Problem 65 - Chemistry: An Atoms-Focused Approach

Question

Why is the \(K_{u_{2}}\) value of phosphoric acid less than its \(K_{u_{1}}\) value but greater than its \(K_{2,}\) value?

Step-by-Step Solution

Answer

The second dissociation constant (Ku2) of phosphoric acid (H3PO4) is less than its first dissociation constant (Ku1) and greater than its third dissociation constant (K2) because of the increasing electrostatic repulsion and decreasing stability of the anions produced at each step of the dissociation process. As more protons are removed, the negative charge on the anions increases, making it less energetically favorable for the acid to dissociate further. Consequently, Ku1 > Ku2 > K2.

Step 1: (Step 1: Understanding phosphoric acid dissociation process)

Phosphoric acid (\(H_3PO_4\)) is a triprotic acid, meaning it can dissociate up to 3 times, releasing three protons in a stepwise process: 1. \(H_3PO_4 \leftrightharpoons H_2PO_4^- + H^+\) 2. \(H_2PO_4^- \leftrightharpoons HPO_4^{2-} + H^+\) 3. \(HPO_4^{2-} \leftrightharpoons PO_4^{3-} + H^+\) Each step has its own dissociation constant, known as \(K_{u_{1}}\), \(K_{u_{2}}\), and \(K_{2}\) respectively.

Step 2: (Step 2: Defining the dissociation constants)

For each step, the dissociation constant represents the ratio of the concentration of the products to the concentration of the reactants at equilibrium: 1. \(K_{u_{1}} = \frac{[H_2PO_4^-][H^+]}{[H_3PO_4]}\) 2. \(K_{u_{2}} = \frac{[HPO_4^{2-}][H^+]}{[H_2PO_4^-]}\) 3. \(K_{2} = \frac{[PO_4^{3-}][H^+]}{[HPO_4^{2-}]}\) Note that higher dissociation constants imply that the dissociation process is more favorable.

Step 3: (Step 3: Explaining the sequential decrease in dissociation constants)

The sequential decrease in dissociation constants (\(K_{u_{1}} > K_{u_{2}} > K_{2}\)) can be explained by examining the structures of the anions formed at each step: 1. In the first step, the most acidic proton is released, forming the \(H_2PO_4^-\) ion. Converting from neutral \(H_3PO_4\) to negatively charged \(H_2PO_4^-\) is energetically favorable due to the stability of the resulting ion, resulting in a relatively high value of \(K_{u_{1}}\). 2. In the second step, the second most acidic proton is released, forming the \(HPO_4^{2-}\) ion. Since the \(H_2PO_4^-\) ion already carries a negative charge, releasing another proton becomes less favorable due to electrostatic repulsion. This leads to a lower value of \(K_{u_{2}}\) compared to \(K_{u_{1}}\). 3. In the final step, the least acidic proton is released, forming the \(PO_4^{3-}\) ion. At this point, removing a proton from an ion already carrying a double negative charge becomes even less energetically favorable due to increased electrostatic repulsion, leading to a much lower value of \(K_{2}\). To summarize, the trend in the dissociation constants is a result of the increasing electrostatic repulsion and decreasing stability of the anions produced at each step of the dissociation process. Therefore, the \(K_{u_{2}}\) value of phosphoric acid is less than its \(K_{u_{1}}\) value and greater than its \(K_{2}\) value.

Key Concepts

Dissociation Constants

Dissociation constants are crucial for understanding how acids like phosphoric acid behave in solution. They quantify the tendency of a chemical compound to dissociate into its ions. Specifically, dissociation constants are represented as a ratio of the concentration of the dissociated species to the concentration of the undissociated species. For phosphoric acid (\(H_3PO_4\)), which is a triprotic acid, there are three dissociation constants:

• \(K_{u_{1}}\) for the first dissociation step, where \(H_3PO_4\) dissociates into \(H_2PO_4^-\) and \(H^+\).
• \(K_{u_{2}}\) for the second dissociation, where \(H_2PO_4^-\) turns into \(HPO_4^{2-}\) and \(H^+\).
• \(K_2\) for the third dissociation, where \(HPO_4^{2-}\) converts to \(PO_4^{3-}\) and \(H^+\).

Higher dissociation constants imply a more favorable dissociation process. For example, \(K_{u_{1}}\) is typically greater than \(K_{u_{2}}\), which is greater than \(K_2\). This trend signifies how the ability of the molecule to release protons diminishes with each successive dissociation step due to factors like electrostatic repulsion.

Triprotic Acid

Phosphoric acid is a classic example of a triprotic acid, meaning it possesses three acidic hydrogens and can undergo three separate dissociation events.
Understanding this characteristic is key to grasping why the dissociation constants vary as they do. In each dissociation step, phosphoric acid releases one proton. Here's how each step unfolds:

• First step: \(H_3PO_4 \rightarrow H_2PO_4^- + H^+\)
• Second step: \(H_2PO_4^- \rightarrow HPO_4^{2-} + H^+\)
• Third step: \(HPO_4^{2-} \rightarrow PO_4^{3-} + H^+\)

As each proton is released, the molecule becomes more negatively charged, affecting the energetic favorability of further dissociation due to increasing electrostatic forces between the negatively charged species. This feature of triprotic acids explains why \(K_{u_{1}} > K_{u_{2}} > K_2\): each step involves more complexity and less favorability compared to the previous one.

Electrostatic Repulsion

Electrostatic repulsion plays a significant role in the dissociation sequence of phosphoric acid. As the name suggests, electrostatic repulsion refers to the force that pushes like charges apart. Negative ions repel each other, making successive dissociation less favorable energetically.In phosphoric acid's dissociation process:

• The first ionization, \(H_3PO_4 \rightarrow H_2PO_4^- + H^+\), leads to the formation of a single negative charge, which is quite stable.
• The second ionization, \(H_2PO_4^- \rightarrow HPO_4^{2-} + H^+\), adds another negative charge, thus experiencing greater electrostatic repulsion. The process is less energetically favorable than the initial release, resulting in a lower \(K_{u_{2}}\).
• The third ionization, \(HPO_4^{2-} \rightarrow PO_4^{3-} + H^+\), involves adding yet another negative charge. The tremendous repulsion among the three negative charges makes this step the least favorable, hence the smallest \(K_2\).

In summary, increased electrostatic repulsion with each proton removal is the primary reason for the observed decrease in dissociation constants (\(K_{u_1} > K_{u_2} > K_2\)) in the phosphoric acid dissociation process.