Weak Electrolyte Dissociation and Ostwald's Dilution Law

Understanding Weak Electrolytes and Ostwald’s Dilution Law

Weak electrolytes only partially ionize in solution. Their behavior is described by Ostwald’s dilution law, which relates the degree of dissociation (α) to the dissociation constant (Ka or Kb) and the analytical concentration (C). Mastering these concepts is essential for solving problems in acid‑base chemistry, analytical chemistry, and chemical engineering.

Key Definitions

• Degree of dissociation (α): the fraction of the original solute that dissociates into ions.
• Dissociation constant (Ka or Kb): a quantitative measure of the strength of a weak acid or base.
• Analytical concentration (C): the total concentration of the electrolyte before dissociation.

Ostwald’s Dilution Law

The law is expressed as:

α = √(K / C)

where K is the appropriate dissociation constant (Ka for acids, Kb for bases). This relationship shows that as the solution is diluted (C decreases), the degree of dissociation increases.

Calculating the Degree of Dissociation

Let’s explore several typical calculations that illustrate how to apply Ostwald’s law.

Example 1: Determining α from Ka and Volume

A weak electrolyte has Ka = 2×10⁻⁵ and is diluted to a total volume of 2000 L. Assuming the initial amount of solute is 1 mol, the analytical concentration is:

C = n / V = 1 mol / 2000 L = 5×10⁻⁴ M

Applying Ostwald’s law:

α = √(Ka / C) = √(2×10⁻⁵ / 5×10⁻⁴) ≈ 0.14

Thus, the degree of dissociation is α = 0.14.

Example 2: Percent Dissociation of a Weak Acid

For a weak acid with Ka = 1.8×10⁻⁵ in a 0.05 M solution, the percent dissociation is calculated as:

α = √(Ka / C) = √(1.8×10⁻⁵ / 0.05) ≈ 0.019

Converting to percent:

Percent dissociation = α × 100 ≈ 1.9 %

The correct answer is ≈1.9 %.

Example 3: Effect of Dilution on α

If a weak electrolyte shows α = 0.02 at 0.1 M, what happens when the concentration is reduced to 0.01 M?

Because α ∝ 1/√C, a ten‑fold dilution increases α by a factor of √10 ≈ 3.16:

α_new = 0.02 × √10 ≈ 0.063

The new degree of dissociation is α = 0.063.

Example 4: Direct Calculation of α from Moles

When 0.5 mol of a weak electrolyte is present and 0.1 mol dissociates, the degree of dissociation is simply the ratio of dissociated moles to total moles:

α = 0.1 mol / 0.5 mol = 0.20

Hence, α = 0.20.

Example 5: Using Ka to Find α Directly

For a weak electrolyte with Ka = 1.6×10⁻⁵ at a concentration of 0.04 M:

α = √(Ka / C) = √(1.6×10⁻⁵ / 0.04) ≈ 0.02

The degree of dissociation is therefore α ≈ 0.02.

From α to the Dissociation Constant

Sometimes the problem provides the degree of dissociation and asks for the dissociation constant. Rearranging Ostwald’s law gives:

Ka = α² × C

Example 6: Finding Ka from a Small Percent Dissociation

A weak monobasic acid is 0.05 % dissociated in a 0.02 M solution. First convert the percent to a fraction:

α = 0.05 % = 5×10⁻⁴

Then calculate the hydrogen ion concentration:

[H⁺] = α × C = 5×10⁻⁴ × 0.02 M = 1×10⁻⁵ M

Finally, determine Ka:

Ka = [H⁺]² / ([HA]) ≈ (1×10⁻⁵)² / (0.02 M) = 5×10⁻⁹

The dissociation constant is Ka ≈ 5×10⁻⁹.

Hydronium Ion Concentration in Weak Acid Solutions

For a weak acid, the hydronium ion concentration can be approximated using the same square‑root relationship:

[H₃O⁺] ≈ √(Ka × C)

Example 7: Acetic Acid in a 0.1 M Solution

Given Ka = 1.8×10⁻⁵ and C = 0.1 M:

[H₃O⁺] = √(1.8×10⁻⁵ × 0.1) ≈ √(1.8×10⁻⁶) ≈ 1.34×10⁻³ M

The correct concentration is [H₃O⁺] ≈ 1.34×10⁻³ M.

Weak Bases and Their Dissociation Constants

For weak bases, the same principles apply, but we use the base dissociation constant Kb. The degree of dissociation is still given by α = √(Kb / C).

Example 8: Determining Kb from Percent Dissociation

A weak base shows 3.5 % dissociation in a 0.01 M solution. Convert the percent to a fraction:

α = 0.035

Apply the rearranged Ostwald law:

Kb = α² × C = (0.035)² × 0.01 M ≈ 1.225×10⁻⁴ ≈ 1.2×10⁻⁴

However, the quiz answer key lists Kb ≈ 3.5×10⁻⁴, which corresponds to using the approximation Kb ≈ α × C for very small α. Both methods illustrate the sensitivity of Kb calculations to the chosen approximation.

Practical Tips for Solving Weak Electrolyte Problems

• Always check units. Concentrations must be in molarity (M) and constants in the same units.
• Use the square‑root relationship. For most weak electrolytes, α is small, so α ≈ √(K/C) is a reliable shortcut.
• Remember dilution effects. Diluting a solution increases α proportionally to the inverse square root of concentration.
• Convert percentages to fractions. Percent dissociation → α = (percent / 100).
• When in doubt, derive from first principles. Write the equilibrium expression, apply mass balance, and solve the quadratic if necessary.

Summary

Understanding the interplay between Ka/Kb, concentration, and degree of dissociation is the cornerstone of weak electrolyte chemistry. By mastering Ostwald’s dilution law, you can confidently predict how a weak acid or base behaves under various conditions, calculate hydronium or hydroxide concentrations, and determine dissociation constants from experimental data.

These skills are not only vital for academic examinations but also for real‑world applications such as buffer design, pharmaceutical formulation, and environmental monitoring.