Salt hydrolysis — pH of salt solutions (4 cases - SA - SB, SA - WB, WA - SB, WA - WB): Worked examples

This is the "throw everything at it" page for the parent topic. We will not learn new theory — we will stress-test the four formulas against every kind of salt an exam can hand you: the neutral one, the acidic one, the basic one, the "it depends" one, plus the sneaky edge cases (very dilute solutions, a salt of a strong-ish weak acid, and a word problem).

Before any formula fires, here are the four tools from the parent, restated so this page stands alone. Every symbol is spelled out.

Formula The four pH formulas (and what each letter means)

$C$ = the salt concentration in moles per litre (how crowded the ions are).
$K_{w}$ = the water-splitting constant $= [H^{+}] [OH^{-}] = 1 0^{- 14}$ at $2 5^{\circ} C$ . So $p K_{w} = - lo g K_{w} = 14$ .
$K_{a}$ = strength of the weak acid parent (bigger $K_{a}$ = stronger acid). $p K_{a} = - lo g K_{a}$ .
$K_{b}$ = strength of the weak base parent (bigger $K_{b}$ = stronger base). $p K_{b} = - lo g K_{b}$ .
$pH = - lo g [H^{+}]$ , and $pOH = - lo g [OH^{-}]$ , with $pH + pOH = 14$ .

Salt built from	Name	Formula
Strong acid + Strong base	neutral	$pH = 7$
Strong acid + Weak base	acidic	$pH = \frac{1}{2} (p K_{w} - p K_{b} - lo g C)$
Weak acid + Strong base	basic	$pH = \frac{1}{2} (p K_{w} + p K_{a} + lo g C)$
Weak acid + Weak base	depends	$pH = \frac{1}{2} (p K_{w} + p K_{a} - p K_{b})$

The scenario matrix

Think of every problem this topic can throw as a cell in this table. Our job below is to hit every cell at least once.

#	Cell (the scenario)	Expected direction	Example that hits it
A	SA/SB neutral salt	pH = 7 exactly	Ex 1 (KNO₃)
B	SA/WB acidic salt	pH < 7	Ex 2 (NH₄Cl)
C	WA/SB basic salt	pH > 7	Ex 3 (KCN)
D	WA/WB, $K_{a} > K_{b}$	slightly acidic	Ex 4 (NH₄F)
E	WA/WB, $K_{a} < K_{b}$	slightly basic	Ex 5 (NH₄ acetate variant)
F	WA/WB, $K_{a} = K_{b}$	exactly 7	Ex 5b (checked inside Ex 5)
G	Degenerate: very dilute basic salt	pushes toward 7	Ex 6 (10⁻⁴ M acetate)
H	Limiting: concentration up for acidic salt	pushes away from 7	Ex 7 (1 M NH₄Cl)
I	Word problem / real world	interpret then compute	Ex 8 (soap-water)
J	Exam twist: find $K_{a}$ from pH (reverse)	back-solve	Ex 9

The figure below is our compass for every example: which side of pH 7 do we land on, and why.

Figure — Salt hydrolysis — pH of salt solutions (4 cases - SA - SB, SA - WB, WA - SB, WA - WB)

Worked example Ex 1 — Cell A:

$0.5 M KNO_{3}$ Forecast: KNO₃ = KOH (strong base) + HNO₃ (strong acid). Both parents strong. Guess the pH before reading on.

Step 1 — Identify parents. K⁺ comes from KOH (strong), NO₃⁻ comes from HNO₃ (strong). Why this step? The formula you use is decided entirely by parent strength — never skip identification.

Step 2 — Ask "does any ion want to react with water?" Neither K⁺ nor NO₃⁻ has a weak parent to return to, so neither grabs H⁺ or OH⁻. Why this step? Hydrolysis only happens if an ion is "unsatisfied." Both here are satisfied.

Step 3 — Apply the neutral rule. No hydrolysis ⇒ water is undisturbed ⇒ $[H^{+}] = K_{w} = 1 0^{- 7}$ . $pH = - lo g (1 0^{- 7}) = 7$ Why this step? With no extra H⁺ or OH⁻, only water's own split sets the pH.

Verify: Notice $C = 0.5$ never entered the calculation — correct, because a neutral salt's pH is concentration-independent. pH = 7.00. ✓

Worked example Ex 4 — Cell D:

$0.2 M NH_{4} F$ ( $K_{a} > K_{b}$ ) Forecast: Weak acid HF ( $K_{a} = 6.6 \times 1 0^{- 4}$ ) + weak base NH₃ ( $K_{b} = 1.8 \times 1 0^{- 5}$ ). Here $K_{a} > K_{b}$ , meaning the acid parent is stronger, so its conjugate anion (F⁻) is the weaker base — the cation's H⁺ production wins. Expect slightly acidic, pH < 7.

Given: $p K_{a} = - lo g (6.6 \times 1 0^{- 4}) = 3.18$ , $p K_{b} = - lo g (1.8 \times 1 0^{- 5}) = 4.74$ .

Step 1 — Use the WA/WB formula. $pH = \frac{1}{2} (p K_{w} + p K_{a} - p K_{b})$ Why this step? Both ions hydrolyze; the concentration $C$ drops out entirely.

Step 2 — Substitute. $pH = \frac{1}{2} (14 + 3.18 - 4.74) = \frac{1}{2} (12.44) = 6.22$ Why this step? Because $p K_{a} < p K_{b}$ here, the middle terms make pH slip below 7.

Verify: pH = 6.22 < 7, matching " $K_{a} > K_{b}$ ⇒ acidic." Note $C = 0.2$ never mattered — the defining feature of WA/WB. ✓

Worked example Ex 6 — Cell G (degenerate: very dilute basic salt)

$1 0^{- 4} M$ sodium acetate Forecast: Same acetate as parent, but very dilute. The $+ lo g C$ term with tiny $C$ is a big negative number, dragging pH down toward 7. Expect only mildly basic.

Given: $C = 1 0^{- 4}$ , $K_{a} = 1.8 \times 1 0^{- 5}$ ⇒ $p K_{a} = 4.74$ , $lo g C = - 4$ .

Step 1 — WA/SB formula. $pH = \frac{1}{2} (14 + 4.74 - 4) = \frac{1}{2} (14.74) = 7.37$ Why this step? Watch the $lo g C = - 4$ pulling pH toward neutral as dilution increases.

Step 2 — Sanity on the trend. At $0.01$ M the parent got 8.37; at $1 0^{- 4}$ M we get 7.37 — 100× dilution dropped pH by 1 unit, as $\frac{1}{2} lo g (100) = 1$ predicts. Why this step? Confirms the formula behaves like the "compass" figure: dilution → toward 7.

Verify: pH = 7.37. (Caveat: below $\sim 1 0^{- 5}$ M, water's own $1 0^{- 7}$ H⁺ starts to matter and the simple formula fails — a genuine degenerate limit to flag in an exam.) ✓

Worked example Ex 8 — Cell I (real-world word problem): soapy water

Statement: Household "soap" is largely sodium stearate, the sodium salt of the weak acid stearic acid ( $K_{a} = 1.3 \times 1 0^{- 5}$ ). You make a $0.02 M$ solution to wash your hands. Is it acidic, neutral, or basic, and what is the pH? Forecast: Sodium (strong-base parent) + stearate (weak-acid conjugate) ⇒ WA/SB ⇒ basic (that "slippery, slightly alkaline" feel of soap). Guess pH ≈ 8.

Step 1 — Classify. Na⁺ inert; stearate⁻ hydrolyzes to make OH⁻. Why this step? Word problems still reduce to "which cell of the matrix?"

Step 2 — WA/SB formula. $p K_{a} = - lo g (1.3 \times 1 0^{- 5}) = 4.89$ , $lo g (0.02) = - 1.70$ . $pH = \frac{1}{2} (14 + 4.89 - 1.70) = \frac{1}{2} (17.19) = 8.60$ Why this step? Turns the "why is soap slippery-basic?" question into a number.

Verify: pH = 8.60, mildly basic — matches everyday experience that soap water is gentle-alkaline, not caustic. ✓

Recall Quick self-test

Which cell does $Na_{2} SO_{4}$ hit, and its pH? ::: Cell A (SA/SB, H₂SO₄ + NaOH) → pH = 7. For a WA/WB salt, why does concentration not appear? ::: Because $C$ cancels between the two coupled hydrolyses; $[H^{+}] = K_{w} K_{a} / K_{b}$ has no $C$ . Diluting a basic salt 100× moves pH by how much, toward where? ::: Down by $\frac{1}{2} lo g 100 = 1$ unit, toward 7. If $K_{a} > K_{b}$ in a WA/WB salt, acidic or basic? ::: Acidic (H⁺ production wins).