Question

Determine the \(\mathrm{pH}\) of each two-component solution. $$\begin{array}{l}{\text { a. } 0.050 \mathrm{M} \mathrm{KOH} \text { and } 0.015 \mathrm{M} \mathrm{Ba}(\mathrm{OH})_{2}} \\ {\text { b. } 0.265 \mathrm{M} \mathrm{NH}_{4} \mathrm{NO}_{3} \text { and } 0.102 \mathrm{M} \mathrm{HCN}} \\ {\text { c. } 0.075 \mathrm{M} \mathrm{RbOH} \text { and } 0.100 \mathrm{M} \mathrm{NaHCO}_{3}} \\ {\text { d. } 0.088 \mathrm{M} \mathrm{HClO}_{4} \text { and } 0.022 \mathrm{M} \mathrm{KOH}} \\ {\text { e. } 0.115 \mathrm{M} \mathrm{NaClO} \text { and } 0.0500 \mathrm{M} \mathrm{KI}}\end{array}$$

Short answer

a. Strong base, calculate using total \([OH^-]\). b. Weak acid(s), use \(\text{Ka}\) and approximations. c. Strong base + amphoteric, calculate net effect. d. Strong acid/base, subtract concentrations for net \([H^+]\). e. Weak base, use \(\text{Kb}\) and approximation.

Step-by-step solution

- Identifying the nature of each component in a.

Determine which components are acids and which are bases. KOH (potassium hydroxide) and Ba(OH)2 (barium hydroxide) are both strong bases. Calculate the total hydroxide ion concentration by adding the contributions from each base.

- Calculating the total \([OH^-]\) concentration for a.

Calculate the total concentration of \([OH^-]\). For KOH, \([OH^-]\) is 0.050 M. For Ba(OH)2, each molecule produces two \([OH^-]\) ions, so the total \([OH^-]\) contribution is 2 * 0.015 M = 0.030 M. Add these to get the total \([OH^-]\) = 0.050 M + 0.030 M = 0.080 M.

- Calculate the \([OH^-]\) and \([H^+]\) concentrations

Use the \([OH^-]\) concentration to find the \([H^+]\) concentration using the water dissociation constant (\([H^+] [OH^-]=10^{-14}\)). \([H^+] = \frac{10^{-14}}{[OH^-]}\).

- Calculate the \(\text{pH}\) for a.

Find the \(\text{pH}\) using the \([H^+]\) concentration. \(\text{pH} = -log[H^+]\). Calculate this using the \([H^+]\) value obtained in the previous step.

- Identifying the nature of each component in b.

Determine which components are acids and which are bases. NH4NO3 is made up of NH4+ (ammonium) and NO3- (nitrate). Ammonium is a weak acid and nitrate is a neutral spectator ion. HCN (hydrocyanic acid) is a weak acid.

- Calculating the \(\text{pH}\) for b.

For a solution with weak acids, use the acid dissociation constant (Ka) to find the \([H^+]\) and calculate \(\text{pH}\) accordingly.

- Identifying the nature of each component in c.

Determine which components are acids and which are bases. RbOH (rubidium hydroxide) is a strong base, and NaHCO3 (sodium bicarbonate) is a weak base and a weak acid.

- Calculating the \(\text{pH}\) for c.

For a solution with a strong base and a substance that can act as both a weak acid and a weak base, calculate the net \([OH^-]\) or \([H^+]\) concentration based on the predominant species and determine the \(\text{pH}\).

- Identifying the nature of each component in d.

HClO4 (perchloric acid) is a strong acid, and KOH is a strong base. Calculate the net \([H^+]\) or \([OH^-]\) concentration based on their reaction.

- Calculating the \(\text{pH}\) for d.

Subtract the concentration of the base from that of the acid to find the net \([H^+]\) if the acid is in excess or net \([OH^-]\) if the base is in excess. Use this net concentration to calculate the \(\text{pH}\).

- Identifying the nature of each component in e.

NaClO (sodium hypochlorite) is a weak base and KI (potassium iodide) does not affect the \(\text{pH}\) since both K+ and I- are spectator ions.

- Calculating the \(\text{pH}\) for e.

Use the concentration of NaClO and its dissociation constant (\(\text{Kb}\)) to find the \([OH^-]\) concentration and then calculate the \(\text{pH}\).

Key concepts

Strong and Weak Acids and Bases

Understanding the difference between strong and weak acids and bases is crucial for pH calculation. Strong acids and bases fully dissociate in water, releasing all their hydrogen (H+) or hydroxide (OH-) ions, respectively. Examples include HCl (hydrochloric acid) for a strong acid and KOH (potassium hydroxide) for a strong base. In contrast, weak acids and bases do not fully dissociate, instead establishing an equilibrium in which only a portion of them releases ions. Acetic acid (CH3COOH) and ammonia (NH3) are common examples of a weak acid and a weak base, respectively. This degree of dissociation affects the resulting pH of a solution.When calculating pH for strong acids or bases, the total concentration of H+ or OH- ions in solution is used directly. However, for weak acids and bases, one must consider the equilibrium constant (Ka for acids and Kb for bases) to calculate the ion concentration.

Acid-Base Equilibria

The concept of acid-base equilibria revolves around the reversible reaction of acids and bases in water. For weak acids, the acid dissociation constant (Ka) expresses the equilibrium between the undissociated acid, the hydronium ions (H3O+ or H+), and the conjugate base (A-). The expression is given by \[Ka = \frac{[H^+][A^-]}{[HA]}\].For weak bases, a similar expression applies for the base dissociation constant (Kb). Acid-base equilibria are dynamic processes where the rate of the forward reaction (dissociation) is equal to the rate of the reverse reaction (association), and the concentrations of reactants and products reach a steady state. This balance is essential for understanding pH in solutions containing weak acids or bases, where the pH is not simply the negative logarithm of the acid or base concentration but calculated using the respective equilibrium constant.

Water Dissociation Constant

The water dissociation constant (Kw) is a fundamental concept in acid-base chemistry. It represents the equilibrium constant for the self-ionization of water and is always at a value of \(1 \times 10^{-14}\) at 25°C. The expression for this constant is\[Kw = [H^+][OH^-]\].This constant implies that the concentrations of hydrogen ions (H+) and hydroxide ions (OH-) are inversely related; as one increases, the other decreases. In a neutral solution at standard temperature, the concentration of both ions is \(1 \times 10^{-7}\) M. To find the hydroxide ion concentration from the hydronium ion concentration (and vice versa), one can simply use this constant. This is a foundational step in pH and pOH calculation.

Hydroxide and Hydronium Ion Concentration

The concentration of hydroxide ions (OH-) and hydronium ions (H3O+ or H+) directly influences the pH and pOH of a solution. The pH is defined as the negative logarithm of the H+ concentration, while pOH is the negative logarithm of the OH- concentration. These two values are inversely related and add up to 14 (pH + pOH = 14) at 25°C.To calculate the concentration of these ions, you may start from the known molarity of a strong acid or base or use the equilibrium expressions for weak acids and bases. In mixtures or reactions between an acid and a base, such as the textbook example involving HClO4 and KOH, stoichiometry determines which substance is in excess and dictates the resulting pH or pOH of the solution. Remember to also account for substances like Ba(OH)2, where multiple OH- ions are produced per formula unit.