Question

The pH of a \(0.129 \mathrm{M}\) solution of a weak acid, \(\mathrm{HB}\), is 2.34. What is \(K_{\mathrm{a}}\) for the weak acid?

Short answer

Answer: The approximate \(K_a\) value for the weak acid (\(\mathrm{HB}\)) is \(3.04 \times 10^{-5}\).

Step-by-step solution

Calculate the concentration of \(\mathrm{H^+}\) ions from the pH

Given that the pH of the solution is 2.34, we can use the formula \(\mathrm{pH} = -\log [\mathrm{H^+}]\) to find the concentration of \(\mathrm{H^+}\) ions in the solution. $$[\mathrm{H^+}] = 10^{-\mathrm{pH}} = 10^{-2.34}$$

Write the equilibrium expression for the weak acid dissociation

The dissociation reaction for the weak acid (\(\mathrm{HB}\)) can be written as: $$\mathrm{HB} \rightleftharpoons \mathrm{H^+} + \mathrm{B^-}$$ At equilibrium, the concentrations of \(\mathrm{HB}\), \(\mathrm{H^+}\), and \(\mathrm{B^-}\) can be represented as: \([\mathrm{HB}] = 0.129 - x\) \([\mathrm{H^+}] = x\) \([\mathrm{B^-}] = x\) The \(K_a\) expression for the weak acid dissociation is: $$K_a = \frac{[\mathrm{H^+}][\mathrm{B^-}]}{[\mathrm{HB}]}$$

Substitute the equilibrium concentrations into the \(K_a\) expression

Since \([\mathrm{H^+}] = x\), let's find the value of x: $$x = 10^{-2.34}$$ Now, substitute the equilibrium concentrations in terms of x: $$K_a = \frac{x \times x}{0.129 - x}$$

Solve for the \(K_a\) value

Given that the \(x\) value is very small, we can assume that \((0.129 - x) \approx 0.129\). Therefore, we can simplify and solve for \(K_a\): $$K_a \approx \frac{x^2}{0.129}$$ $$K_a = \frac{(10^{-2.34})^2}{0.129}$$ After calculating, we get: $$K_a \approx 3.04 \times 10^{-5}$$ The \(K_a\) value for the weak acid (\(\mathrm{HB}\)) is approximately \(3.04 \times 10^{-5}\).

Key concepts

pH Calculation

When chemists talk about pH, they're discussing the acidity or basicity of a solution.
The pH scale measures how acidic or basic a substance is; the scale ranges from 0 to 14. A pH less than 7 is acidic, a pH of 7 is neutral, and a pH greater than 7 is basic.

To calculate pH, you can use the simple formula \( \mathrm{pH} = -\log [\mathrm{H^+}] \).
This formula depends on the concentration of hydrogen ions (\( \mathrm{H^+} \) ions) in the solution. Lower pH values correlate with higher concentrations of hydrogen ions, indicating a more acidic solution.

For instance, in the given exercise, to find the concentration of hydrogen ions from the pH, we use the reciprocal log function: \( [\mathrm{H^+}] = 10^{-\mathrm{pH}} = 10^{-2.34} \). This calculation is the first step towards understanding the chemical properties of the acid solution involved, setting the stage for further analysis like determining the acid dissociation constant.

Equilibrium Expression

Understanding Equilibrium
In a chemical reaction, equilibrium is the state where the rate of the forward reaction equals the rate of the reverse reaction; thus, the concentrations of reactants and products remain unchanged over time.

Writing Equilibrium Expressions

Chemists use an equilibrium expression to relate the concentrations of reactants and products at equilibrium. For a weak acid dissociation, like our weak acid \( \mathrm{HB} \), the reaction can be represented as: \( \mathrm{HB} \rightleftharpoons \mathrm{H^+} + \mathrm{B^-} \).
The equilibrium expression for this reaction, known as the acid dissociation constant and represented by \( K_a \), is calculated with: \( K_a = \frac{[\mathrm{H^+}][\mathrm{B^-}]}{[\mathrm{HB}]} \). Here, \( [\mathrm{H^+}] \) and \( [\mathrm{B^-}] \) denote the equilibrium concentrations of the ion products, while \( [\mathrm{HB}] \) represents the equilibrium concentration of the undissociated weak acid.

Acid Dissociation Constant

The acid dissociation constant, \( K_a \), is a quantitative measure of the strength of an acid in solution.
It's used to describe the extent to which an acid can donate hydrogen ions in an aqueous solution. Stronger acids have larger \( K_a \) values, meaning they dissociate more in water, leading to higher concentrations of hydrogen ions and thus a lower pH.

Specifically, \( K_a \) helps us understand how a weak acid dissociates to form hydrogen ions and its conjugate base. By knowing the \( K_a \) value, we can make predictions about the behavior of the acid under different conditions and calculate other essential properties, such as equilibrium concentrations and pH.

Equilibrium Concentrations

Equilibrium concentrations are the amounts of reactants and products present in a reaction mixture when the reaction has reached equilibrium.
These concentrations do not change unless the reaction conditions are altered.

To find equilibrium concentrations in a weak acid solution, we begin by assuming that the initial concentration of the acid is the sum of the concentrations of the undissociated acid and the products at equilibrium. Using a simplified expression when \( x \) (representing the change in concentration of \( \mathrm{HB} \) as it dissociates) is small relative to the initial concentration of the acid; we can make an approximation that lets us solve for \( K_a \) more straightforwardly, as illustrated in the given exercise.