Problem 43 - Chemistry: Principles and Reactions

Question

The \(\mathrm{pH}\) 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?

Step-by-Step Solution

Answer

Based on the given step by step solution, the short answer to the problem is: 1. Calculate the hydrogen ion concentration, [H+], using the formula [H+] = 10^(-pH) with pH = 2.34. 2. Set up an ICE table for the weak acid equilibrium reaction. 3. Write the Ka expression: \[K_a= \frac{[H+][B−]}{[HB]}\] 4. Substitute the equilibrium concentration values from the ICE table into the Ka expression and approximate: \[K_a= \frac{([H+] from \ step \ 1)(x)}{0.129}\] 5. Solve for Ka using the calculated value of [H+] from step 1.

Step 1: Convert the pH value to the hydrogen ion concentration

First, we will use the pH formula, pH = -log[H+], to calculate the hydrogen ion concentration. The formula can be rewritten as [H+] = 10^(-pH). Calculate [H+] for the given pH. [H+] = 10^(-2.34)

Step 2: Use the ICE table to calculate the change in concentrations

Next, set up an ICE table for the weak acid equilibrium reaction, HB <=> H+ + B-. The initial concentrations are [HB]= 0.129 M, [H+]= calculated in step 1, and [B-]= 0. Initial: [HB]= 0.129 M, [H+]= [H+] from step 1, [B-]= 0 Change: [HB]= -x, [H+]= +x, [B-]= +x Equilibrium: [HB]= 0.129-x, [H+]= [H+] from step 1 + x, [B-]= x In the equilibrium, change in the concentration of H+ (x) is smaller compared to its initial concentration ([H+] from step 1), which allows us to make the assumption that [H+]=[H+] from step 1.

Step 3: Write the Ka expression

The Ka expression for the given equilibrium is: \[K_a= \frac{[H+][B−]}{[HB]}\]

Step 4: Substitute the equilibrium concentrations into the Ka expression

Substitute the equilibrium concentration values from the ICE table into the Ka expression and solve for Ka. \[K_a= \frac{([H+] from \ step \ 1)(x)}{0.129-x}\] We assume that x is very small compared to 0.129, so we can approximate this as: \[K_a= \frac{([H+] from \ step \ 1)(x)}{0.129}\] Referring back to the ICE table, x is equal to [B-] and also approximately equal to [H+] - [H+] from step 1. Therefore: \[K_a= \frac{([H+] from \ step \1)([H+]-[H+] from \ step \1)}{0.129}\]

Step 5: Solve for Ka

Now solve the equation to find Ka for the weak acid. Replace [H+] from step 1 with the value calculated in step 1 and solve for Ka.

Key Concepts

Weak Acids

Weak acids are those acids that do not fully dissociate in water. This means when a weak acid, like \( ext{HB}\) from our example, is dissolved in water, not all of it turns into hydrogen ions \( ext{H}^+ \) and conjugate base ions \( ext{B}^- \).
This incomplete dissociation is what classifies it as a weak acid.Weak acids have an associated equilibrium where the dissociation of the acid into ions is represented, and the acid's strength is often quantified by its acid dissociation constant \(K_a\). More heavily dissociated acids result in larger \(K_a\) values, indicating greater acid strength, even among weak acids.
An example of this dissociation would look like this: \[ ext{HB} ightleftharpoons ext{H}^+ + ext{B}^- \]
Notably, most organic acids like acetic acid and many inorganic acids are weak acids, which makes understanding them crucial in both everyday chemistry and advanced studies.

pH and pOH

Understanding \(pH\) and \(pOH\) is essential to grasp how acidic or basic a solution is. \(pH\) is a measure of the concentration of hydrogen ions \( ext{H}^+ \) in a solution. In the problem provided, the \(pH\) of the weak acid solution was given as \( ext{2.34}\).
We can use the formula \( ext{pH} = - ext{log}[ ext{H}^+] \) to find the hydrogen ion concentration \[ ext{H}^+ \], which is essential to determining the extent of dissociation of the acid. This conversion helps bridge the parameter of \(pH\) into something actionable for equilibrium calculations.
Similarly, \(pOH\) represents the concentration of hydroxide ions \( ext{OH}^- \), and is given by the formula: \( ext{pOH} = - ext{log}[ ext{OH}^-] \). The sum of \( pH \) and \( pOH \) is always \( ext{14}\) at room temperature, making them tightly intertwined in their uses across chemistry.

Equilibrium Calculations

Equilibrium calculations are fundamental to understand chemical dynamics in solutions, particularly when dealing with weak acids.
These calculations often involve creating an ICE table, which stands for Initial, Change, and Equilibrium, to track the concentrations of the species involved through the reaction.

• Initial: Denotes the starting concentrations of each species before any reaction takes place.
• Change: Describes how the concentrations of species change as the reaction moves toward equilibrium.
• Equilibrium: Shows the concentrations at equilibrium, allowing for ultimate determination of \( K_a \).

For weak acid equilibrium, these tables make it easier to visualize how exactly the dissociation products, in terms of hydrogen ions and conjugate base ions, come into play. Assumptions are often made (e.g., that changes in concentration are minimal), which help simplify calculations without significantly affecting accuracy.
Once these concentrations are identified, the acid dissociation constant \( K_a \) can be calculated. \( K_a \) encapsulates the extent of the acid dissociation and is a critical value in assessing the relative strength of an acid within the weak acid category, thus combining multiple equilibrium principles into one quantifiable measure.