Question

Calculate the \(\mathrm{pH}\) of each solution: (a) \(0.0155 \mathrm{MHBr}\) (b) \(1.28 \times 10^{-3} \mathrm{MKOH}\) (c) \(1.89 \times 10^{-3} \mathrm{M} \mathrm{HNO}_{3}\) (d) \(1.54 \times 10^{-4} \mathrm{M} \mathrm{Sr}(\mathrm{OH})_{2}\)

Short answer

The pH for the solutions is: (a) 1.81, (b) 14.00 - 2.89 = 11.11, (c) 2.72, (d) 14.00 - (2 x 1.19) = 11.62.

Step-by-step solution

Identifying the Nature of Each Solution

First, identify whether each compound is an acid or a base. HBr and HNO3 are strong acids, KOH and Sr(OH)2 are strong bases.

Calculating pH for Strong Acids (HBr and HNO3)

For strong acids, the concentration of the acid equals the concentration of H+ ions because they dissociate completely in water. The pH is then calculated using the formula pH = -log[H+].

Step 2.1: Calculating pH of 0.0155 M HBr

The [H+] is 0.0155 M. Use the formula pH = -log[0.0155] to find the pH.

Step 2.2: Calculating pH of 1.89 x 10^-3 M HNO3

The [H+] is 1.89 x 10^-3 M. Use the formula pH = -log[1.89 x 10^-3] to find the pH.

Calculating pOH for Strong Bases (KOH and Sr(OH)2)

For strong bases, the concentration of the base equals the concentration of OH- ions. KOH dissociates completely into K+ and OH-, with a 1:1 ratio. Sr(OH)2 dissociates into Sr2+ and 2 OH-, with a 1:2 ratio.

Step 3.1: Calculating pOH of 1.28 x 10^-3 M KOH

The [OH-] is 1.28 x 10^-3 M. Use the formula pOH = -log[1.28 x 10^-3] to find the pOH.

Step 3.2: Calculating pOH of 1.54 x 10^-4 M Sr(OH)2

The [OH-] is 2 times 1.54 x 10^-4 M because there are two OH- ions per molecule. First calculate the [OH-], then use the formula pOH = -log[2 x 1.54 x 10^-4] to find the pOH.

Converting pOH to pH

The relationship between pH and pOH is pH + pOH = 14. To find the pH from the pOH, subtract the pOH from 14.

Final Step: Summarize All pH Values

Combine the calculated pH values from steps 2.1, 2.2, 4.3.1, and 4.3.2 for each solution to obtain a final answer.

Key concepts

Strong Acids and Bases

Understanding the characteristics of strong acids and bases is essential for calculating pH. Strong acids, such as hydrobromic acid (HBr) and nitric acid (HNO3), fully dissociate in water, meaning they release all of their hydrogen ions (H+) into the solution. Similarly, strong bases like potassium hydroxide (KOH) and strontium hydroxide (Sr(OH)2) also dissociate completely, but they release hydroxide ions (OH-) instead.

This total dissociation simplifies the pH calculation, because the concentration of the acid or base directly equals the concentration of H+ or OH- ions in the solution. For example, a 0.0155 M solution of HBr results in a 0.0155 M concentration of H+ ions. Similarly, a 1.28 x 10^-3 M KOH solution has the same concentration of OH- ions.

pH and pOH Relationship

The concepts of pH and pOH are related to the hydronium (H3O+) and hydroxide (OH-) ion concentrations, respectively, and they have a reciprocal relationship in water at 25°C. The sum of pH and pOH in any aqueous solution always equals 14, a principle that proves most helpful when needing to convert between pH and pOH.

For instance, after calculating the pOH of a strong base like KOH, you can easily find its pH by subtracting the pOH from 14. Consequently, knowing the pOH of a solution can give you the acidity or basicity with just simple subtraction, which is an invaluable tool when you are presented with calculations involving strong bases.

Hydrogen Ion Concentration

The hydrogen ion concentration [H+] is fundamental to understanding acidity in a solution. When calculating pH, which stands for 'power of hydrogen', the formula involves taking the negative logarithm of the hydrogen ion concentration: pH = -log[H+].

Because strong acids such as HBr and HNO3 fully dissociate in aqueous solutions, the initial concentration of the acid directly translates to the concentration of H+ ions. This makes calculating the pH straightforward—a higher concentration of H+ ions means a lower pH value, which indicates a more acidic solution. For example, a 1.89 x 10^-3 M solution of HNO3 will have a pH calculated by plugging the [H+] value directly into the pH formula.

Hydroxide Ion Concentration

Much like hydrogen ion concentration is linked to acids and pH, the hydroxide ion concentration [OH-] relates to bases and pOH. The pOH is calculated as the negative logarithm of the [OH-] concentration: pOH = -log[OH-].

For strong bases, such as KOH and Sr(OH)2, which fully dissociate, calculating [OH-] is just as straightforward as calculating [H+] for strong acids. Remember that polyhydroxide bases, like Sr(OH)2, provide more than one OH- ion per molecule. This factor must be considered when determining [OH-]—as seen in the case of Sr(OH)2, where the concentration of OH- ions is twice that of the compound's molecular concentration due to the two OH- ions released per molecule of Sr(OH)2.