Titrations

Overview

Titration is a quantitative analytical technique to determine the concentration of an unknown solution by reacting it with a solution of known concentration. The equivalence point is reached when stoichiometrically equivalent amounts have reacted.

Key Terms

Term	Definition
Titrant	Solution of known concentration (in burette)
Analyte	Solution of unknown concentration
Equivalence Point	Moles of acid = Moles of base (stoichiometric)
End Point	Point where indicator changes color
Indicator	Substance that changes color at a specific pH

Basic Calculation

At equivalence point:

$$n_{\text{acid}} \times \text{coeff}_{\text{base}} = n_{\text{base}} \times \text{coeff}_{\text{acid}}$$

For 1:1 reactions:

$$M_aV_a = M_bV_b$$

Types of Titration Curves

Strong Acid + Strong Base

• Initial pH: Low (acidic)
• Equivalence point: pH = 7
• Final pH: High (basic)

Weak Acid + Strong Base

• Initial pH: Moderate (weak acid)
• Half-equivalence: pH = pKa
• Equivalence point: pH > 7
• Buffer region: Gradual rise

Strong Acid + Weak Base

• Initial pH: High (weak base)
• Equivalence point: pH < 7

Weak Acid + Weak Base

• Less distinct equivalence point
• No sharp pH change
• Difficult to titrate accurately

Important Points on Titration Curves

Initial Point

Before adding any titrant.

Half-Equivalence Point

Half the volume needed to reach equivalence.

$$\text{pH} = \text{p}K_a \quad \text{(for weak acid)}$$ $$[\text{HA}] = [\text{A}^-]$$

Equivalence Point

Stoichiometric amounts have reacted.

Titration Type	pH at Equivalence
Strong acid + Strong base	7.0
Weak acid + Strong base	> 7
Strong acid + Weak base	< 7

Indicators

Common Indicators

Indicator	pH Range	Color Change
Methyl orange	3.1-4.4	Red → Yellow
Methyl red	4.4-6.2	Red → Yellow
Bromothymol blue	6.0-7.6	Yellow → Blue
Phenolphthalein	8.2-10.0	Colorless → Pink
Thymolphthalein	9.3-10.5	Colorless → Blue

Choosing an Indicator

Select an indicator whose color change range includes the equivalence point pH.

Titration	Equivalence pH	Suitable Indicator
Strong/Strong	7	Bromothymol blue
Weak acid/Strong base	8-10	Phenolphthalein
Strong acid/Weak base	4-6	Methyl red

Example Calculations

Example 1: Strong Acid-Strong Base

25.0 mL of HCl requires 32.5 mL of 0.100 M NaOH. Find [HCl].

$$M_aV_a = M_bV_b$$ $$M_a \times 25.0 = 0.100 \times 32.5$$ $$M_a = 0.130 \text{ M}$$

Example 2: Finding Equivalence Point pH

Titrating 50.0 mL of 0.100 M acetic acid with 0.100 M NaOH.

At equivalence:

$$\text{mol CH}_3\text{COOH} = \text{mol NaOH} = 0.050 \times 0.100 = 0.00500 \text{ mol}$$ $$\text{Volume} = 50.0 + 50.0 = 100.0 \text{ mL}$$

All acetic acid → acetate ion

$$[\text{CH}_3\text{COO}^-] = \frac{0.00500}{0.100} = 0.0500 \text{ M}$$

Acetate is a weak base:

$$K_b = \frac{K_w}{K_a} = \frac{10^{-14}}{1.8 \times 10^{-5}} = 5.6 \times 10^{-10}$$ $$[\text{OH}^-] = \sqrt{K_b \times C} = \sqrt{5.6 \times 10^{-10} \times 0.0500} = 5.3 \times 10^{-6} \text{ M}$$ $$\text{pOH} = 5.28$$ $$\text{pH} = 8.72$$

Example 3: pH During Titration

25.0 mL of 0.100 M weak acid ($K_a = 1.0 \times 10^{-5}$) titrated with 0.100 M NaOH.

After adding 10.0 mL NaOH:

$$\text{Initial mol HA} = 0.0250 \times 0.100 = 0.00250 \text{ mol}$$ $$\text{mol NaOH added} = 0.0100 \times 0.100 = 0.00100 \text{ mol}$$ $$\text{mol HA remaining} = 0.00250 - 0.00100 = 0.00150 \text{ mol}$$ $$\text{mol A}^- \text{ formed} = 0.00100 \text{ mol}$$ $$\text{pH} = \text{p}K_a + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)$$ $$\text{pH} = 5.00 + \log\left(\frac{0.00100}{0.00150}\right)$$ $$\text{pH} = 5.00 + \log(0.667) = 5.00 - 0.18 = 4.82$$

Polyprotic Acid Titrations

Multiple equivalence points for each ionizable hydrogen.

H₃PO₄ Titration

• First equivalence: $\text{H}_3\text{PO}_4 \rightarrow \text{H}_2\text{PO}_4^-$
• Second equivalence: $\text{H}_2\text{PO}_4^- \rightarrow \text{HPO}_4^{2-}$
• Third equivalence: $\text{HPO}_4^{2-} \rightarrow \text{PO}_4^{3-}$

Back Titration

Used when direct titration is difficult.

1. Add excess reagent to analyte
2. Titrate excess with another solution
3. Calculate analyte by difference

Example

To find % CaCO₃ in antacid:

1. Add excess HCl to antacid
2. Back-titrate excess HCl with NaOH
3. Calculate CaCO₃ from HCl consumed