Question

If a solution of \(\mathrm{HF}\left(K_{a}=6.8 \times 10^{-4}\right)\) has a pH of \(3.65,\) calculate the concentration of hydrofluoric acid.

Short answer

The concentration of hydrofluoric acid (HF) in the solution is \( 0.147 \mathrm{mol/L} \).

Step-by-step solution

Define the acidity constant equation.

We will use the acidity constant equation, which relates the concentrations of the reactants and products in a weak acid equilibrium. For a weak acid, HA, the Ka equation is: \[ K_a = \frac{[H^+][A^-]}{[HA]} \] In our case, the weak acid is hydrofluoric acid (HF), so the equation becomes: \[ K_{a} = 6.8 \times 10^{-4} = \frac{[H^+][F^-]}{[HF]} \]

Convert pH to H+ ion concentration.

The pH is given as 3.65. Recall that the pH is the negative base-10 logarithm of the hydrogen ion concentration. So, we can express the H+ ion concentration as: \[ [H^+] = 10^{-pH} \] Now, we can plug in the given pH value: \[ [H^+] = 10^{-3.65} \]

Calculate the [H+] concentration.

To find the H+ concentration, we use a calculator to evaluate the expression: \[ [H^+] = 10^{-3.65} = 2.24 \times 10^{-4} \mathrm{mol/L} \]

Set up the Ka equation with known values.

From the equilibrium equation, we know that: \[ K_{a} = \frac{[H^+][F^-]}{[HF]} \] We can express [F-] as equal to [H+] since they will be produced in a 1:1 ratio. Thus, we have: \[ 6.8 \times 10^{-4} = \frac{(2.24 \times 10^{-4})^2}{[HF]} \]

Solve for [HF].

We want to find the concentration of hydrofluoric acid, [HF]. To do that, we can rearrange the equation and solve for [HF]: \[ [HF] = \frac{(2.24 \times 10^{-4})^2}{6.8 \times 10^{-4}} \] Now, we can calculate the value of [HF]: \[ [HF] = \frac{(2.24 \times 10^{-4})^2}{6.8 \times 10^{-4}} = 0.147 \mathrm{mol/L} \]

Express the final answer.

The concentration of hydrofluoric acid (HF) in the solution is 0.147 mol/L.

Key concepts

Understanding the Acid Dissociation Constant (Ka)

The acid dissociation constant, known as Ka, is a fundamental concept in chemistry that measures the strength of a weak acid in solution. It is defined as the equilibrium constant for the dissociation reaction of the acid into its anion and hydrogen ion. A weak acid partially dissociates in solution, establishing an equilibrium between the undissociated acid (HA) and the ions produced (H+ and A-).

The Ka equation is expressed as:
\[ K_a = \frac{[H^+][A^-]}{[HA]} \]

In the equation:

• \([H^+]\) signifies the concentration of hydrogen ions
• \([A^-]\) represents the concentration of the acid's anion
• \([HA]\) is the concentration of the undissociated acid

The larger the value of Ka, the stronger the acid — meaning it donates hydrogen ions more readily. The Ka value is a constant at a given temperature, so it can be used to predict the extent of dissociation given the pH of the solution or to calculate the pH when the concentrations are known.

From pH to H+ Ion Concentration

The pH scale is a measure of the acidity or basicity of a solution, inversely proportional to the concentration of hydrogen ions (H+). The pH is calculated as the negative logarithm (base 10) of the concentration of H+ ions in moles per liter. Mathematically, it is represented by the following equation:

\[ \text{pH} = -\log[H^+] \]

To find the hydrogen ion concentration from pH, the equation can be rearranged:

• \([H^+] = 10^{-\text{pH}}\)

For instance, if a solution has a pH of 3.65, the concentration of H+ ions can be calculated as:\[ [H^+] = 10^{-3.65} \]

Knowing the concentration of hydrogen ions is crucial in acid-base chemistry as it aids in understanding the acidity of the solution, calculating the degree of dissociation of weak acids, and finding the equilibrium concentrations in acid-base reactions.

Weak Acid Equilibrium and Its Constants

Weak acids do not completely ionize in water—they establish an equilibrium between the non-ionized acid and its ions. The equilibrium constant for this process, Ka, helps predict the concentration of products and reactants in solution at equilibrium.

For a generic weak acid (HA), the equilibrium can be represented as:
\[ HA \rightleftharpoons H^+ + A^- \]

At equilibrium, the rate of the forward reaction (dissociation of HA) is equal to the rate of the reverse reaction (recombination of H+ and A-), leading to stable concentrations of HA, H+, and A-. The equilibrium expression is:
\[ K_a = \frac{[H^+][A^-]}{[HA]} \]

In practice, to determine the concentration of a weak acid at equilibrium, you would use the Ka value and the initial concentration of the acid, along with the change in concentration of H+ and A- ions. This is often approached by setting up an ICE table (Initial, Change, Equilibrium) to organize and solve the concentrations at equilibrium. However, in some cases, simplifications can be made if the acid is weak and the ionization is small, allowing the change in the concentration of the undissociated acid to be approximated as negligible.