Ionic Equilibrium – Chemistry Study Notes

Ostwald’s Dilution Law and Degree of Dissociation

When dealing with weak electrolytes like acetic acid or ammonium hydroxide, molecules exist in a dynamic equilibrium with their respective ions in solution. Ostwald’s dilution law applies the Law of Mass Action to weak electrolytes, establishing a mathematical relationship between the degree of dissociation ($\alpha$), the dissociation constant ($K$), and the concentration or dilution of the solution.

Consider a weak monobasic acid represented as $HA$ with initial concentration $c$ and degree of dissociation $\alpha$. At equilibrium, the concentrations are $[HA] = c(1-\alpha)$, $[H^+] = c\alpha$, and $[A^-] = c\alpha$. Substituting these into the equilibrium constant expression yields the acid dissociation constant: $K_a = \frac{c\alpha^2}{1-\alpha}$.

For very weak electrolytes where $\alpha \ll 1$, the term $(1-\alpha)$ approximates to $1$, simplifying the expression to $\alpha = \sqrt{\frac{K_a}{c}}$ or $\alpha = \sqrt{K_a \cdot V}$, where $V$ is the dilution volume.

This reveals a crucial physical insight: the degree of dissociation of a weak electrolyte increases upon dilution. As you add more water, the system shifts forward to counteract the volume change, leading to more ionization. However, the total number of ions increases while the concentration of individual ions drops, a balance that governs many analytical titrations.

pH, pOH Scales and Properties of Water

Pure water undergoes self-ionization to a very small extent, producing hydronium ions and hydroxide ions: $2H_2O_{(l)} \rightleftharpoons H_3O^+_{(aq)} + OH^-_{(aq)}$. The equilibrium constant for this process is termed the ionic product of water, designated as $K_w$. At $25^\circ C$, the numerical value of $K_w$ is fixed at $1.0 \times 10^{-14} \, M^2$, defined mathematically as $K_w = [H^+][OH^-]$.

Because $K_w$ is an equilibrium constant, it is strictly dependent on temperature. Endothermic autoionization means that as temperature increases, $K_w$ increases, making neutral water slightly more acidic in terms of absolute ion concentration, though it remains electrically neutral because $[H^+] = [OH^-]$. To handle these extremely wide ranges of concentration conveniently, Sörensen introduced the logarithmic scale known as pH:

• pH = $-\log_{10}[H^+]$
• pOH = $-\log_{10}[OH^-]$
• pH + pOH = $pK_w = 14$ (at $25^\circ C$)

“At standard temperature, a neutral solution maintains a pH of $7$, acidic solutions measure below $7$, and basic solutions register above $7$.”

pH Calculations for Acids, Bases, and Mixtures

Calculating the pH of various chemical formulations requires tracking dominant species and applying mass-action approximations. For strong acids and strong bases, dissociation is assumed to be $100\%$, meaning the hydronium or hydroxide ion concentration directly equals the normality of the strong electrolyte. For instance, a $0.001 \, M \, HCl$ solution yields $[H^+] = 10^{-3} \, M$, resulting in a straightforward pH = $3$.

Weak acids and weak bases require accounting for their dissociation constants, $K_a$ and $K_b$. For a weak acid, the hydronium ion concentration is computed using $[H^+] = \sqrt{K_a \cdot c}$, followed by taking the negative logarithm. When dealing with polyprotic acids like phosphoric acid ($H_3PO_4$), consecutive dissociation constants ($K_{a1}, K_{a2}, K_{a3}$) typically differ by factors of $10^4$ or more, allowing chemists to calculate the primary pH contribution almost entirely from the first dissociation step.

Mixtures of strong acids or strong bases demand conservation of moles rather than simple averages. When mixing two strong solutions, calculate the total milliequivalents of $H^+$ or $OH^-$ and divide by the total final volume in milliliters to find the resultant normality:

• $[H^+]_{result} = \frac{N_1V_1 + N_2V_2}{V_1 + V_2}$ (for acid-acid mixtures)
• If mixing an acid and a base, neutralization occurs first, and the pH is determined by the reactant present in excess.

Salt Hydrolysis

When salts dissolve in water, their constituent ions may react with water molecules to reform weak acids or weak bases, altering the neutral pH of the solution. This process is known as salt hydrolysis. Salts are categorized into four standard types based on the strength of their parent acids and bases:

• Salts of Strong Acid and Strong Base: (e.g., $NaCl$) Do not hydrolyze. The solution remains strictly neutral with a pH of $7$.
• Salts of Weak Acid and Strong Base: (e.g., $CH_3COONa$) Undergo anionic hydrolysis, yielding a basic solution. The hydrolysis constant is $K_h = \frac{K_w}{K_a}$, and the pH is calculated via pH = $7 + \frac{1}{2}pK_a + \frac{1}{2}\log c$.
• Salts of Strong Acid and Weak Base: (e.g., $NH_4Cl$) Undergo cationic hydrolysis, yielding an acidic solution. The pH is given by pH = $7 – \frac{1}{2}pK_b – \frac{1}{2}\log c$.
• Salts of Weak Acid and Weak Base: (e.g., $CH_3COONH_4$) Both ions hydrolyze. The pH depends entirely on the relative strengths of the parent acid and base, calculated using pH = $7 + \frac{1}{2}pK_a – \frac{1}{2}pK_b$.

Buffer Solutions

A buffer solution resists drastic changes in pH upon the addition of small amounts of strong acid or strong base. They are vital in biological systems, such as blood, and industrial chemical processes. Buffers are generally classified into two categories: acidic buffers (a weak acid combined with its salt of a strong base) and basic buffers (a weak base combined with its salt of a strong acid).

The quantitative behavior of an acidic buffer is described by the Henderson-Hasselbalch equation:

$\text{pH} = \text{p}K_a + \log \left( \frac{[\text{Salt}]}{[\text{Acid}]} \right)$

Similarly, for a basic buffer, the equation calculates pOH:

$\text{pOH} = \text{p}K_b + \log \left( \frac{[\text{Salt}]} {[\text{Base}]} \right)$

The efficiency of a buffer is measured by its buffer capacity, which is defined as the number of moles of strong acid or strong base added to one liter of the buffer to change its pH by one unit. A buffer operates at maximum efficiency when the concentration of the salt equals the concentration of the weak acid or weak base.

Solubility Product and Precipitation Reactions

For sparingly soluble salts like silver chloride ($AgCl$) or barium sulfate ($BaSO_4$), a dynamic equilibrium is established between the solid solute and its constituent ions in a saturated solution: $AxBy_{(s)} \rightleftharpoons xA^{y+}_{(aq)} + yB^{x-}_{(aq)}$. The equilibrium constant expression for this equilibrium is called the solubility product constant ($K_{sp}$):

$K_{sp} = [A^{y+}]^x [B^{x-}]^y$

To predict whether a precipitate will form when two solutions are mixed, chemists compare the ionic product ($IP$) to the solubility product ($K_{sp}$):

• $IP < K_{sp}$: The solution is unsaturated, and no precipitation occurs. More solute can dissolve.
• $IP = K_{sp}$: The solution is saturated and at dynamic equilibrium.
• $IP > K_{sp}$: The solution is supersaturated, and excess solute precipitates out until the ionic product drops to equal $K_{sp}$.

A classic application of this concept is the common ion effect, where the solubility of a sparingly soluble salt decreases significantly when a strong electrolyte containing a common ion is added to the solution. For instance, adding $NaCl$ to a saturated solution of $AgCl$ introduces excess $Cl^-$ ions, driving the equilibrium backward and precipitating more $AgCl$.

Important Facts / Formulas

Key Points to Remember

• The degree of dissociation ($\alpha$) increases with dilution, approaching unity at infinite dilution for weak electrolytes.
• $K_w$ and $pK_w$ vary with temperature; a neutral solution at high temperatures may have a pH different from $7$, though it remains neutral because $[H^+] = [OH^-]$.
• Polyprotic acids dissociate stepwise, and $K_{a1} \gg K_{a2} \gg K_{a3}$ in almost all standard cases.
• Salts of strong acids and strong bases do not undergo hydrolysis; their aqueous solutions maintain a neutral pH of $7$.
• Maximum buffer capacity occurs when the concentration of the weak acid equals the concentration of its conjugate salt ($\text{pH} = \text{p}K_a$).
• The common ion effect suppresses the ionization of weak electrolytes and reduces the solubility of sparingly soluble salts.
• Precipitation happens strictly when the ionic product exceeds the solubility product ($IP > K_{sp}$).

Previous Year Question Hints

• Buffer Calculations: Aspirants frequently encounter problems requiring the calculation of pH changes when a specific volume of strong acid or base is added to an established buffer mixture. Always update the moles of acid/salt using stoichiometry before applying the Henderson equation.
• Solubility Product Relationships: Questions often ask to correlate the molar solubility ($s$) of salts with unusual stoichiometric coefficients (such as $A_2B_3$ or $AB_3$) to their $K_{sp}$ values. Set up algebraic expressions using variables like $2s$ and $3s$ based on dissociation stoichiometry.
• Mixed Acid/Base Titrations: Expect numerical problems on the titration curve points (like the half-equivalence point where $\text{pH} = \text{p}K_a$). Recognizing these shortcut identities saves critical calculation time during exams.

Quick Revision Summary

• Ostwald’s dilution law links dissociation constants with concentration and degree of dissociation for weak electrolytes.
• pH and pOH scales provide a logarithmic method to express hydronium and hydroxide ion concentrations compactly.
• Autoionization of water is endothermic, meaning $K_w$ rises alongside temperature increases.
• Strong electrolytes dissociate completely, whereas weak electrolytes establish dynamic ionic equilibria.
• Salt hydrolysis shifts the pH away from $7$ depending on the relative strengths of the parent ions.
• Buffers resist pH fluctuations by neutralizing added hydronium or hydroxide ions using conjugate acid-base pairs.
• The Henderson-Hasselbalch equation is the primary tool for calculating buffer pH and pOH values.
• Solubility product ($K_{sp}$) and ionic product ($IP$) comparisons determine whether a precipitate will form in saturated solutions.
• The common ion effect diminishes both weak electrolyte dissociation and sparingly soluble salt solubility.