Question

The \(\mathrm{p} K_{\mathrm{a}}\) of an acid is a measure of its proton-group- transfer potential. (a) Derive a relation between \(\Delta G^{\circ \prime}\) and \(\mathrm{p} K_{\mathrm{a}}\) (b) What is the \(\Delta G^{\circ}\) ' for the ionization of acetic acid, which has a \(\mathrm{p} K_{\mathrm{a}}\) of \(4.8 ?\)

Short answer

\( \Delta G^{\circ'} \) for acetic acid is approximately \( 7.99 \ kJ/mol \).

Step-by-step solution

Understanding the Relation

We start by relating the equilibrium constant \( K_a \) of the ionization process to \( pK_a \). The equilibrium constant \( K_a \) is related to the Gibbs free energy change by the equation \( \Delta G^{\circ'} = -RT \ln K_a \). The \( pK_a \) is defined as \( pK_a = -\log_{10} K_a \).

Converting Logs and Constants

We use the conversion \( \ln x = 2.303 \log_{10} x \) to relate \( \ln K_a \) to \( pK_a \):\[\Delta G^{\circ'} = -RT \ln K_a = -RT (2.303 \log_{10} K_a) = -2.303 RT (-pK_a)\]Thus, the relation becomes \( \Delta G^{\circ'} = 2.303 RT \times pK_a \).

Substitute Known Values

At standard temperature \( 25^\circ C \) or \( 298 \ K \), the gas constant \( R = 8.314 \ J/(mol\cdot K) \). Substitute these values into the equation:\[\Delta G^{\circ'} = 2.303 \times 8.314 \times 298 \times pK_a\]This simplifies to \( \Delta G^{\circ'} = 5.708 \times 298 \times pK_a \).

Calculate for Acetic Acid

Substitute \( pK_a = 4.8 \) for acetic acid:\[\Delta G^{\circ'} = 5.708 \times 298 \times 4.8\]This yields \( \Delta G^{\circ'} \approx 7990 \ J/mol \) or \( 7.99 \ kJ/mol \).

Key concepts

Gibbs free energy

Gibbs free energy (\( \Delta G \)) is a fundamental concept in thermodynamics that helps us understand the spontaneity of chemical reactions. A negative \( \Delta G \) indicates that a reaction can occur spontaneously, meaning that it does not require additional energy to proceed. If \( \Delta G \) is positive, the reaction is non-spontaneous and requires energy input to take place.

The Gibbs free energy for a process at standard conditions is often represented as \( \Delta G^{\circ'} \). One of the key equations connecting Gibbs free energy with the reaction's equilibrium state is:- \( \Delta G^{\circ'} = -RT \ln K \) where:

• \( R \) is the universal gas constant \( \approx 8.314 \ J/(mol\cdot K) \)
• \( T \) is the temperature in Kelvin
• \( K \) is the equilibrium constant of the reaction

Understanding this relationship helps to derive \( pK_a \) values and estimate whether ionization reactions are favorable under specific conditions.

equilibrium constant

The equilibrium constant (\( K_a \)) plays a crucial role in chemical reactions, particularly in determining the extent of acid-base reactions. For the ionization of acids like acetic acid, \( K_a \) quantifies the balance between ions and non-ionized molecules at equilibrium.

The relationship between the equilibrium constant and \( pK_a \) is given by:- \( pK_a = -\log_{10} K_a \) This equation shows that a lower \( pK_a \) corresponds to a larger \( K_a \), indicating stronger acidity since more acid is ionized. Conversely, a higher \( pK_a \) suggests weaker acidity.

By understanding \( K_a \), one can predict how an acid will behave under various conditions, making it an essential tool in both academic and industrial chemistry settings. It's a measure of how far a reaction goes to completion at equilibrium, and thus, crucial for predicting the yield of products.

ionization of acetic acid

The ionization of acetic acid is an important example of acid-base chemistry that students often encounter. Acetic acid \( (CH_3COOH) \), a weak acid, ionizes in water to form acetate ions \( (CH_3COO^-) \) and hydrogen ions \( (H^+) \).

This ionization can be represented by the chemical equation:- \( CH_3COOH \rightleftharpoons CH_3COO^- + H^+ \) Knowing the \( pK_a \) value of acetic acid, which is 4.8, helps determine how readily the ionization occurs. A higher \( pK_a \) signifies a weak acid which does not ionize completely in water, indicating that the solution will not have a high concentration of \( H^+ \) ions.

By calculating the \( \Delta G^{\circ'} \) for this ionization process, one can understand the energy changes involved and predict whether the reaction occurs spontaneously under standard conditions. This kind of problem is foundational for students seeking to master acid-base balance and thermodynamics in chemistry.