Question

Calculate the \(\mathrm{pH}\) of each of these solutions: (a) \(0.0010 \mathrm{MHCl},\) (b) \(0.76 \mathrm{M} \mathrm{KOH}\) (c) \(2.8 \times 10^{-4} M\) \(\mathrm{Ba}(\mathrm{OH})_{2},(\mathrm{~d}) 5.2 \times 10^{-4} \mathrm{M} \mathrm{HNO}_{3}\).

Short answer

The pH of (a) 0.0010 M HCl would be approximately 3.00. The pH of (b) 0.76 M KOH would be approximately 0.12. The pH of (c) 2.8 × 10^-4 M Ba(OH)2 would be approximately 13.45. The pH of (d) 5.2 × 10^-4 M HNO3 would be approximately 3.28. Please note, these are approximate values and might differ slightly based on the logarithmic calculations.

Step-by-step solution

Calculating pH of 0.0010 MHCl

As HCl is a strong acid, it will ionize completely in water to form H+ ions. So, the concentration of H+ ions, [H+], will be equal to the concentration of the HCl, which is 0.0010 M. Using the formula, pH = -log[H+], substitute [H+] = 0.0010 M into the formula to calculate the pH of the HCl solution.

Calculating pH of 0.76 M KOH

KOH is a strong base which completely ionizes in water to form OH- ions. The concentration of OH- ions, will be equal to the concentration of the KOH, which is 0.76 M. Using the formula, pOH = -log[OH-], substitute [OH-] = 0.76 M into the formula to calculate the pOH of the KOH solution. Once the pOH is calculated, use the formula pH + pOH = 14 (at 25°C), to find the pH of the KOH solution.

Calculating pH of 2.8 × 10^-4 M Ba(OH)2

Ba(OH)2 is a strong base that completely ionizes in water to form OH- ions. However, each formula unit of Ba(OH)2 produces 2 OH- ions, so the concentration of OH- ions is twice that of the Ba(OH)2. Thus, [OH-] = 2 × (2.8 × 10^-4 M). Use this to calculate pOH = -log[OH-] and then use pH + pOH = 14 (at 25°C) to find the pH of the Ba(OH)2 solution.

Calculating pH of 5.2 × 10^-4 M HNO3

As HNO3 is a strong acid, it will ionize completely in water to form H+ ions. So, the concentration of H+ ions, [H+], will be equal to the concentration of the HNO3, which is 5.2 × 10^-4 M. Use pH = -log[H+] to calculate the pH of the HNO3 solution.

Key concepts

pH scale

Understanding the pH scale is crucial in chemistry, especially when dealing with solutions and their acidity or basicity. The pH scale is a numeric scale used to specify the acidity or basicity (alkalinity) of an aqueous solution. It ranges from 0 to 14, with 7 being neutral. A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic solution.

The pH value is a direct measure of the hydrogen ion concentration in a solution and is calculated using the formula \( pH = -\log[H^+] \), where \( [H^+] \) is the molar concentration of hydrogen ions in the solution. The logarithmic scale means that each one-unit change in pH represents a tenfold change in hydrogen ion concentration, making it easier to work with the very small numbers involved in these calculations.

strong acids and bases

Strong acids and bases are substances that completely ionize in solution, which means they fully dissociate into their ions when dissolved in water. This complete ionization is what characterizes them as strong, unlike weak acids and bases, which only partially ionize.

Examples of strong acids include hydrochloric acid (HCl), nitric acid (HNO3), and sulfuric acid (H2SO4), while examples of strong bases include potassium hydroxide (KOH) and barium hydroxide (Ba(OH)2). When dealing with strong acids and bases, it is safe to assume that the concentration of hydrogen ions (for acids) or hydroxide ions (for bases) in solution will equal the initial concentration of the acid or base respectively. This assumption significantly simplifies pH and pOH calculations.

acid-base ionization

Acid-base ionization refers to the process by which acids and bases dissolve in water to produce ions. An acid will release hydrogen ions (H\textsuperscript{+}) into the solution, whereas a base will generate hydroxide ions (OH\textsuperscript{-}). The extent to which this ionization occurs depends on whether the substance is a strong or weak acid or base, as previously mentioned.

The ionization of a strong acid like HCl can be represented by the chemical equation HCl \(\rightarrow\) H\textsuperscript{+} + Cl\textsuperscript{-}. Similarly, strong bases like KOH ionize in water according to the equation KOH \(\rightarrow\) K\textsuperscript{+} + OH\textsuperscript{-}. This concept is pivotal when interpreting the ion concentrations necessary for calculating pH and pOH values.

pOH and pH calculations

The calculations of pOH and pH are interrelated and vital for understanding the properties of aqueous solutions. The pOH of a solution is a measure of the hydroxide ion concentration and is calculated using the formula \( pOH = -\log[OH^-] \). Just like the pH, the pOH scale also runs from 0 to 14.

In a solution at 25°C, the sum of pH and pOH is always 14 (pH + pOH = 14). This relation is based on the self-ionization of water and the product of hydrogen ion concentration and hydroxide ion concentration always being 1×10\textsuperscript{-14} at 25°C (\begin{small}[H^+] \times [OH^-] = 1 \times 10^{-14}\textmd{mol}^2/\textmd{L}^2\textmd{ at 25°C}\textmd{)}\textmd{. Knowing one of the values allows calculation of the other.}

When dealing with strong bases, such as in the provided exercise, one must first calculate the pOH using the base's hydroxide ion concentration, and then use the relation between pH and pOH to find the pH. Remember, for strong bases that release more than one hydroxide ion per molecule, like Ba(OH)2, this must be accounted for in the pOH calculation by multiplying the base's concentration by the number of hydroxide ions produced per formula unit.