Question

One way to determine the predominant species at equilibrium for an acid-base reaction is to say that the reaction arrow points to the acid with the higher value of \(\mathrm{p} K_{\mathrm{a}}\). For example, $$ \begin{array}{cr} \mathrm{NH}_{4}^{+}+\mathrm{H}_{2} \mathrm{O} \longleftrightarrow \mathrm{NH}_{3}+\mathrm{H}_{3} \mathrm{O}^{+} \\ \mathrm{p} K_{\mathrm{a}} 9.24 & \mathrm{p} K_{\mathrm{a}}-1.74 \\ \mathrm{NH}_{4}^{+}+\mathrm{OH}^{-} \longrightarrow \mathrm{NH}_{3}+\mathrm{H}_{2} \mathrm{O} \\ \mathrm{p} K_{\mathrm{a}} 9.24 & \mathrm{p} K_{\mathrm{a}} 15.7 \end{array} $$ Explain why this rule works.

Short answer

In an acid-base equilibrium reaction, the reaction arrow points to the acid with the higher pKa value because the higher pKa value represents a weaker acid. The reaction favors the side with the weaker acid, as it is less likely to donate its proton. This is due to the relation between pKa and the equilibrium constant (Keq), where higher pKa values indicate weaker acids, and the reaction proceeds towards the side with the weaker acid in order to stabilize the reaction and establish equilibrium.

Step-by-step solution

Understanding pKa

pKa is the negative logarithm of the acid dissociation constant (Ka), which is a measure of the strength of an acid. The higher the pKa value, the weaker the acid, as it indicates that the acid is less willing to donate a proton (H⁺). In an acid-base reaction, the equilibrium lies in favor of the side with the weaker acid, as it is less likely to donate its proton.

Understanding Keq

In an acid-base reaction, the equilibrium constant (Keq) is the ratio of the concentrations of products to reactants at equilibrium. The equation for the equilibrium constant is: $$Keq = \frac{[A-][H^+]}{[HA]}$$ Where [A-] is the concentration of the conjugate base, [H⁺] is the concentration of protons, and [HA] is the concentration of the acid. The higher the Keq value, the more the reaction lies towards the products side, while a lower Keq value indicates that the reaction lies towards the reactants side.

Comparing pKa values

In the given reaction: $$\mathrm{NH}_{4}^{+}+\mathrm{H}_{2} \mathrm{O} \longleftrightarrow \mathrm{NH}_{3}+\mathrm{H}_{3} \mathrm{O}^{+}$$ $$\mathrm{p} K_{\mathrm{a}} 9.24 \quad \quad \quad \mathrm{p} K_{\mathrm{a}}-1.74$$ The equilibrium will lie on the side of the acid with a higher pKa value, which in this case is NH4⁺ with pKa of 9.24. The reaction will proceed as follows: $$\mathrm{NH}_{4}^{+}+\mathrm{OH}^{-} \longrightarrow \mathrm{NH}_{3}+\mathrm{H}_{2} \mathrm{O}$$ $$\mathrm{p} K_{\mathrm{a}} 9.24 \quad \quad \quad \mathrm{p} K_{\mathrm{a}} 15.7$$ The equilibrium will lie on the side of the acid with a higher pKa value, which in this case is NH3 (pKa = 15.7). The reaction will, therefore, proceed in the direction of NH4⁺ and OH⁻ acting as reactants.

Relation between pKa and equilibrium

The rule that the reaction arrow points to the acid with the higher pKa value works because of the relation between pKa and the equilibrium constant, Keq. The higher pKa value indicates a weaker acid, and the reaction will proceed in favor of the weaker acid side to stabilize the reaction and establish equilibrium. In this way, the system aims to minimize the free energy and achieve a state of maximum stability.

Key concepts

pKa and Acid Strength

The concept of pKa is pivotal in understanding acid strength and plays a crucial role in acid-base chemistry. pKa is a numerical scale used to quantify the strength of an acid; it is the negative logarithm of the acid dissociation constant (Ka).

When we deal with acids, it's synonymous to refer to their propensity to donate a proton. The strength of an acid is inversely related to its pKa value: the lower the pKa, the stronger the acid. Why? Because a low pKa value indicates a high Ka, which means the acid readily dissociates to form its conjugate base and a proton. Conversely, a high pKa value signifies a weak acid, as it suggests that the acid has less tendency to release a proton.

To shed light on the significance of pKa in equilibrium, consider this analogy: a strong acid is like someone eagerly giving away a hot potato, whereas a weak acid is like someone reluctantly holding on to it. With this in mind, in acid-base reactions, the equilibrium tends to favor the formation of weaker acids and bases, as per Le Chatelier's principle, which aims for a state of balance.

Equilibrium Constant (Keq)

Understanding the equilibrium constant, often denoted as Keq, is essential for comprehending how chemical reactions reach balance. In the context of acid-base reactions, Keq provides insight into the relative concentrations of reactants and products when a reaction reaches equilibrium. The Keq is defined by the following expression: \[\begin{equation}Keq = \frac{[A-][H^+]}{[HA]}\end{equation}\]where [A-] represents the concentration of the conjugate base, [H+] denotes the proton concentration, and [HA] is the concentration of the acid.

This ratio, Keq, is fixed for a given reaction at a specific temperature and indicates the position of equilibrium. A high Keq value suggests that the reaction favors product formation (forward reaction), while a low Keq implies a preference for reactant preservation (reverse reaction).

Let's digest this: a high Keq means a strong acid (low pKa) as reactant tends to fulfill its destiny — to become ions in solution. A low Keq, in contrast, tells us that we're dealing with a weak acid (high pKa) that's content remaining mostly un-ionized as reactants.

Acid-Base Reaction Mechanism

Diving deeper into the mechanics of acid-base reactions unveils the procedural aspect of how acids and bases interact. The reaction mechanism of an acid-base process generally involves the transfer of a proton from the acid (proton donor) to the base (proton acceptor).

This transfer is typically visualized in a stepwise fashion, where the acid loses a proton, forming its conjugate base, and the base gains a proton, becoming its conjugate acid.

Example in Action:

Considering the reaction between NH4+ and OH-, NH4+ acts as an acid, donating a proton to OH-, which acts as a base. This leads to the formation of NH3 and H2O.

When it comes to mechanisms, several factors influence how and to what extent these reactions occur, such as the pKa values of the reactants, the solvent, temperature, and the presence of catalysts or inhibitors. Thus, the mechanism is the narrative of how reactants become products — the story of a dynamic equilibrium dance, choreographed by the innate properties of the molecules involved.