ionic strength on $\gamma$.

Example: The activity coefficient of Ca2+ in 3.3 mM CaCl2 can be found using tables or the Debye-Hückel equation.

pH Revisited

Activity and pH Measurement

pH meters measure the negative logarithm of the hydrogen ion activity, not its concentration:

$pH = -\log a_{\text{H}^+} = -\log([\text{H}^+] \gamma_{\text{H}^+})$

At higher ionic strengths, the activity coefficient of H+ decreases, so the measured pH differs from the value calculated using concentration alone.

Systematic Treatment of Equilibrium

General Approach

The systematic treatment of equilibrium provides a structured method to solve complex equilibrium problems by writing as many independent equations as there are unknowns. The key steps are:

1. Write all relevant chemical equilibrium expressions.

2. Write the charge balance equation: the sum of positive charges equals the sum of negative charges in solution.

3. Write one or more mass balance equations: the total amount of each atom or group delivered to the solution equals the sum of all its forms in solution.

4. Solve the resulting system of equations, often using algebraic or numerical methods.

Charge Balance Example

For a solution of CaCl2:

$2[\text{Ca}^{2+}] + [\text{H}^+] = [\text{OH}^-] + [\text{Cl}^-]$

Mass Balance Example

For a 0.1 M Na2S solution:

$0.1 = [\text{S}^{2-}] + [\text{HS}^-] + [\text{H}_2\text{S}]$

Applying the Systematic Treatment of Equilibrium

Stepwise Solution

To solve for the concentrations of all species in a system:

1. List all pertinent chemical reactions and their equilibrium constants.

2. Write the charge balance equation.

3. Write the mass balance equation(s).

4. Write the equilibrium constant expressions (using activities if necessary).

5. Count the number of equations and unknowns to ensure the system is solvable.

6. Solve the equations, using approximations or computational tools as needed.

Example: For a solution containing 0.0100 mol NH3 in 1.000 L, the following equations are set up:

• NH3 + H2O ↔ NH4+ + OH-

• Charge balance: [NH4+] + [H+] = [OH-]

• Mass balance: [NH3] + [NH4+] = 0.0100 M

• Equilibrium expressions for $K_b$ and $K_w$

Summary Table: Factors Affecting Activity Coefficient

Factor	Effect on Activity Coefficient ($\gamma$)
Ionic Strength ($\mu$)	Increasing $\mu$ decreases $\gamma$
Ion Charge ($|z|$)	Higher $|z|$ decreases $\gamma$
Hydrated Radius ($a$)	Smaller $a$ decreases $\gamma$

Additional info: The notes also include worked examples and graphical illustrations of how ionic strength and activity coefficients affect equilibrium, as well as step-by-step procedures for applying the systematic treatment to real chemical systems.