Question

A solution contains a mixture of acids: 0.50\(M\) HA \(\left(K_{\mathrm{a}}=1.0\right.\) \(\times 10^{-3} ), 0.20 M \mathrm{HB}\left(K_{\mathrm{a}}=1.0 \times 10^{-10}\right),\) and 0.10 \(\mathrm{MHC}\left(K_{\mathrm{a}}=\right.\) \(1.0 \times 10^{-12} ) .\) Calculate the \(\left[\mathrm{H}^{+}\right]\) in this solution.

Short answer

The total concentration of H+ ions in the solution is \([H+]_{total} = [H+]_{HA} + [H+]_{HB} + [H+]_{HC} = \sqrt{(1.0 \times 10^{-3})(0.50)} + \sqrt{(1.0 \times 10^{-10})(0.20)} + \sqrt{(1.0 \times 10^{-12})(0.10)}\). Calculate the individual values and add them up to find the total [H+].

Step-by-step solution

Determine dissociation of HA

First, let's determine how much HA will dissociate in the solution. The Ka of HA is given as \(1.0 \times 10^{-3}\). To find the H+ concentration contributed by HA, we will use the expression for Ka: Ka = \(\frac{[H+][A-]}{[HA]}\) Where: - [H+] is the concentration of hydrogen ions - [A-] is the concentration of the conjugate base - [HA] is the concentration of the acid Since HA is a weak acid, its dissociation will be relatively small, so we can make the assumption: [H+] = [A-] and [HA] ≈ 0.50M which is the initial concentration Ka = \(\frac{[H+]^2}{0.50}\) Solving for [H+], we get: [H+] = \(\sqrt{Ka \times [HA]}\) [H+] = \(\sqrt{(1.0 \times 10^{-3})(0.50)}\) #Step 2: Calculate [H+] from HB#

Determine dissociation of HB

Next, we will determine how much HB will dissociate in the solution. The Ka of HB is given as \(1.0 \times 10^{-10}\). Using the same process as in Step 1: Ka = \(\frac{[H+][B-]}{[HB]}\) Making the assumption that [H+] = [B-] and [HB] ≈ 0.20M : Ka = \(\frac{[H+]^2}{0.20}\) Solving for [H+]: [H+] = \(\sqrt{Ka \times [HB]}\) [H+] = \(\sqrt{(1.0 \times 10^{-10})(0.20)}\) #Step 3: Calculate [H+] from HC#

Determine dissociation of HC

Lastly, we will determine how much HC will dissociate in the solution. The Ka of HC is given as \(1.0 \times 10^{-12}\). Using the same process as in previous steps: Ka = \(\frac{[H+][C-]}{[HC]}\) Making the assumption that [H+] = [C-] and [HC] ≈ 0.10M : Ka = \(\frac{[H+]^2}{0.10}\) Solving for [H+]: [H+] = \(\sqrt{Ka \times [HC]}\) [H+] = \(\sqrt{(1.0 \times 10^{-12})(0.10)}\) #Step 4: Calculate the total [H+]#

Adding the H+ contributions from HA, HB, and HC

Now that we have calculated the H+ concentration contributed by each of the acids HA, HB, and HC, we can simply add them up to determine the total H+ concentration in the solution: \([H+]_{total} = [H+]_{HA} + [H+]_{HB} + [H+]_{HC}\) Calculate the values obtained from the previous steps and add them up to find the total concentration of H+ ions in the solution.

Key concepts

Understanding Weak Acids

Weak acids do not fully dissociate into ions in water, which is why they are called "weak." This partial dissociation results in an equilibrium between the undissociated acid and its ions. For the acids HA, HB, and HC in our exercise, each has a specific dissociation constant, known as the acid dissociation constant ( Ka ). Even at the same molarity, weak acids behave differently depending on their Ka value.
For example, HA with a higher Ka dissociates more compared to HB and HC, which have lower Ka values. The strength of an acid is directly related to its Ka value:

• Higher Ka indicates a stronger weak acid (more dissociation).
• Lower Ka means less dissociation, making it a weaker acid.

Recognizing weak acids and their varying dissociation helps in calculating their contribution to the overall hydrogen ion concentration in a solution.

Importance of Equilibrium Constants

Equilibrium constants (K_a) are crucial for understanding how much of a weak acid will dissociate in a solution. For any weak acid: HA ↔ H⁺ + A⁻, K_a is expressed in the equation:\[K_a = \frac{[H^+][A^-]}{[HA]}\]This constant value is unique to each acid and indicates the balance between the reactants and products at equilibrium. In our problem, each acid—HA, HB, and HC—has its own equilibrium constant, revealing how much hydrogen ion concentration each one contributes.
Generally, equilibrium constants allow predictions about how acids act in a solution:

• If K_a is large, the acid tends to donate more H⁺ ions, indicating a more extensive dissociation.
• If K_a is small, there will be less donation of H⁺ ions, meaning minimal dissociation.

Equilibrium constants provide insight into the behavior of acids in solution, helping to predict their respective contributions to the overall H^+ concentration.

How to Calculate pH from H+ Concentration

To calculate the pH of a solution from the hydrogen ion concentration, we utilize the formula:\[pH = -\log [H^+]\]This logarithmic scale helps quantify how acidic a solution is. Lower pH values mean the solution is more acidic, while higher values indicate basicity. The total H^+ concentration, calculated by adding the concentrations contributed by each acid (HA, HB, and HC), is essential for determining the pH.
Steps to calculate pH from multiple acids:

• Calculate individual H^+ contributions using the appropriate K_a expression for each weak acid.
• Add the H^+ results together to get the total H^+ concentration.
• Use the pH formula to find the pH of the whole solution.

This method ensures accurate pH measurement of solutions containing multiple weak acids, crucial for understanding their acidity and potential chemical behavior in various contexts.