Question

Calculate the \(\mathrm{pH}\) at the equivalence point in titrating \(0.100 \mathrm{M}\) solutions of each of the following with \(0.080 \mathrm{M}\) \(\mathrm{NaOH}:\) (a) hydrobromic acid \((\mathrm{HBr})\), (b) lactic acid \(\left[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\right]\), (c) sodium hydrogen chromate \(\left(\mathrm{NaHCrO}_{4}\right)\)

Short answer

At the equivalence point, the pH for (a) hydrobromic acid (HBr) is \(7\). For (b) lactic acid \(\left[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\right]\), calculate the pH by first finding the pOH using the Ka expression for lactic acid, and then using the relationship pH = 14 - pOH. For (c) sodium hydrogen chromate \(\left(\mathrm{NaHCrO}_{4}\right)\), calculate the pH using the Kb expression for HCrO4⁻ and the formula pH = -log[H3O⁺].

Step-by-step solution

(a) Calculate the moles of HBr and NaOH at the equivalence point

At the equivalence point, the moles of acid (HBr) equal the moles of base (NaOH). We can use the formula, moles = volume × molarity. Let's assume that the volume of both HBr and NaOH solutions is V (in liters). Then, the moles of HBr and NaOH at the equivalence point are: Moles of HBr: \(0.100 \mathrm{M} \cdot V\) Moles of NaOH: \(0.080 \mathrm{M} \cdot V\) Since they are equal at the equivalence point, \(0.100 \mathrm{M} \cdot V = 0.080 \mathrm{M} \cdot V\)

(a) Calculate the pH of the resulting solution

Since HBr is a strong acid, it will completely dissociate into H+ and Br- ions. The reaction between HBr and NaOH will form NaBr and H2O. The resulting solution will only have Na+ and Br- ions, which will not affect the pH of the solution. Thus, the pH at the equivalence point will be 7.

(b) Calculate the moles of lactic acid and NaOH at the equivalence point

At the equivalence point, the moles of lactic acid equal the moles of NaOH. We can use the same approach as in step (a). Let's assume the volume of both lactic acid and NaOH solutions is V (in liters). Then, the moles of lactic acid and NaOH at the equivalence point are: Moles of lactic acid: \(0.100 \mathrm{M} \cdot V\) Moles of NaOH: \(0.080 \mathrm{M} \cdot V\) Since they are equal at the equivalence point, \(0.100 \mathrm{M} \cdot V = 0.080 \mathrm{M} \cdot V\)

(b) Calculate the pH of the resulting solution

When NaOH reacts with lactic acid, a lactate ion and water are formed. The dissociation of lactic acid after reaction with NaOH is as follows: \(\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COO}^- + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH} + \mathrm{OH}^-\) We can describe the above equilibrium using the Ka expression for lactic acid: Ka = \(\frac{[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}][\mathrm{OH}^-]}{[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH})\mathrm{COO}^-]}\) Given Ka for lactic acid = 1.37 × 10⁻⁴, we can solve for pOH, and then calculate pH using the relationship: pH = 14 - pOH

(c) Calculate the moles of NaHCrO4 and NaOH at the equivalence point

At the equivalence point, the moles of NaHCrO4 equal the moles of NaOH. Let's assume the volume of both NaHCrO4 and NaOH solutions is V (in liters). Then, the moles of NaHCrO4 and NaOH at the equivalence point are: Moles of NaHCrO4: \(0.100 \mathrm{M} \cdot V\) Moles of NaOH: \(0.080 \mathrm{M} \cdot V\) Since they are equal at the equivalence point, \(0.100 \mathrm{M} \cdot V = 0.080 \mathrm{M} \cdot V\)

(c) Calculate the pH of the resulting solution

When NaOH reacts with NaHCrO4, water and sodium chromate (Na2CrO4) are formed. The dissociation of NaHCrO4 after the reaction with NaOH is as follows: HCrO4⁻ + H2O ↔ CrO42⁻ + H3O+ The above equilibrium can be described using the Kb expression for HCrO4⁻: Kb = \(\frac{[\mathrm{CrO}_{4}^{2-}][\mathrm{H}_{3}\mathrm{O}^+]}{[\mathrm{HCrO}_{4}^-]}\) Since we know the Ksp of CrO42- is 2.15 × 10⁻¹³, we can find Kb using the relationship: Kw = Ka × Kb Once we have Kb, we can solve for the concentration of H3O+ and then calculate the pH using the formula: pH = -log[H3O⁺]

Key concepts

Equivalence Point

In a titration, the equivalence point is reached when the amount of titrant added is exactly enough to completely neutralize the analyte solution. At this juncture, the moles of acid are equivalent to the moles of base, forming products such as water and a salt. The solution is typically devoid of any free hydrogen ions (H+) or hydroxide ions (OH-), which would otherwise affect the pH.

• The equivalence point for a strong acid-strong base titration, like hydrobromic acid (HBr) with sodium hydroxide (NaOH), results in a neutral solution with a pH of 7.
• When a weak acid is involved, like lactic acid, the product is a weak conjugate base affecting the pH positively.

Understanding the equivalence point is crucial for calculating the pH of a solution, as it is where the reactants are perfectly balanced.

Strong Acids and Bases

Strong acids and bases completely ionize in water. This means that every molecule of a strong acid, like HBr, releases one hydrogen ion (H+) while a strong base like NaOH releases one hydroxide ion (OH-). This complete dissociation is what makes strong acids and bases predictable in titrations.

• When titrating a strong acid with a strong base, as with HBr and NaOH, the result is a neutralization reaction where the pH at the equivalence point is neutral, or about 7.
• Because both partners in a strong acid-strong base titration fully dissociate, it is one of the simplest titration scenarios to comprehend.

This property makes them particularly important in basic analytical chemistry where precision in expected outcomes is essential.

Weak Acid Dissociation

Weak acids, such as lactic acid, do not ionize completely in solution. Only a small fraction dissociates to release H+ ions. This partial dissociation makes calculations involving weak acids slightly more intricate than those involving strong acids.

• The degree of dissociation is described by the acid dissociation constant (Ka). For lactic acid, the Ka is 1.37 × 10⁻⁴, indicating a relatively weak ionization.
• In a titration involving a weak acid, like lactic acid with a strong base, the equivalence point will not be at pH 7 but higher, due to the production of the weak conjugate base.

The equilibrium established by a weak acid in a reaction must be mathematically evaluated to determine pH, typically requiring knowledge of initial concentration and the Ka value.

Acid-Base Titration

Acid-base titration is a laboratory method used to determine the concentration of an acid or base in a solution. This is accomplished by neutralizing the acid or base with a titrant—the solution of known concentration.

• The endpoint of a titration is indicated by a color change, often due to an added indicator that changes at a specific pH range.
• For strong acid-strong base titrations (like HBr and NaOH), the endpoint and equivalence point are often very close.
• Weak acid-strong base titrations require careful calculation, as seen with lactic acid, to account for the contribution of the conjugate base to the pH.

Titrations are pivotal in both educational and practical chemistry settings, providing a method for precise measurement and analysis of chemical solutions.