14.2 pH and pOH

pH and pOH are the standard scales chemists use to express how acidic or basic a solution is. Rather than working with tiny ion concentrations like $1.0 \times 10^{-9} \text{ M}$, pH and pOH compress those numbers into a simple 0–14 scale that's much easier to compare and communicate.

Both values depend on the concentrations of hydronium ions ($H_3O^+$) and hydroxide ions ($OH^-$) in solution. Understanding how to calculate and convert between pH, pOH, and ion concentrations is one of the most-tested skills in acid-base chemistry.

pH and pOH

Acidic, basic, and neutral solutions

The pH scale runs from 0 to 14 (at 25°C) and tells you the balance between $H_3O^+$ and $OH^-$ ions in a solution.

• Neutral (pH = 7): The concentrations of $H_3O^+$ and $OH^-$ are equal. Pure water at 25°C is the classic example, with both ions at $1.0 \times 10^{-7} \text{ M}$.
• Acidic (pH < 7): $H_3O^+$ concentration is greater than $OH^-$ concentration. The lower the pH, the more acidic the solution.
  • Examples: lemon juice (pH ≈ 2), vinegar (pH ≈ 3), black coffee (pH ≈ 5)
• Basic (pH > 7): $OH^-$ concentration is greater than $H_3O^+$ concentration. The higher the pH, the more basic the solution.
  • Examples: baking soda solution (pH ≈ 8.3), milk of magnesia (pH ≈ 10.5), bleach (pH ≈ 12.6)

One thing that trips people up: the pH scale is logarithmic, not linear. A solution with pH 3 is ten times more acidic than one with pH 4, and a hundred times more acidic than pH 5. Each whole-number step represents a tenfold change in $H_3O^+$ concentration.

Conversion of ion concentrations and pH

pH is defined as the negative base-10 logarithm of the hydronium ion concentration:

$pH = -\log[H_3O^+]$

To go the other direction and find the ion concentration from a known pH:

$[H_3O^+] = 10^{-pH}$

• Example: If pH = 4, then $[H_3O^+] = 10^{-4} = 1.0 \times 10^{-4} \text{ M}$

pOH works the same way, but for hydroxide ions:

$pOH = -\log[OH^-]$

$[OH^-] = 10^{-pOH}$

• Example: If pOH = 3, then $[OH^-] = 10^{-3} = 1.0 \times 10^{-3} \text{ M}$

The "p" in pH and pOH just means "take the negative log of." This convention shows up elsewhere in chemistry too (like $pK_a$), so it's worth remembering.

Relationship between pH and pOH

At 25°C, pH and pOH always add up to 14:

$pH + pOH = 14$

This means you can convert freely between the two:

• $pOH = 14 - pH$
  • Example: If pH = 5.5, then $pOH = 14 - 5.5 = 8.5$
• $pH = 14 - pOH$
  • Example: If pOH = 2.7, then $pH = 14 - 2.7 = 11.3$

Why does this work? It comes from the ion product of water ($K_w$), which is the equilibrium constant for water's autoionization:

$K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14} \text{ at 25°C}$

If you take the negative log of both sides, you get $pH + pOH = 14$. This relationship also lets you convert directly between $H_3O^+$ and $OH^-$ concentrations:

1. Given $[H_3O^+]$, find $[OH^-]$:

   • $[OH^-] = \frac{K_w}{[H_3O^+]}$
   • Example: If $[H_3O^+] = 1.0 \times 10^{-6} \text{ M}$, then $[OH^-] = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-6}} = 1.0 \times 10^{-8} \text{ M}$

2. Given $[OH^-]$, find $[H_3O^+]$:

   • $[H_3O^+] = \frac{K_w}{[OH^-]}$
   • Example: If $[OH^-] = 1.0 \times 10^{-3} \text{ M}$, then $[H_3O^+] = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-3}} = 1.0 \times 10^{-11} \text{ M}$

A quick way to check your work: the exponents of $[H_3O^+]$ and $[OH^-]$ should add up to -14 (when both concentrations are exact powers of 10).

Advanced pH Concepts

These topics build on pH and pOH and will come up later in the unit:

• Conjugate acid-base pairs are the products formed when an acid donates a proton or a base accepts one. They're central to understanding how acid-base equilibria shift.
• Buffer solutions resist changes in pH when small amounts of acid or base are added. They work because they contain both a weak acid and its conjugate base (or vice versa).
• The Henderson-Hasselbalch equation relates the pH of a buffer to the ratio of conjugate base to weak acid concentration: $pH = pK_a + \log\frac{[A^-]}{[HA]}$
• Titration is a lab technique for determining the concentration of an unknown acid or base by gradually neutralizing it with a solution of known concentration and tracking the pH change.