Problem 183 - Introductory Chemistry Atoms First

Question

In each of the following pairs, which is the stronger acid? (a) \(\mathrm{HPO}_{4}^{2-}\) and \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) (b) \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{H}_{3} \mathrm{O}^{+}\) (c) \(\mathrm{HCN}\left(K_{\mathrm{eq}}=6.2 \times 10^{-10}\right)\) and \(\mathrm{HCO}_{2} \mathrm{H}\left(K_{\mathrm{eq}}=1.8 \times 10^{-4}\right)\) (d) HI and HF

Step-by-Step Solution

Answer

The stronger acids in each pair are: (a) \(H_2PO_4^{-}\) (b) \(H_3O^{+}\) (c) \(HCO_2H\) (d) \(HI\)

Step 1: Recall the dissociation constants of each acid in pair (a) to (d)

We can search for K values for different acids from textbooks or online sources. We'll use these approximate K values for our analysis: (a) \(HPO_4^{2-}: K_{a3} = 4.2 \times 10^{-13}\) and \(H_2PO_4^{-}: K_{a2} = 6.3 \times 10^{-8}\) (b) \(H_2O: K_{w} = 1 \times 10^{-14}\) and \(H_3O^{+}: K_{a} = 1\) (c) Already given: \(HCN: K_{eq} = 6.2 \times 10^{-10}\) and \(HCO_2H: K_{eq} = 1.8 \times 10^{-4}\) (d) \(HI: K_{a} = 1.0 \times 10^1\) and \(HF: K_{a} = 7.2 \times 10^{-4}\)

Step 2: Compare the K values of each acid in the respective pairs#a b c and d

(a) \(HPO_4^{2-}\) has a K value of \(4.2 \times 10^{-13}\) and \(H_2PO_4^{-}\) has a K value of \(6.3 \times 10^{-8}\). Since \(6.3 \times 10^{-8} > 4.2 \times 10^{-13}\), \(H_2PO_4^{-}\) is the stronger acid. (b) \(H_2O\) has a K value of \(1 \times 10^{-14}\) and \(H_3O^{+}\) has a K value of \(1\). Since \(1 > 1 \times 10^{-14}\), \(H_3O^{+}\) is the stronger acid. (c) \(HCN\) has a K value of \(6.2 \times 10^{-10}\) and \(HCO_2H\) has a K value of \(1.8 \times 10^{-4}\). Since \(1.8 \times 10^{-4} > 6.2 \times 10^{-10}\), \(HCO_2H\) is the stronger acid. (d) \(HI\) has a K value of \(1.0 \times 10^1\) and \(HF\) has a K value of \(7.2 \times 10^{-4}\). Since \(1.0 \times 10^1 > 7.2 \times 10^{-4}\), \(HI\) is the stronger acid.

Step 3: Summarize the results

We can now list the stronger acids in each pair: (a) \(H_2PO_4^{-}\) is the stronger acid (b) \(H_3O^{+}\) is the stronger acid (c) \(HCO_2H\) is the stronger acid (d) \(HI\) is the stronger acid

Key Concepts

Dissociation Constants

Dissociation constants, often denoted as \( K_a \), are crucial to understanding the strength of an acid. These constants measure the extent to which an acid dissociates into its ions in a solution. A higher \( K_a \) value means the acid dissociates more completely, making it stronger. This occurs because the acid releases more hydrogen ions (\( H^+ \)) into the solution.
In contrast, a low \( K_a \) value indicates limited dissociation, characteristic of a weaker acid. By comparing the \( K_a \) values of solutions, one can determine which acid is stronger. This is fundamental for accurately comparing acids, especially when dealing with weak acids that do not fully dissociate. Understanding these values helps in predicting the behavior of acids in various chemical environments.

Comparing Acids

Comparing acids involves looking at their dissociation constants to determine their strengths. The stronger acid in a pair is the one with the larger \( K_a \) value because it indicates more extensive dissociation and greater availability of \( H^+ \) ions.
For instance, in the exercise pairs, the acid with the higher dissociation constant was identified as the stronger acid:

• \( H_2PO_4^- \) is stronger than \( HPO_4^{2-} \) because \( 6.3 \times 10^{-8} > 4.2 \times 10^{-13} \).
• \( H_3O^+ \) is stronger than \( H_2O \) because \( 1 > 1 \times 10^{-14} \).
• \( HCO_2H \) is stronger than \( HCN \) because \( 1.8 \times 10^{-4} > 6.2 \times 10^{-10} \).
• \( HI \) is stronger than \( HF \) because \( 1.0 \times 10^{1} > 7.2 \times 10^{-4} \).

These comparisons are based solely on \( K_a \) values, offering a clear and quantitative method for evaluating acid strength.

pH and Acid Dissociation

pH is a measure of the acidity or basicity of an aqueous solution. It's calculated as \( \text{pH} = -\log[H^+] \). In acidic solutions, the pH is less than 7, reflecting high concentrations of hydrogen ions. The dissociation of acids into \( H^+ \) is what contributes to this decrease in pH.
The relationship between pH and dissociation constants helps us understand acid strength. A strong acid, with a high dissociation constant, will have a higher concentration of \( H^+ \) and a lower pH. Conversely, a weak acid results in a higher pH due to fewer \( H^+ \) ions in solution.
By considering both \( K_a \) and pH, chemists can gauge the acidic environment's impact on reactions and anticipate how a solution's chemistry might change with varying acidity.

Chemical Equilibrium

Chemical equilibrium refers to the state in which the forward and reverse reactions occur at the same rate, leading to stable concentrations of reactants and products. In the context of acid dissociation, equilibrium involves the balance between the undissociated acid and its ions in solution.
For weak acids, this equilibrium is dynamic, with the extent of dissociation dictated by \( K_a \). When an acid reaches equilibrium in water, the concentration of ions and undissociated molecules remains constant. However, this balance can shift if external conditions, such as temperature or concentration of substances, change.
Understanding chemical equilibrium in acid-base reactions is vital for predicting the outcomes of chemical processes and for manipulating conditions to achieve desired reaction extents. This is especially important in laboratory and industrial settings where precise control over chemical compositions is required.