Question

Consider the dissociation of a weak acid \(\mathrm{HA}\left(K_{\mathrm{a}}=4.5 \times 10^{-3}\right)\) in water: $$\mathrm{HA}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{A}^{-}(a q)$$ Calculate \(\Delta G^{\circ}\) for this reaction at \(25^{\circ} \mathrm{C}\)

Short answer

The standard Gibbs free energy change, \(\Delta G^{\circ}\), for the dissociation reaction of the weak acid HA in water at \(25^{\circ}C\) is approximately 20.66 kJ/mol.

Step-by-step solution

Convert temperature to Kelvin

To convert the given temperature from Celsius to Kelvin, use the formula below: $$T(K) = T(^{\circ}C) + 273.15$$ Plug in the given value \(T(^{\circ}C) = 25^{\circ}C\): $$T(K) = 25 + 273.15 = 298.15\,K$$

Calculate the standard Gibbs free energy change

Now we can use the formula to calculate \(\Delta G^{\circ}\): $$\Delta G^{\circ} = -RT\ln{K_a}$$ Plug in the values: R = 8.314 J/mol·K, T = 298.15 K, and \(K_a = 4.5 \times 10^{-3}\): $$\Delta G^{\circ} = - (8.314 \,\text{J/mol·K})(298.15 \,\text{K})\ln{(4.5 \times 10^{-3})}$$

Simplify and find the answer

Calculate the value of \(\Delta G^{\circ}\) by using the values from the previous step: $$\Delta G^{\circ} = -(8.314 \,\text{J/mol·K})(298.15 \,\text{K})\ln{(4.5 \times 10^{-3})} \approx 20662 \,\text{J/mol}$$ Since we usually express Gibbs free energy in kJ/mol, divide the result by 1000: $$\Delta G^{\circ} = \dfrac{20662 \,\text{J/mol}}{1000} \approx 20.66 \, \text{kJ/mol}$$ Therefore, \(\Delta G^{\circ}\) for the dissociation reaction of the weak acid HA in water at \(25^{\circ} \mathrm{C}\) is approximately 20.66 kJ/mol.

Key concepts

Dissociation of weak acids

In chemistry, weak acids are those that only partially dissociate into ions when dissolved in water. This means they do not completely ionize, making the equilibrium position of the dissociation reaction essential in determining their properties. For a weak acid like \(\text{HA}\), the dissociation can be represented as:

• \(\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\)

Here, \(\text{HA}\) is the weak acid, \(\text{H}^+\) is the hydrogen ion, and \(\text{A}^-\) is the conjugate base of the acid.
The degree to which a weak acid dissociates in water is typically quantified by its acid dissociation constant, \(K_{\text{a}}\).
This constant helps us understand how much of the acid will donate protons in solution: the larger the \(K_{\text{a}}\), the stronger the acid.
In the provided exercise, \(K_{\text{a}} = 4.5 \times 10^{-3}\), indicating a relatively weak, but notable tendency to dissociate. Calculating the Gibbs free energy change, \(\Delta G^\circ\), for the reaction helps us understand its spontaneity under standard conditions.

Thermodynamics

Thermodynamics is the study of energy changes, especially in chemical reactions. One important concept is Gibbs free energy (\(\Delta G\)), which determines whether a chemical reaction will occur spontaneously. For the standard change in Gibbs free energy (\(\Delta G^\circ\)), the formula is given by:

• \(\Delta G^\circ = -RT\ln{K}\)

where \(R\) is the universal gas constant (8.314 J/mol·K), \(T\) is the temperature in Kelvin, and \(K\) is the equilibrium constant.

For reactions like the dissociation of weak acids, \(\Delta G^\circ\) tells us how favorable the reaction is under standardized conditions (1 atm, 298.15 K).
A negative \(\Delta G^\circ\) means the reaction will proceed spontaneously in the forward direction, while a positive \(\Delta G^\circ\) suggests it is non-spontaneous under standard conditions. In calculations, converting temperatures from degrees Celsius to Kelvin ensures that all units match, which is essential for accurate results.

Equilibrium constant

The equilibrium constant (\(K\)) is a dimensionless value that measures the relative amounts of products and reactants at equilibrium for a given chemical reaction. For weak acids, this is specifically called the acid dissociation constant \(K_{\text{a}}\).

\(K_{\text{a}}\) provides insight into the extent of acid dissociation: a higher \(K_{\text{a}}\) value indicates more products relative to reactants, showing stronger acidity.
In thermodynamic terms, \(K\) is linked to \(\Delta G^\circ\) through the relation:

• \(\Delta G^\circ = -RT\ln{K}\)

Inserting the particular \(K_{\text{a}}\) value into this equation helps calculate the Gibbs free energy change, showing the spontaneity under standard conditions.
For the dissociation of the weak acid \(\text{HA}\) in the example, \(K_{\text{a}} = 4.5 \times 10^{-3}\), is used to deduce that while the acid does dissociate, it does not do so fully without external interventions or specific conditions.