Question

Calculate the \(\mathrm{pH}\) of each solution: (a) \(1.34 \times 10^{-3} \mathrm{M} \mathrm{HClO}_{4}\) (b) \(0.0211 \mathrm{M} \mathrm{NaOH}\) (c) \(0.0109 \mathrm{MHBr}\) (d) \(7.02 \times 10^{-5} \mathrm{M} \mathrm{Ba}(\mathrm{OH})_{2}\)

Short answer

The pH of each solution is: (a) 2.87 (b) 12.33 (c) 2.06 (d) 11.15

Step-by-step solution

- Calculate the pH for HClO4

HClO4 is a strong acid which dissociates completely in water. The concentration of H3O+ ions is equal to the concentration of HClO4. Calculate the pH using the formula pH = -log[H3O+].

- Calculate the pH for NaOH

NaOH is a strong base which dissociates completely in water. The concentration of OH- ions is equal to the concentration of NaOH. Calculate the pOH using the formula pOH = -log[OH-], then use pH + pOH = 14 to find the pH.

- Calculate the pH for HBr

HBr is a strong acid which dissociates completely in water. The concentration of H3O+ ions is equal to the concentration of HBr. Calculate the pH using the formula pH = -log[H3O+].

- Calculate the pH for Ba(OH)2

Ba(OH)2 is a strong base that dissociates completely in water, and it provides two OH- ions for each molecule. Calculate the concentration of OH- ions by multiplying the Ba(OH)2 concentration by 2. Then, calculate the pOH using the formula pOH = -log[OH-], and use pH + pOH = 14 to find the pH.

Key concepts

Strong Acids and Bases

Understanding the behavior of strong acids and bases is crucial when performing pH calculations. These substances dissociate completely in aqueous solution, meaning that they release all of their hydrogen ions (H+) or hydroxide ions (OH-) into the solution. This is a key characteristic that sets them apart from weak acids and bases, which only partially dissociate.

For instance, HClO4 (perchloric acid) and HBr (hydrobromic acid) are strong acids. When HClO4 is dissolved in water, it separates completely into H3O+ ions and ClO4- ions. Similarly, in the case of HBr, it fully dissociates into H3O+ and Br- ions. In both cases, the resulting concentration of hydronium ions (H3O+) is equal to the original concentration of the acid, making the pH calculation straightforward using the formula \( \text{pH} = -\text{log}[H_3O^+] \).

On the other hand, NaOH (sodium hydroxide) and Ba(OH)2 (barium hydroxide) are examples of strong bases. They completely dissociate into Na+ and OH- ions, and Ba2+ and 2 OH- ions, respectively. With strong bases, the pH is found by first calculating the pOH using the hydroxide ion concentration and then determining the pH from the relationship that pH + pOH = 14.

pH and pOH Relationship

The pH and pOH of a solution are interconnected by the relation pH + pOH = 14 at 25°C, which is the typical temperature for these measurements. This relationship is derived from the ion-product constant for water (Kw), which at 25°C is \( 1 \times 10^{-14} \). The pH scale is a measure of the acidity or basicity of a solution, and it ranges from 0 to 14, with 7 being neutral.

The pH is defined as the negative logarithm of the concentration of hydronium ions, \( \text{pH} = -\text{log}[H_3O^+] \), whereas the pOH is defined as the negative logarithm of the concentration of hydroxide ions, \( \text{pOH} = -\text{log}[OH^-] \). When a solution contains a strong base, such as NaOH, the direct pH calculation is not possible; instead, we calculate the pOH and use the relationship to find the pH value. This step is essential when dealing with strong bases and ensures students can accurately determine the pH of any solution, whether acidic or basic.

Dissociation of Electrolytes

Electrolytes are substances that produce ions when dissolved in water, and their dissociation is a fundamental concept in understanding how to calculate pH. Strong electrolytes like the strong acids and bases we have discussed dissociate completely into their ions. Understanding the extent of dissociation helps us in quantifying the ion concentrations in solution, which is directly related to the pH and pOH.

For strong acids and bases, dissociation is straightforward: one mole of the strong acid or base will produce one mole of H3O+ or OH-, respectively. For bases that yield more than one hydroxide per formula unit, such as Ba(OH)2, the dissociation will produce twice the concentration of OH- compared to the original concentration of the compound. In this example, Ba(OH)2 produces two moles of OH- for every mole of Ba(OH)2 dissolved. Therefore, the resulting concentration of hydroxide ions must be doubled before calculating the pOH and subsequently the pH.

Understanding the complete dissociation of strong electrolytes is vital for accurately calculating the pH of strong acid and base solutions. It's imperative for students to recognize that the concentrations of ions are proportional to the strength of the dissociation, and for strong acids and bases, this proportion is one-to-one or two-to-one, greatly simplifying pH calculations.