Question

HCN has \(\mathrm{pK}_{\mathrm{a}}=9.21 ;\) acetic acid has \(\mathrm{pK}_{\mathrm{a}}=4.76 .\) (a) What is the difference in the standard free energies \(\left(\Delta \Delta \mathrm{G}^{\circ}\right)\) for these two acid-base equilibria? (b) What is the equilibrium constant and \(\Delta \mathrm{G}^{\circ}\) for the reaction \(\mathrm{HCN}+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-} \rightarrow \mathrm{CN}^{-}+\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\)

Short answer

(a) The difference in standard free energies (ΔΔG°) for HCN and acetic acid can be calculated by first converting their pKa values to Ka values, then calculating their individual ΔG° values, and finally subtracting them. The calculation gives: \(ΔΔG° = ΔG_{1}° - ΔG_{2}°\). (b) The equilibrium constant (K) for the given reaction is the ratio of Ka values of products and reactants: \(K = \cfrac{K_{a2}}{K_{a1}}\). The standard free energy change (ΔG°) for this reaction can be calculated using the relation: \(ΔG_{\text{reaction}}° = -8.314 \cdot 298 \cdot \ln \left(\cfrac{K_{a2}}{K_{a1}}\right)\).

Step-by-step solution

Convert pKa to Ka

Using the relation, \(pK_a = -\log_{10}(K_a)\), we convert pKa values to Ka values for both HCN and acetic acid. - For HCN: \(K_{a1} = 10^{-9.21}\) - For acetic acid: \(K_{a2} = 10^{-4.76}\)

Calculate ΔG° for both acids

Using the relation, \(ΔG° = -RT \ln K_a\), calculate the ΔG° values for both HCN and acetic acid. The value of R is 8.314 J/mol·K and the temperature T is assumed to be 298 K. - For HCN: \(ΔG_{1}° = -8.314 \cdot 298 \cdot \ln(10^{-9.21})\) - For acetic acid: \(ΔG_{2}° = -8.314 \cdot 298 \cdot \ln(10^{-4.76})\)

Calculate ΔΔG° for both acids

To find the difference in standard free energies, subtract the ΔG° values obtained in step 2. \(ΔΔG° = ΔG_{1}° - ΔG_{2}°\) Part (b):

Find the equilibrium constant for the given reaction

For the given reaction, we can write the equilibrium constant as the ratio of Ka values of the products and reactants. \(K = \cfrac{K_{a2}}{K_{a1}}\)

Calculate ΔG° for the given reaction

Using the relation, \(ΔG° = -RT \ln K\), calculate the ΔG° value for the given reaction. - For the given reaction: \(ΔG_{\text{reaction}}° = -8.314 \cdot 298 \cdot \ln \left(\cfrac{K_{a2}}{K_{a1}}\right)\)

Key concepts

pKa and Ka relationship

Understanding the relationship between pKa and Ka is essential for mastering acid-base chemistry. The pKa of a substance is the negative logarithm (base 10) of its acid dissociation constant (Ka). In mathematical terms, this relationship is defined as:

\( pKa = -\log_{10}(Ka) \)

By knowing the pKa value, we can calculate the strength of an acid; the smaller the pKa, the stronger the acid, and vice versa. In practical terms, converting pKa to Ka (and vice versa) is achieved through logarithmic calculations. For example, a pKa of 9.21 implies a Ka value exponentially smaller than a pKa of 4.76, which indicates the acid with a pKa of 4.76 dissociates more readily and is therefore a stronger acid.

Standard free energy

The standard free energy change (\( \Delta G^\circ \)) of a chemical reaction indicates the spontaneity of the process under standard conditions (1 bar pressure and reference temperatures, usually taken as 298 K). Thermodynamics in chemistry allows us to predict whether a reaction will proceed forward or reverse. The relationship between standard free energy change and the equilibrium constant (Ka) is depicted by the formula:

\( \Delta G^\circ = -RT \ln(Ka) \)

Where R is the universal gas constant and T is the temperature in Kelvin. If \( \Delta G^\circ \) is negative, the reaction proceeds spontaneously. The more negative the \( \Delta G^\circ \), the more strongly the reaction favors product formation. This concept is pivotal in understanding the energetic favorability of acid-base reactions and other chemical equilibria.

Equilibrium constants

Equilibrium constants, such as the acid dissociation constant (Ka), play a crucial role in chemical equilibrium. They quantify the extent to which a chemical reaction proceeds before reaching a state of balance. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant.

The equilibrium constant for an acid-base reaction is directly related to the pKa values of the reactants and products involved. Specifically, for the reaction of weak acids (such as HCN) with weak bases (like acetic acid), the equilibrium constant (K) can be expressed as the ratio of their respective Ka values. This gives us a clear picture of the direction in which the reaction will be favored. A larger equilibrium constant indicates a reaction that favors the formation of products.

Thermodynamics in chemistry

Thermodynamics is the study of energy transformations, and in chemistry, it is fundamental in predicting reaction behavior. It encompasses concepts such as enthalpy, entropy, and free energy, which collectively determine whether a chemical process is energetically viable. Specifically, it tells us whether energy must be supplied to make a reaction proceed ('non-spontaneous') or whether the reaction releases energy ('spontaneous').

The aforementioned standard free energy change (\( \Delta G^\circ \)) is a key thermodynamic quantity that combines enthalpy and entropy to predict the spontaneity of chemical reactions at constant temperature and pressure. This helps chemists understand how different factors, such as temperature or the nature of the substances involved, affect reaction equilibria and enables them to optimize conditions for desired reactions to occur.

Logarithmic calculations in chemistry

Logarithmic calculations are ubiquitous in chemistry, particularly in the study of acid-base equilibria and other processes governed by exponential relationships. Logarithms allow us to easily work with the very large or small numbers that arise when dealing with concentration terms and equilibrium constants. The log to the base 10 is particularly useful because it directly relates to the concept of 'p' values, like pH and pKa, which denote the negative log of concentration-related constants.

When it comes to calculating standard free energies from equilibrium constants, the natural logarithm (base e) is used due to its mathematical properties in physical equations. Using logarithms, we can elegantly transform the multiplicative relationships inherent in equilibrium expressions into additive relationships that are more straightforward to handle quantitatively.