Question

At \(25^{\circ} \mathrm{C}, K_{\mathrm{a}}\) for acid dissociation of aspirin \(\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\right)\) is \(3.0 \times 10^{-4}\). Calculate \(\Delta G^{\circ}\) for the reaction \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}(a q)+\) \(\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{C}_{9} \mathrm{H}_{7} \mathrm{O}_{4}^{-}(a q)\).

Short answer

ΔG° for the reaction is approximately 20.106 kJ/mol.

Step-by-step solution

Understand the Relationship between ΔG° and K_a

The change in Gibbs free energy, \(\Delta G^{\circ}\), for a reaction is related to its equilibrium constant, \(K_{a}\), at a given temperature by the equation: \[\Delta G^{\circ} = -RT\ln K_{a}\], where R is the universal gas constant (8.314 J/mol·K), and T is the temperature in Kelvin.

Convert Temperature to Kelvin

The temperature is given as 25°C. Convert this temperature to Kelvin by adding 273.15: \[T = 25 + 273.15 = 298.15\,\text{K}.\]

Plug Values into the ΔG° Formula

Substitute the values of R (8.314 J/mol·K), T (298.15 K), and \(K_{a} = 3.0 \times 10^{-4}\) into the equation for \(\Delta G^{\circ}\): \[\Delta G^{\circ} = -8.314 \times 298.15 \times \ln(3.0 \times 10^{-4}).\]

Calculate ln(K_a)

Calculate the natural logarithm of the equilibrium constant: \[\ln(3.0 \times 10^{-4}) \approx -8.1116.\]

Solve for ΔG°

Multiply the values: \[\Delta G^{\circ} = -8.314 \times 298.15 \times (-8.1116).\] This results in \[\Delta G^{\circ} \approx 20106\,\text{J/mol}.\]

Convert J/mol to kJ/mol

Since Gibbs free energy is often expressed in kJ/mol, convert the result by dividing by 1000: \[\Delta G^{\circ} \approx 20.106\,\text{kJ/mol}.\]

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K_{a} \) in the context of acids, is pivotal in understanding how reversible reactions behave at equilibrium. For chemical reactions, the equilibrium constant is defined based on the concentration of the products and reactants when the reaction reaches equilibrium. This constant provides insight into how far a reaction has proceeded before reaching equilibrium.

• For a simple reaction \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant \( K \) is given by: \[ K = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
• It shows the ratio of the concentration of products to reactants.
• In acid dissociation, \( K_{a} \) specifically refers to the equilibrium constant for an acid releasing a proton, forming its conjugate base and hydronium ion in solution.

Unlike reaction rates, which can vary with time, the equilibrium constant is constant for a given temperature, indicating a fixed balance point for the reaction's concentrations. Increasing the value of \( K_a \) suggests a stronger acid, with more products formed at equilibrium.

Acid Dissociation

Acid dissociation is a fundamental process in chemistry, characterizing how acids release hydrogen ions (protons) into a solution. When an acid liquid dissolves in water, it dissociates into its respective ions. The degree to which an acid dissociates can be quantified by its acid dissociation constant, \( K_{a} \).

• In aqueous solutions, generic acid \( HA \) will dissociate into \( H^+ \) and its conjugate base \( A^- \).
• This is represented by the equation: \( HA + H_2O \rightleftharpoons H_3O^+ + A^- \).
• The larger the \( K_{a} \), the stronger the acid, as it signifies a greater tendency to lose a proton.

Aspirin (\( C_9H_8O_4 \)) in water undergoes such dissociation. The significance of \( K_{a} \) is crucial for understanding the drug's behavior in biological systems, influencing factors like solubility and absorption. Understanding aspirin's dissociation helps in predicting and controlling its medicinal effects.

Aspirin Chemistry

Aspirin, known chemically as acetylsalicylic acid, represents a classic compound in medicinal chemistry with applications far beyond simple pain relief. It is derived from salicylic acid, with an added acetyl group enhancing its pharmacokinetic properties. The chemistry of aspirin is intriguing as it plays a dual role in both medicinal efficacy and as an example of acid dissociation.

• The molecular formula of aspirin is \( C_9H_8O_4 \).
• It dissociates in water, forming hydronium ion \( H_3O^+ \) and the conjugate base \( C_9H_7O_4^- \).
• This reaction is essential for its biological activity, as the ionized form helps in crossing mucous membranes.

Beyond its pharmacological effects, aspirin's chemistry illustrates broader principles like equilibrium dynamics and acid-base behavior in solution. The dissociation constant \( K_{a} \) plays a critical role in determining aspirin's solubility and thus its bioavailability. Understanding these concepts helps chemists and pharmacologists optimize and predict the drug's actions within the human body.