Weak Acids and Their Equilibria: Study Notes for General Chemistry

Chapter 15: Acids and Bases

Weak Acids

Weak acids are a fundamental concept in acid-base chemistry, characterized by their partial ionization in water. Understanding their behavior is essential for predicting pH and equilibrium in aqueous solutions.

• Definition: Weak acids react only partially with water, producing a limited amount of hydronium ions (H3O+).

• Ionization: Unlike strong acids, weak acids do not ionize completely when dissolved in water.

General equation for a weak acid (HA) in water:

$\mathrm{HA\ (aq) + H_2O\ (l) \rightleftharpoons H_3O^+\ (aq) + A^-\ (aq)}$

From this equation, we can write the equilibrium expression:

$K_a = \frac{[H_3O^+][A^-]}{[HA]}$

where $K_a$ is the equilibrium constant for a weak acid reacting with water.

Notes about $K_a$ Values

• Magnitude: The value of $K_a$ indicates the strength of a weak acid; larger $K_a$ means a stronger weak acid.

• Comparison: Weak acids have much smaller $K_a$ values than strong acids (which are considered to have very large $K_a$ values).

• Equilibrium: The position of equilibrium for weak acids lies far to the left (favoring the non-ionized form).

p$K_a$ Values

p$K_a$ is a logarithmic measure of acid strength, similar to how pH measures acidity.

• Definition: $\mathrm{p}K_a = -\log K_a$

• Interpretation: Smaller p$K_a$ values correspond to stronger acids; larger p$K_a$ values indicate weaker acids.

Examples:

• $K_a = 1.8 \times 10^{-5} \Rightarrow \mathrm{p}K_a = 4.74$

• $K_a = 2.0 \times 10^{-2} \Rightarrow \mathrm{p}K_a = 1.70$

Differences Between Strong Acids and Weak Acids

The distinction between strong and weak acids is primarily based on the degree of dissociation and the position of equilibrium.

Property	Strong Acid	Weak Acid
$K_a$ value	Very large	Small
Position of dissociation equilibrium	Far to the right (almost complete dissociation)	Far to the left (partial dissociation)
Equilibrium concentration of $H_3O^+$	Nearly equal to initial acid concentration	Much less than initial acid concentration

Some Classes of Weak Acids

• Organic Weak Acids: Examples include carboxylic acids such as acetic acid (CH3COOH), propanoic acid, etc.

• Inorganic Weak Acids:

  • Hydrofluoric acid: $\mathrm{HF(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + F^-(aq)}$, $K_a = 7.3 \times 10^{-4}$

  • Hydrocyanic acid: $\mathrm{HCN(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + CN^-(aq)}$, $K_a = 6.2 \times 10^{-10}$

• Conjugate Acids of Weak Bases: For example, ammonium ion ($\mathrm{NH_4^+}$) is the conjugate acid of ammonia ($\mathrm{NH_3}$):

  • $\mathrm{NH_4^+(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + NH_3(aq)}$, $K_a = 5.56 \times 10^{-10}$

• Certain Metal Ions (Small Size, High Charge): e.g., $\mathrm{Al^{3+}}$, $\mathrm{Fe^{3+}}$, $\mathrm{Cr^{3+}}$

Mathematical Treatment of Weak Acid Equilibria

To solve weak acid equilibrium problems, we use an ICE (Initial, Change, Equilibrium) table to track concentrations throughout the reaction.

• Initial: Initial concentrations before any ionization occurs.

• Change: The change in concentrations, defined as $-x$ for reactants and $+x$ for products.

• Equilibrium: The sum of the initial and change values.

Example ICE Table for HA dissociation:

	HA (aq)	H2O (l)	H3O+ (aq)	A- (aq)
Initial	$C_{HA}$	--	0	0
Change	$-x$	--	$+x$	$+x$
Equilibrium	$C_{HA} - x$	--	$x$	$x$

Substitute equilibrium values into the $K_a$ expression to solve for $x$ (which equals $[H_3O^+]$).

Types of Weak Acid Equilibrium Problems

• Given $[H_3O^+]$ and $C_{HA}$, find $K_a$

• Given $K_a$ and $C_{HA}$, find $[H_3O^+]$

• Given $K_a$ and $[H_3O^+]$, find $C_{HA}$

Remember: $[H_3O^+]$ can be found from pH, pOH, or $[OH^-]$ as needed.

Worked Examples

• Example 1: The pH of 0.100 mol/L lactic acid is 2.08. What is the $K_a$ for lactic acid?

  1. Identify the acid type (weak acid).

  2. Use pH to find $[H_3O^+]$:

     • $[H_3O^+] = 10^{-pH}$

  3. Complete the ICE table for $\mathrm{HA (aq) + H_2O (l) \rightleftharpoons H_3O^+ (aq) + A^- (aq)}$.

  4. Substitute values into the $K_a$ expression and solve.

• Example 2: What concentration of formic acid ($\mathrm{HCO_2H}$, $K_a = 2.1 \times 10^{-4}$) will yield a solution of pH 2.47?

  1. Use pH to determine $[H_3O^+]$.

  2. Fill in the ICE table for $\mathrm{HCO_2H (aq) + H_2O (l) \rightleftharpoons H_3O^+ (aq) + HCO_2^- (aq)}$.

  3. Substitute values into the $K_a$ expression and solve for the initial concentration.

• Example 3: Find the pH of a 0.10 mol/L solution of acetic acid ($K_a = 1.8 \times 10^{-5}$).

  1. Write the equation and fill in the ICE table for $\mathrm{CH_3COOH (aq) + H_2O (l) \rightleftharpoons H_3O^+ (aq) + CH_3COO^- (aq)}$.

  2. Solve for $x$ using the $K_a$ expression. This may require solving a quadratic equation, but approximations can often be used if $x$ is small compared to the initial concentration.

Quadratic Equation and Approximations

• Solving for $x$ in the $K_a$ expression may lead to a quadratic equation.

• To avoid solving the quadratic, check if $x$ is small compared to the initial concentration ($C_{HA}$). If so, the approximation $C_{HA} - x \approx C_{HA}$ can be used.

• Always verify the validity of the approximation by checking if $x/C_{HA} < 5\%$.

Summary Table: Key Relationships

Relationship	Equation
Acid dissociation constant	$K_a = \frac{[H_3O^+][A^-]}{[HA]}$
p$K_a$ definition	$\mathrm{p}K_a = -\log K_a$
pH definition	$\mathrm{pH} = -\log [H_3O^+]$

Additional info: These notes expand on the provided lecture slides by including definitions, worked examples, and summary tables for clarity and completeness.