Calculating pH and pOH with Weak Acids and Bases

🧪 Understanding Weak Acids and Bases

Weak acids and bases don't fully dissociate in water, meaning they don't completely break apart into ions. This partial dissociation is described by an equilibrium, where both the undissociated acid/base and its ions are present. Because of this equilibrium, we use equilibrium constants ($K_a$ for acids and $K_b$ for bases) to calculate the pH or pOH.

📜 History and Background

The concept of pH was first introduced by Søren Peder Lauritz Sørensen in 1909 to provide a convenient way to express acidity. Later, the understanding of weak acids and bases evolved with the development of chemical equilibrium principles. The Bronsted-Lowry theory (acids as proton donors, bases as proton acceptors) helped further define acid-base behavior in solution.

⚗️ Key Principles for Calculating pH and pOH

• ⚖️ Equilibrium Constants ($K_a$ and $K_b$): These constants indicate the strength of the acid or base. A smaller $K_a$ or $K_b$ means a weaker acid or base.
• 📝 ICE Tables: Use ICE (Initial, Change, Equilibrium) tables to determine the equilibrium concentrations of the acid/base and its ions.
• 🧮 Approximations: If the acid or base is weak enough, you can often simplify the calculation by assuming that the change in concentration ($x$) is negligible compared to the initial concentration.
• 💧 Autoionization of Water ($K_w$): Remember that water itself can act as both an acid and a base, and its autoionization is described by $K_w = [H^+][OH^-] = 1.0 \times 10^{-14}$ at 25°C.
• ⚗️ Relationship between $K_a$, $K_b$, and $K_w$: For a conjugate acid-base pair, $K_a \times K_b = K_w$. This helps calculate $K_b$ if $K_a$ is known, and vice versa.
• ➕ Calculating pH and pOH: Once you find the $[H^+]$ concentration, calculate pH using $pH = -\log[H^+]$. Similarly, find the $[OH^-]$ concentration and calculate pOH using $pOH = -\log[OH^-]$. Then, use $pH + pOH = 14$ to find the pH if you know the pOH, or vice versa.

⚲ Real-World Examples

Acetic Acid ($\text{CH}_3\text{COOH}$)

Acetic acid is a weak acid found in vinegar. Let's calculate the pH of a 0.1 M solution of acetic acid, given $K_a = 1.8 \times 10^{-5}$.

1. 📝 Set up the ICE table:

   	$\text{CH}_3\text{COOH}$	$\text{H}^+$	$\text{CH}_3\text{COO}^-$
   Initial (I)	0.1	0	0
   Change (C)	-$x$	+$x$	+$x$
   Equilibrium (E)	0.1-$x$	$x$	$x$

2. ⚗️ Write the $K_a$ expression:

   $K_a = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]} = \frac{x^2}{0.1-x}$

3. 💡 Approximate:

   Since $K_a$ is small, assume $x$ is much smaller than 0.1, so $0.1 - x \approx 0.1$.

   $1.8 \times 10^{-5} = \frac{x^2}{0.1}$
4. ➕ Solve for $x$:

   $x^2 = 1.8 \times 10^{-6}$

   $x = \sqrt{1.8 \times 10^{-6}} = 1.34 \times 10^{-3}$

5. 💧 Calculate pH:

   $pH = -\log(1.34 \times 10^{-3}) = 2.87$

Ammonia ($\text{NH}_3$)

Ammonia is a weak base. Let's calculate the pH of a 0.1 M solution of ammonia, given $K_b = 1.8 \times 10^{-5}$.

1. 📝 Set up the ICE table:

   	$\text{NH}_3$	$\text{NH}_4^+$	$\text{OH}^-$
   Initial (I)	0.1	0	0
   Change (C)	-$x$	+$x$	+$x$
   Equilibrium (E)	0.1-$x$	$x$	$x$

2. ⚗️ Write the $K_b$ expression:

   $K_b = \frac{[NH_4^+][OH^-]}{[NH_3]} = \frac{x^2}{0.1-x}$
3. 💡 Approximate:

   Since $K_b$ is small, assume $x$ is much smaller than 0.1, so $0.1 - x \approx 0.1$.

   $1.8 \times 10^{-5} = \frac{x^2}{0.1}$

4. ➕ Solve for $x$:

   $x^2 = 1.8 \times 10^{-6}$

   $x = \sqrt{1.8 \times 10^{-6}} = 1.34 \times 10^{-3}$

5. 💧 Calculate pOH:

   $pOH = -\log(1.34 \times 10^{-3}) = 2.87$

6. ➕ Calculate pH:

   $pH = 14 - pOH = 14 - 2.87 = 11.13$

🔑 Conclusion

Calculating pH and pOH for weak acids and bases involves understanding equilibrium principles and using approximations when appropriate. By setting up ICE tables and carefully applying the $K_a$ and $K_b$ expressions, you can accurately determine the pH of these solutions. Remember to consider the autoionization of water and the relationship between $K_a$ and $K_b$ for conjugate acid-base pairs.