Question

Calculate the $\mathrm{pH}$ of each of the following solutions. a. $0.12\mathrm{M}{\mathrm{KNO}}_2$ c. $0.40\mathrm{M}{\mathrm{NH}}_4{\mathrm{ClO}}_4$ b. $0.45\mathrm{M}\mathrm{NaOCl}$

Short answer

The pH of the given solutions are: a. Solution with $0.12 \mathrm{M} \mathrm{KNO}_{2}$: pH $\approx 8.22$ c. Solution with $0.40 \mathrm{M} \mathrm{NH}_{4} \mathrm{ClO}_{4}$: pH $\approx 5.22$ b. Solution with $0.45 \mathrm{M} \mathrm{NaOCl}$: pH $\approx 7.52$

Step-by-step solution

Calculate the concentration of the conjugate acid/base

For each of the solutions, we will first determine the concentration of their respective conjugate acid/base, required for the calculation of pH. For Solution a: ${\mathrm{KNO}}_2$ dissociates in water to form ${\mathrm{OH}}^{-}$ ions and ${\mathrm{NO}}_2^{-}$ ions. Since the concentration of ${\mathrm{KNO}}_2$ is given, the concentration of ${\mathrm{NO}}_2^{-}$ ions will also be $0.12$M. For Solution c: ${\mathrm{NH}}_4{\mathrm{ClO}}_4$ dissociates in water to form ${\mathrm{NH}}_4^{+}$ ions and ${\mathrm{ClO}}_4^{-}$ ions. The concentration of ${\mathrm{NH}}_4^{+}$ ions will be $0.40$M. For Solution b: $\mathrm{NaOCl}$ dissociates in water to form ${\mathrm{Na}}^{+}$ ions and ${\mathrm{OCl}}^{-}$ ions. The concentration of ${\mathrm{OCl}}^{-}$ ions will be $0.45$M.

Determine the ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$ concentration

We will now find the concentration of ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$ ions in each solution using the equilibrium constants (Ka or Kb) of their respective conjugate acid/base. For Solution a: The equilibrium constant for the dissociation of ${\mathrm{NO}}_2^{-}$ ions, Kb, is given by: $$K_b=\frac{[{\mathrm{HNO}}_{2+}][{\mathrm{OH}}^{-}]}{[{\mathrm{NO}}_2^{-}]}$$ The ${\mathrm{OH}}^{-}$ concentration can be calculated as: $${\mathrm{OH}}^{-}=\sqrt{K_b\times [{\mathrm{NO}}_2^{-}]}$$ For Solution c: The equilibrium constant for the dissociation of ${\mathrm{NH}}_4^{+}$ ions, Ka, is given by: $$K_a=\frac{[{\mathrm{NH}}_3][{\mathrm{H}}^{+}]}{[{\mathrm{NH}}_4^{+}]}$$ The ${\mathrm{H}}^{+}$ concentration can be calculated as: $${\mathrm{H}}^{+}=\sqrt{K_a\times [{\mathrm{NH}}_4^{+}]}$$ For Solution b: The equilibrium constant for the dissociation of ${\mathrm{OCl}}^{-}$ ions, Kb, is given by: $$K_b=\frac{[\mathrm{HOCl}+][{\mathrm{OH}}^{-}]}{[{\mathrm{OCl}}^{-}]}$$ The ${\mathrm{OH}}^{-}$ concentration can be calculated as: $${\mathrm{OH}}^{-}=\sqrt{K_b\times [{\mathrm{OCl}}^{-}]}$$

Calculate the pH

Finally, we will calculate the pH for each solution using the ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$ concentrations. For Solution a: Calculate the pOH using the ${\mathrm{OH}}^{-}$ concentration: $$pOH=-\log ([{\mathrm{OH}}^{-}])$$ Then, calculate pH using the relationship: $$pH=14-pOH$$ For Solution c: Calculate pH using the ${\mathrm{H}}^{+}$ concentration: $$pH=-\log ([{\mathrm{H}}^{+}])$$ For Solution b: Calculate the pOH using the ${\mathrm{OH}}^{-}$ concentration: $$pOH=-\log ([{\mathrm{OH}}^{-}])$$ Then, calculate pH using the relationship: $$pH=14-pOH$$

Key concepts

acid-base equilibrium

Acid-base equilibrium is a key concept in the field of chemistry that describes the balance between acid and base forms of a compound in solution. An acid-base equilibrium is established when a weak acid partially dissociates in water to donate protons (${\mathrm{H}}^{+}$), forming its conjugate base. Similarly, a weak base can accept protons, forming its conjugate acid. The equilibrium state is dynamic, meaning reactions proceed in both directions at equal rates:

• The forward reaction involves the dissociation of the acid into its conjugate base and protons.
• The reverse reaction involves the recombination of the conjugate base and protons to form the acid again.

This balance is crucial for calculating the pH of solutions, as the concentrations of ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$ ions directly affect pH values. Understanding this equilibrium helps students solve problems like the pH calculations in various solutions such as ${\mathrm{KNO}}_2$, ${\mathrm{NH}}_4{\mathrm{ClO}}_4$, and $\mathrm{NaOCl}$ shown in the exercise.

conjugate acid-base pairs

Conjugate acid-base pairs are fundamental to understanding acid-base reactions. When an acid donates a proton, the remaining part is called its conjugate base, while when a base accepts a proton, it becomes a conjugate acid.Each conjugate pair is related through a simple equilibrium:

• The acid in the pair is capable of donating a proton.
• The base is capable of accepting a proton.

This relationship is crucial for predicting the behavior of species in solution. For example, in the exercise solutions:

• ${\mathrm{NO}}_2^{-}$ is the conjugate base of ${\mathrm{HNO}}_2$, contributing ${\mathrm{OH}}^{-}$ ions in solution when ${\mathrm{KNO}}_2$ is dissolved.
• ${\mathrm{NH}}_4^{+}$ is the conjugate acid of ${\mathrm{NH}}_3$, releasing ${\mathrm{H}}^{+}$ ions when ${\mathrm{NH}}_4{\mathrm{ClO}}_4$ is in solution.
• ${\mathrm{OCl}}^{-}$ is the conjugate base of $\mathrm{HOCl}$, affecting $\mathrm{pH}$ through its dissociation in water.

These pairs help us understand the shifts in pH as well as the reaction paths that balance the acid and base forms in equilibrium.

equilibrium constants

Equilibrium constants are pivotal in calculating the extent of reactions in solutions, particularly acid-base reactions. Defined as $K_a$ for acids and $K_b$ for bases, these constants help determine the concentration of ${\mathrm{H}}^{+}$ or ${\mathrm{OH}}^{-}$ ions at equilibrium.The formulas for these constants are as follows:

• For acid dissociation:$K_a=\frac{[{\mathrm{H}}^{+}][\text{Conjugate Base}]}{[\text{Acid}]}$
• For base dissociation:$K_b=\frac{[\text{Conjugate Acid}][{\mathrm{OH}}^{-}]}{[\text{Base}]}$

Equilibrium constants indicate the strength of acids and bases; the larger the $K_a$ or $K_b$ value, the stronger the acid or base. In the exercise, these constants are used to calculate the concentration of ions in solutions of ${\mathrm{KNO}}_2$, ${\mathrm{NH}}_4{\mathrm{ClO}}_4$, and $\mathrm{NaOCl}$, which are essential steps in determining $\mathrm{pH}$.

solution concentration

Solution concentration is a measure of the amount of solute present in a given volume of solvent. In acid-base chemistry, understanding the concentration of ions, such as ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$, is crucial for calculating $\mathrm{pH}$.Concentration determines the degree to which the acid or base can affect the pH of a solution. Here are some key points:

• Concentration is usually expressed in molarity ($\mathrm{M}$), which is the number of moles of solute per liter of solution.
• Knowing the initial concentration of solutes like ${\mathrm{KNO}}_2$, ${\mathrm{NH}}_4{\mathrm{ClO}}_4$, and $\mathrm{NaOCl}$ is important for starting calculations.
• Equilibrium concentrations are used in conjunction with $K_a$ or $K_b$ to find the concentrations of ions like ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$.

Ultimately, the concentration of the solution influences the resulting $\mathrm{pH}$, making it one of the foundational aspects to understand in chemistry.