Question

Consider two carboxylic acids (acids that contain the \(-\) COOH group \(): \mathrm{CH}_{3} \mathrm{COOH}\) (acetic acid, \(\left.K_{\mathrm{a}}=1.8 \times 10^{-5}\right)\) and \(\mathrm{CH}_{2} \mathrm{ClCOOH}\) (chloroacetic acid, \(K_{\mathrm{a}}=1.4 \times 10^{-3}\) ). (a) Calculate \(\Delta G^{\circ}\) for the ionization of these acids at \(25^{\circ} \mathrm{C}\). (b) From the equation \(\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ},\) we see that the contributions to the \(\Delta G^{\circ}\) term are an enthalpy term \(\left(\Delta H^{\circ}\right)\) and a temperature times entropy term \(\left(T \Delta S^{\circ}\right)\). These contributions are listed here for the two acids: \begin{tabular}{lcc} & \(\Delta \boldsymbol{H}^{\circ}(\mathbf{k} \mathbf{J} / \mathbf{m o l})\) & \(T \Delta S^{\circ}(\mathbf{k} \mathbf{J} / \mathbf{m o l})\) \\ \hline \(\mathrm{CH}_{3} \mathrm{COOH}\) & -0.57 & -27.6 \\\ \(\mathrm{CH}_{2} \mathrm{ClCOOH}\) & -4.7 & -21.1 \end{tabular} Which is the dominant term in determining the value of \(\Delta G^{\circ}\) (and hence \(K_{\mathrm{a}}\) of the acid)? (c) What processes contribute to \(\Delta H^{\circ} ?\) (Consider the ionization of the acids as a Bronsted acid-base reaction.) (d) Explain why the \(T \Delta S^{\circ}\) term is more negative for \(\mathrm{CH}_{3} \mathrm{COOH} .\)

Short answer

(a) \( \Delta G^{\circ}(\text{CH}_3\text{COOH}) \approx 28.88 \text{ kJ/mol} \), \( \Delta G^{\circ}(\text{CH}_2\text{ClCOOH}) \approx 18.99 \text{ kJ/mol} \). (b) \( T \Delta S^{\circ} \) is dominant. (c) Breaking O-H bond, forming solvent interactions. (d) More negative due to lesser solvation disorder.

Step-by-step solution

Calculate \( \Delta G^{\circ} \) using \( K_a \)

To calculate \( \Delta G^{\circ} \) for each acid, use the formula:\[ \Delta G^{\circ} = -RT \ln K_a \]Where \( R = 8.314 \) J/(mol·K) is the gas constant and \( T = 298 \) K since we're discussing standard conditions at 25°C.For acetic acid, \( \Delta G^{\circ} = - (8.314) (298) \ln (1.8 \times 10^{-5}) \).For chloroacetic acid, \( \Delta G^{\circ} = - (8.314) (298) \ln (1.4 \times 10^{-3}) \).

Evaluate \( \Delta G^{\circ} \) for \( \text{CH}_3\text{COOH} \)

Calculate\[ \Delta G^{\circ} = - (8.314) (298) \ln (1.8 \times 10^{-5}) \approx 28.88 \text{ kJ/mol} \] for acetic acid.

Evaluate \( \Delta G^{\circ} \) for \( \text{CH}_2\text{ClCOOH} \)

Calculate\[ \Delta G^{\circ} = - (8.314) (298) \ln (1.4 \times 10^{-3}) \approx 18.99 \text{ kJ/mol} \] for chloroacetic acid.

Determine Dominant Term in \( \Delta G^{\circ} \)

From the equation \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \), plug in the values for each acid.For acetic acid:\[ \Delta G^{\circ} = -0.57 - (-27.6) \approx 28.17 \text{ kJ/mol} \]and for chloroacetic acid:\[ \Delta G^{\circ} = -4.7 - (-21.1) \approx 16.4 \text{ kJ/mol} \]The dominant term affecting \( \Delta G^{\circ} \) is \( T \Delta S^{\circ} \) since it provides a larger absolute value than \( \Delta H^{\circ} \).

Identify Processes Contributing to \( \Delta H^{\circ} \)

For a Bronsted acid-base reaction, \( \Delta H^{\circ} \) generally involves breaking the O-H bond in the acid and forming new interactions with the solvent. The energy absorbed or released in this process contributes to \( \Delta H^{\circ} \).

Explain Higher Negativity of \( T \Delta S^{\circ} \) for Acetic Acid

The more negative \( T \Delta S^{\circ} \) for acetic acid \( \text{CH}_3\text{COOH} \) indicates a greater decrease in disorder when it ionizes compared to chloroacetic acid \( \text{CH}_2\text{ClCOOH} \). This may be due to less solvation disorder or a smaller volume increase upon ionizing, influencing entropy negatively.

Key concepts

Gibbs Free Energy

Gibbs Free Energy, denoted as \( \Delta G^{\circ} \), is essential for determining the spontaneity of chemical reactions. When calculating \( \Delta G^{\circ} \) for a reaction, a negative value suggests that the process is spontaneous under standard conditions, whereas a positive value implies non-spontaneity.
In the context of carboxylic acids, such as acetic acid \((\text{CH}_3\text{COOH})\) and chloroacetic acid \((\text{CH}_2\text{ClCOOH})\), \( \Delta G^{\circ} \) is calculated using the acid dissociation constant \( K_a \).
The formula used is:

• \( \Delta G^{\circ} = -RT \ln K_a \)

Here, \( R \) is the gas constant (8.314 J/mol·K), and \( T \) is temperature in Kelvin. This expression ties together the equilibrium constant with thermodynamic data, bridging chemistry with the concepts of energy and spontaneity.

Enthalpy

Enthalpy, represented as \( \Delta H^{\circ} \), describes the heat content change within a system during a chemical reaction. For many reactions, it is a measure of bond-breaking and bond-formation energies.
In the case of a Bronsted acid-base reaction like the ionization of carboxylic acids, we consider the breaking of the O-H bond and the subsequent solvation interactions.

• A negative \( \Delta H^{\circ} \) means the process releases heat and is exothermic.
• A positive \( \Delta H^{\circ} \) implies the system absorbs heat, indicating an endothermic process.

In the given exercise, acetic acid and chloroacetic acid both have negative \( \Delta H^{\circ} \), indicating that they release energy upon ionization. This release is partly due to the formation of new interactions between ions and solvent molecules.

Entropy

Entropy \((\Delta S^{\circ})\) is a vital concept that describes the degree of disorder or randomness in a system. The term \( T\Delta S^{\circ} \) in the Gibbs Free Energy equation represents the entropy component, scaled by temperature.
For the carboxylic acids, it's noteworthy that acetic acid has a more negative entropy term than chloroacetic acid.
This hints at a reduced increase in disorder upon ionization, possibly due to differences in solvation or spatial constraints.

• More negative \( T \Delta S^{\circ} \) suggests higher systematic order or decreased volume change during ionization.
• The molecular structure and solvent interactions greatly influence \( \Delta S^{\circ} \).

Understanding entropy helps explain why reactions may not be entirely driven by energy changes alone, but also by changes in the order of particles.

Bronsted Acid-Base Reaction

Bronsted acid-base reactions are fundamental chemical processes where a proton (H\(^+\)) is transferred from an acid to a base. This perspective is essential for understanding the ionization of carboxylic acids, which involves the donation of a proton from the \(-\text{COOH}\) group.
During the ionization process, the O-H bond in the carboxylic acid is broken, releasing a proton to the surrounding environment.

• The remaining structure becomes a negatively charged carboxylate ion.
• Water or other solvents may act as the base, accepting the released proton.
• This reaction is further classified by equilibrium constants, reflecting how readily proton transfer occurs.

Understanding these dynamics provides insight into the factors that control acidity and reactivity, key determinants in chemical equilibrium and reaction spontaneity.