Problem 60 - Chemistry The Central Science

Question

Determine the \(\mathrm{pH}\) of each of the following solutions \(\left(K_{a}\right.\) and \(K_{b}\) values are given in Appendix \(D\) ): (a) \(0.095 M\) hypochlorous acid, (b) \(0.0085 \mathrm{M}\) hydrazine, (c) \(0.165 \mathrm{M}\) hydroxylamine.

Step-by-Step Solution

Answer

The pH values of the three solutions can be determined as follows: (a) For the given 0.095 M hypochlorous acid, use the Ka value from Appendix D to find the concentration of H+ ions and then use the formula \(pH = -\log[H^{+}]\) to calculate the pH. (b) For the given 0.0085 M hydrazine, use the Kb value from Appendix D to find the concentration of OH- ions. Calculate the pOH using the formula \(pOH = -\log[OH^{-}]\) and then use the relationship between pH and pOH: \(pH + pOH = 14\) to determine the pH. (c) For the given 0.165 M hydroxylamine, similarly to step (b), use the Kb value from Appendix D to find the concentration of OH- ions, calculate the pOH, and then determine the pH of the solution.

Step 1: (a) Hypochlorous Acid

Hypochlorous acid (HClO) is a weak acid. Its dissociation can be represented as: \[HClO \rightleftharpoons H^{+} + ClO^{-}\] Given the concentration of HClO as 0.095 M, we can use the Ka value from Appendix D to find the concentration of H+ ions. Then, use the formula for pH: \(pH = -\log[H^{+}]\) to calculate the pH of the solution.

Step 2: (b) Hydrazine

Hydrazine (N2H4) is a weak base. Its dissociation can be represented as: \[N_{2}H_{4} + H_{2}O \rightleftharpoons N_{2}H_{5}^{+} + OH^{-}\] Given the concentration of N2H4 as 0.0085 M, we can use the Kb value from Appendix D to find the concentration of OH- ions. First, calculate the pOH using the formula: \(pOH = -\log[OH^{-}]\). Then, use the relationship between pH and pOH: \(pH + pOH = 14\) to determine the pH of the solution.

Step 3: (c) Hydroxylamine

Hydroxylamine (NH2OH) is also a weak base. Its dissociation can be represented as: \[NH_{2}OH + H_{2}O \rightleftharpoons NH_{3}OH^{+} + OH^{-}\] Given the concentration of NH2OH as 0.165 M, we can use the Kb value from Appendix D to find the concentration of OH- ions. Similar to step (b), calculate the pOH and then determine the pH of the solution. By following these steps, you will be able to calculate the pH of each given solution.

Key Concepts

Acid-Base Equilibrium

Understanding the concept of acid-base equilibrium is essential when calculating the pH of a solution. Acid-base equilibrium refers to the state in which the rate of the forward reaction (acid dissociating into ions) is equal to the rate of the reverse reaction (ions combining to form the acid), resulting in a stable concentration of acid, base, and ions in solution.

For a weak acid, this equilibrium can be represented as \[HA \rightleftharpoons H^{+} + A^{-}\].Here, \(HA\) is the weak acid, \(H^{+}\) represents the proton, and \(A^{-}\) is the conjugate base. The equilibrium constant for this reaction is known as the acid dissociation constant \(K_a\), which gives us an idea of the strength of the acid. A similar equilibrium exists for weak bases, where the base accepts a proton from water to form a conjugate acid and hydroxide ions.The pH is ultimately a measure of the concentration of hydrogen ions \(H^{+}\) in a solution. A low pH indicates a high concentration of hydrogen ions (acidic solution), while a high pH indicates a low concentration of hydrogen ions (basic solution). When solving for pH, one must consider the equilibrium concentrations of all species in the solution.

Weak Acid Dissociation

Weak acids do not dissociate completely in solution; instead, they reach a state of equilibrium between undissociated acid and dissociated ions. To calculate the pH of a weak acid solution, it's important to understand its dissociation reaction and apply the acid dissociation constant \(K_a\).The general equation for the dissociation of a weak acid \(HA\) is:\[HA \rightleftharpoons H^{+} + A^{-}\]Given this equilibrium, \(K_a\) is expressed as:\[K_{a} = \frac{[H^{+}][A^{-}]}{[HA]}\]When the initial concentration of the acid and the \(K_a\) value are known, one can set up an ICE table (Initial, Change, Equilibrium) to find the equilibrium concentrations of \(H^{+}\) and \(A^{-}\), from which the pH can be calculated using the formula:\(pH = -\text{log}[H^{+}]\).The process involves some approximations, such as assuming the change in concentration of the acid is negligible compared to the initial concentration, especially when \(K_a\) is very small.

Weak Base Dissociation

Similar to weak acids, weak bases also partially dissociate in water, and their pH can be calculated by understanding their dissociation and equilibrium. A typical weak base \(B\) will react with water to form a hydroxide ion \(OH^{-}\) and its conjugate acid \(BH^{+}\):\[B + H_2O \rightleftharpoons BH^{+} + OH^{-}\]The equilibrium constant for this reaction is the base dissociation constant \(K_b\), defined as:\[K_{b} = \frac{[BH^{+}][OH^{-}]}{[B]}\]In practice, when calculating the pH of a weak base solution, one must first find the concentrations of \(OH^{-}\) using the \(K_b\) value and the initial concentration. An ICE table may again be utilized here. Once \(OH^{-}\) is found, the pOH can be calculated using:\(pOH = -\text{log}[OH^{-}]\).The final step involves converting pOH to pH using the relationship \(pH + pOH = 14\) to arrive at the desired value. Throughout the steps, approximations and assumptions tailored to weak bases must be made, similar to the approach for weak acids.

pOH and pH Relationship

The pOH of a solution is a measure of the hydroxide ion concentration and is related to pH by the simple equation \(pH + pOH = 14\), which stems from the water dissociation constant \(K_w\) at 25°C. This relationship is crucial when dealing with weak bases, as it allows the conversion from pOH to pH.When the concentration of \(OH^{-}\) is known, pOH can be calculated as follows:\(pOH = -\text{log}[OH^{-}]\).Then, by knowing that the sum of pH and pOH must equal 14, the pH can be determined. It's important to remember that calculations using this relationship are most accurate at 25°C since the value of \(K_w\) varies with temperature.As an example, when calculating the pH of a weak base, after finding \(OH^{-}\) and pOH, one would simply subtract the pOH from 14 to get the pH. This step is integral in ensuring that textbook solutions for weak base dissociations yield a comprehensive understanding of the pH level of the solution.