Question

Consider the titration of a \(25.0-\mathrm{mL}\) sample of \(0.175 \mathrm{M} \mathrm{CH}_{3} \mathrm{NH}_{2}\) with \(0.150 \mathrm{M}\) HBr. Determine each quantity. a. the initial \(\mathrm{pH}\) b. the volume of added acid required to reach the equivalence point c. the \(\mathrm{pH}\) at \(5.0 \mathrm{~mL}\) of added acid d. the \(\mathrm{pH}\) at one-half of the equivalence point e. the \(\mathrm{pH}\) at the equivalence point f. the pH after adding \(5.0 \mathrm{~mL}\) of acid beyond the equivalence point

Short answer

The initial pH is calculated using the base dissociation constant. The equivalence point is reached when the moles of HBr equals the moles of CH3NH2. The pH at 5.0 mL of added acid, at the half-equivalence point, at the equivalence point, and after adding 5.0 mL of acid beyond the equivalence point are calculated considering the reactions occurring at each stage, using appropriate formulas such as the Henderson-Hasselbalch equation and the hydrolysis of the conjugate acid.

Step-by-step solution

- Calculate initial pH

The initial pH of the CH3NH2 solution can be calculated using Kb (base dissociation constant) of CH3NH2. Since CH3NH2 is a weak base, we use the formula pH = 14 - pOH, where pOH is calculated from Kb and the concentration of the base. The Kb value is not given, so one would normally look it up in a table of dissociation constants. For this example, let’s assume Kb for CH3NH2 is 4.4 x 10^-4. First, calculate pOH using the approximation method, ignoring the contribution of CH3NH2 to the equilibrium, then calculate pH using the equation pH = 14 - pOH.

- Determine volume of acid to reach equivalence point

To reach the equivalence point, the moles of HBr added must equal the moles of CH3NH2 present initially. Calculate the moles of CH3NH2 using its concentration and volume, then use the concentration of HBr to find the volume required to provide an equal number of moles.

- Calculate pH at 5.0 mL of added acid

Calculate the moles of HBr added after 5.0 mL of it is introduced to the solution. Determine the moles of CH3NH2 that reacted with HBr. Subtract the number of moles of HBr from the initial moles of CH3NH2 to find the excess moles. Divide the excess moles by the new total volume of the solution to find the new concentration. Use this concentration to calculate the new pH, considering the solution may now be a buffer.

- Determine pH at half-equivalence point

At half the equivalence point, the number of moles of CH3NH2 equals the number of moles of its conjugate acid, CH3NH3+. This is a buffer solution, and the pH can be found using the Henderson-Hasselbalch equation. pH = pKa + log([base]/[acid]), where [base] = [acid] for the half-equivalence point.

- Calculate pH at the equivalence point

At the equivalence point, all the base has been converted to its conjugate acid, CH3NH3+. The pH is determined by the hydrolysis of the conjugate acid, which is now the main species in solution. Using the Ka of the conjugate acid, obtained from Kb by using the relationship Kw = Ka * Kb, calculate the pH of the solution.

- Determine pH after adding 5.0 mL of acid beyond the equivalence point

Calculate the moles of excess HBr after adding 5.0 mL beyond the equivalence point. Divide these moles by the new total volume to get the concentration of excess HBr. Since HBr is a strong acid, the pH of the solution is simply calculated from the concentration of excess HBr using the equation pH = -log[H+].

Key concepts

Acid-Base Titration

Acid-base titration is an analytical technique used to determine the concentration of an acid or base in a solution. During the titration, a solution of known concentration, called the titrant, is slowly added to a solution of the substance being analyzed until the reaction between them is complete. This point of completion is known as the equivalence point, where the number of moles of hydrogen ions equals the number of moles of hydroxide ions. For example, titrating the weak base CH₃NH₂ with the strong acid HBr involves the base reacting with the acid to form the conjugate acid CH₃NH₃⁺ as well as water. The initial pH of the base solution and the pH changes throughout the titration can be calculated to provide a better understanding of the reaction's progression.

Equivalence Point Calculation

The equivalence point in a titration is reached when the amount of titrant added is stoichiometrically equivalent to the amount of substance initially present in the solution. To calculate the volume of acid needed to reach the equivalence point, you must first determine the moles of the base in the solution. Once the moles are known, you can use the molarity of the titrant acid to find the volume needed. For instance, with a CH₃NH₂ solution, you'd use its initial volume and molarity to find the moles of base. Then, divide those moles by the molarity of the HBr to obtain the required volume of HBr for equivalence.

The equation is: Volume of acid = Moles of base / Molarity of acid. This step is crucial for understanding the titration's progress and determining the subsequent pH at various stages.

pH Calculation

Calculating the pH of a solution during a titration involves understanding the nature of the substances involved. The initial pH of a weak base solution can be estimated using its base dissociation constant (K_b) and the given concentration. As the strong acid is added, the pH can be calculated at any point based on the amounts of acid and base present. At points before the equivalence, you may have a buffer solution, necessitating the use of the Henderson-Hasselbalch equation. At equivalence, the pH is determined by the hydrolysis of the conjugate acid formed. If an excess of acid is added beyond the equivalence point, the solution's pH depends on the concentration of the excess strong acid, utilizing the pH equation pH = -log[H⁺].

Understanding these phases of pH calculation allows for precise control and prediction of results during the titration process.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a formula that relates the pH of a buffer solution to the pKa of the acid and the concentrations of the acid and its conjugate base. The equation is given by: pH = pKa + log([A^-]/[HA]), where [A^-] is the concentration of the conjugate base and [HA] is the concentration of the acid. At the half-equivalence point in a titration, the concentrations of the weak base and its conjugate acid are equal, and therefore, the pH equals the pKa of the conjugate acid. This equation is particularly helpful when dealing with buffer solutions, as it allows for the estimation of pH when the ratio of base to acid concentrations is known.

Base Dissociation Constant (Kb)

The base dissociation constant (K_b) is an expression of the extent to which a base dissociates in water to form hydroxide ions and the conjugate acid. It is a measure of the strength of the base. The higher the K_b value, the stronger the base. For instance, the K_b for CH₃NH₂ must be known or looked up in order to determine the initial pH of the base solution. The relationship K_w = K_a * K_b (where K_w is the ion-product constant for water) can also be used to find the K_a of the conjugate acid, which is necessary for calculating the pH at the equivalence point in the titration of a weak base with a strong acid.

Hydrolysis of Conjugate Acid

Hydrolysis of the conjugate acid refers to the reaction of the conjugate acid with water to produce hydronium ions and the original base. At the equivalence point of a weak base-strong acid titration, all of the base has reacted to form the conjugate acid, which can undergo hydrolysis. The extent of hydrolysis and the resulting pH can be calculated using the K_a value of the conjugate acid. Here's the process: K_w (ion-product constant for water) is used to find the K_a, as it is the inverse of K_b for the base. With K_a, we then calculate the concentration of H⁺ ions from hydrolysis and use this to find the pH. Understanding hydrolysis is essential for predicting the pH at the equivalence point for weak bases titrated with strong acids.