Consider two carboxylic acids (acids that contain the \(-\mathrm{COOH}\) 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 below for the two acids: $$ \begin{array}{lcc} \hline & \Delta H^{\circ}(\mathrm{k} \mathrm{J} / \mathrm{mol}) & T \Delta S^{\circ}(\mathrm{k} \mathrm{J} / \mathrm{mol}) \\ \hline \mathrm{CH}_{3} \mathrm{COOH} & -0.57 & -27.6 \\ \mathrm{CH}_{2} \mathrm{ClCOOH} & -4.7 & -21.1 \\ \hline \end{array} $$ 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 Brønsted acid-base reaction.) (d) Explain why the \(T \Delta S^{\circ}\) term is more negative for \(\mathrm{CH}_{3} \mathrm{COOH}\)

Step by step solution

01

Calculating ΔG°

The equation for calculating ΔG° is given by \(\Delta G^{\circ} = -RT \ln K_a\)where \(R\) is the universal gas constant \(R = 8.314 \, J/(K⋅mol)\) and \(T\) is the temperature in Kelvin. At \(25\,°C\), \(T = 298.15 \, K\).For acetic acid:\[\Delta G^{\circ}_{acetic} = -8.314 \times 298.15 \times \ln (1.8 \times 10^{-5}) \]For chloroacetic acid:\[\Delta G^{\circ}_{chloroacetic} = -8.314 \times 298.15 \times \ln (1.4 \times 10^{-3}) \]

02

Determine the dominant term in ΔG°

We can find this by comparing the numerical values of \(\Delta H^{\circ}\) and \(T \Delta S^{\circ}\) for each acid. For acetic acid, the dominant term is \(T \Delta S^{\circ}\) as the magnitude of this value is significantly larger than \(\Delta H^{\circ}\). The same applies to chloroacetic acid.

03

Identify the processes contributing to ΔH°

ΔH°, the enthalpy change, primarily includes breaking of initial (H-OH) bonds and forming new (H-O^-, H_3O+) bonds during acid ionization.

04

Explain the larger TΔS° for acetic acid

The entropy contribution \(T \Delta S^{\circ}\) relay on the disorder of the molecules involved in a reaction. On the ionization state, acetic acid ions are more disordered due its extra CH3 group, in contrast to the more rigid Cl atom in chloroacetic acid. Therefore, acetic acid has a larger \(T \Delta S^{\circ}\) value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acid Ionization Constant (Ka)

Understanding the acid ionization constant, usually symbolized as Ka, is crucial when studying carboxylic acids and their behaviors in aqueous solutions. The constant gives us an idea of the acid's strength, which refers to its tendency to donate protons (H+) to water molecules, forming a hydronium ion (H3O+) and the acid's conjugate base. A larger Ka value indicates a stronger acid, which ionizes more in solution.
The process can be represented by the equation for a general carboxylic acid (HA):
\[ HA + H2O \leftrightarrow H3O^+ + A^- \]
A high Ka value, as is the case with chloroacetic acid (\(1.4 \times 10^{-3}\)), signifies a greater extent of ionization compared to acids with a lower Ka, such as acetic acid (\(1.8 \times 10^{-5}\)). Consequently, this difference in ionization also affects the Gibbs free energy of the reaction, as seen in the exercise.

Gibbs Free Energy (ΔG°)

Gibbs free energy, denoted as ΔG°, determines the spontaneity of a chemical reaction at constant temperature and pressure. When ΔG° is negative, the process is spontaneous; conversely, a positive ΔG° indicates non-spontaneity.
The equation linking ΔG° to the acid ionization constant is:
\[ \Delta G^{\circ} = -RT \ln K_a \]
In this formula, R is the universal gas constant, T the temperature in Kelvin, and Ka the acid ionization constant. For the carboxylic acids mentioned, the ionization process involves transferring a ΔG° that is negative, indicating that ionization is a spontaneous process at room temperature. The larger the value of Ka, reflecting a stronger acid, relates to a more negative ΔG°, solidifying the connection between acid strength and spontaneity of ionization.

Enthalpy (ΔH°)

The change in enthalpy, or ΔH°, during a chemical reaction reflects the heat absorbed or released when the reaction takes place under constant pressure. It's an intrinsic property of the substances involved and is a direct measure of the bond energies that are broken and formed during the reaction.
In the context of ionizing carboxylic acids, ΔH° accounts for the energy needed to break the O-H bond of the acid and the energy released when new bonds form in the hydronium ion and the conjugate base. A negative ΔH°, as seen with both acetic and chloroacetic acid, means that the reaction is exothermic, releasing heat into the surroundings.
It should be noted that although ΔH° is essential for understanding reaction energetics, it does not solely determine spontaneity. As shown in the exercise, entropy holds significant sway in affecting the Gibbs free energy.

Entropy (ΔS°)

Entropy, represented by ΔS°, is a measure of the disorder or randomness in a system. Chemical processes tend to proceed in a direction that increases the overall entropy of the universe.
When carboxylic acids ionize in water, there is a change in entropy due to the different degrees of molecular disorder before and after the reaction. The entropy change, which is influenced by the structure and bonding of the acid molecules, can either be positive, if disorder increases, or negative, if the system becomes more ordered.
The term \(T \Delta S^{\circ}\) includes the effect of temperature on the system's entropy. In our exercise, acetic acid exhibits a more significant increase in entropy upon ionization than chloroacetic acid. This is because the removal of a hydrogen atom from acetic acid's \(-\mathrm{COOH}\) group increases the molecule's freedom to move, hence its disorder, more than in the case of chloroacetic acid.