What is being titrated?
We titrate a weak base B (analytical solution) with a strong acid (titrant).
The principal stoichiometric reaction is:
B(aq) + H+(aq) → BH+(aq)
Before the equivalence point, the mixture is a buffer (conjugate pair B/BH+).
At the equivalence point, only the conjugate acid BH+ remains and it undergoes acidic hydrolysis.
Beyond equivalence, excess strong acid dictates the pH.
Inputs & symbols
Cb: initial concentration of weak base B (M)Vb: initial volume of the weak base solution (mL)Ca: concentration of the strong acid (M)Kb or pKb of B (at 25 °C)Va: volume of acid added (mL) — can be multiple custom values
Internally, the calculator uses Kw = 1.0×10−14 at 25 °C and converts mL → L where needed.
Key volumes
- Equivalence volume:
\[
V_e = \frac{C_b\,V_b}{C_a}\quad (\text{in the same volume units as }V_b)
\]
At
Va=Ve, moles of acid added equal initial moles of base.
- Halfway volume: \(\;V_{1/2} = \tfrac{1}{2}V_e\).
At this point \([\text{B}] = [\text{BH}^+]\) and \( \mathrm{pH} \approx \mathrm{p}K_a(\mathrm{BH}^+) \).
Equilibrium relationships
- Base dissociation:
\[
\mathrm{B} + \mathrm{H_2O} \rightleftharpoons \mathrm{BH^+} + \mathrm{OH^-},\quad
K_b = \frac{[\mathrm{BH^+}][\mathrm{OH^-}]}{[\mathrm{B}]}
\]
- Conjugate-acid dissociation:
\[
\mathrm{BH^+} + \mathrm{H_2O} \rightleftharpoons \mathrm{B} + \mathrm{H_3O^+},\quad
K_a = \frac{K_w}{K_b},\quad \mathrm{p}K_a = 14 - \mathrm{p}K_b
\]
pH in each titration region
1) Initial solution (Va=0) — weak base only
Solve the exact weak-base equilibrium for \([\mathrm{OH}^-]\) (set \(x=[\mathrm{OH}^-]\)):
\[
K_b = \frac{x^2}{C_b - x}\;\Rightarrow\;
x = \frac{-K_b + \sqrt{K_b^2 + 4K_b C_b}}{2},\quad
\mathrm{pH} = 14 - \mathrm{pOH} = 14 + \log_{10} x
\]
(When \(x \ll C_b\), the approximation \(x \approx \sqrt{K_b C_b}\) is often valid, but the calculator uses the exact root.)
2) Buffer region (0 < Va < Ve)
Stoichiometry converts part of B to BH+:
\[
n_{\text{B}} = n_{\text{B0}} - n_{\text{H}^+},\quad
n_{\text{BH}^+} = n_{\text{H}^+}
\]
With the conjugate-acid acidity \(K_a = K_w/K_b\) and \( \mathrm{p}K_a = 14 - \mathrm{p}K_b\),
the buffer pH uses the Henderson–Hasselbalch form (mole ratio is fine since both species share the same volume):
\[
\mathrm{pH} = \mathrm{p}K_a + \log_{10}\!\left(\frac{n_{\mathrm{B}}}{n_{\mathrm{BH}^+}}\right)
\]
At the halfway point (\(n_{\mathrm{B}} = n_{\mathrm{BH}^+}\)), \(\mathrm{pH} \approx \mathrm{p}K_a\).
3) Equivalence point (Va=Ve) — acidic hydrolysis of BH+
After complete neutralization, only the salt of the conjugate acid remains at
\( C_{\text{salt}} = \dfrac{n_{\text{B0}}}{V_{\text{tot}}} \).
The hydrolysis of BH+ gives:
\[
K_a = \frac{x^2}{C_{\text{salt}} - x}\;\Rightarrow\;
[\mathrm{H}^+] = \frac{-K_a + \sqrt{K_a^2 + 4K_a C_{\text{salt}}}}{2},\quad
\mathrm{pH} = -\log_{10}[\mathrm{H}^+]
\]
(If \(x \ll C_{\text{salt}}\), the approximation \(x \approx \sqrt{K_a C_{\text{salt}}}\) is acceptable.)
4) Beyond equivalence (Va > Ve) — excess strong acid
\[
[\mathrm{H}^+] = \frac{n_{\mathrm{H}^+} - n_{\mathrm{B0}}}{V_{\text{tot}}},\quad
\mathrm{pH} = -\log_{10}[\mathrm{H}^+]
\]
Titration curve & special points
- Start (0 mL): basic pH from weak-base equilibrium.
- Halfway: \(\mathrm{pH} \approx \mathrm{p}K_a(\mathrm{BH}^+)\).
- Equivalence: pH < 7 (acidic), set by hydrolysis of
BH+.
- ±1 mL around eq: computed to capture the sharp pH change near the inflection.
- Large excess acid: pH approaches that of strong acid at the diluted concentration.
How the calculator proceeds (algorithm)
- Read
Cb, Vb, Ca and either Kb or pKb;
compute Kb, then Ka=Kw/Kb and pKa.
- Compute
Ve = (CbVb)/Ca, and V1/2=Ve/2.
- For each requested
Va, determine the regime:
Va=0: solve weak-base equilibrium exactly.0<Va<Ve: buffer → Henderson–Hasselbalch with mole ratio.Va=Ve: hydrolysis of BH+ (quadratic for [H+]).Va>Ve: strong-acid excess.
- Also evaluate the curve near equivalence (±1 mL) and generate a smooth pH–volume plot with highlights.
Assumptions & notes
- Standard conditions: \(K_w = 1.0\times10^{-14}\) (25 °C). Temperature changes shift \(K_w\), \(K_a\), \(K_b\).
- Ionic strength, activity coefficients, and dilution effects beyond simple mixing are not explicitly modeled.
- All concentrations are formal molarities; volumes are additive; gases and volatility are ignored.
- For very dilute/extreme pH cases, water autoionization may need explicit treatment; the model is adequate
for typical analytical titrations.
Quick checks
- At halfway: pH ≈ pKa of
BH+ (powerful diagnostic).
- Equivalence pH must be < 7 (acidic) for weak base titrated by strong acid.
- Far beyond equivalence, pH should match a diluted strong acid solution:
\([\mathrm{H}^+] \approx (n_{\mathrm{H}^+}-n_{\mathrm{B0}})/V_{\text{tot}}\).
