Question

For the autoionization of water at \(25^{\circ} \mathrm{C}:\) $$ \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q) $$ \(K_{\mathrm{w}}\) is \(1.0 \times 10^{-14}\). What is \(\Delta G^{\circ}\) for the process?

Short answer

\(\Delta G^{\circ} = 184 \, \text{kJ/mol}\) for water autoionization at 25°C.

Step-by-step solution

Write the Formula for ΔG°

The standard Gibbs free energy change (abla G^{\circ}) is related to the equilibrium constant (K) by the formula:\[ \Delta G^{\circ} = -RT \ln K\]where R is the universal gas constant (8.314 J/mol·K) and T is the temperature in Kelvin.

Convert Temperature to Kelvin

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

Insert Values into the Formula

Now use the known values:- R = 8.314 J/mol·K- T = 298.15 K- K_w = 1.0 \times 10^{-14}Insert these into the formula:\[ \Delta G^{\circ} = -(8.314 \, \text{J/mol·K})(298.15 \, \text{K}) \ln (1.0 \times 10^{-14})\]

Calculate Natural Logarithm

Calculate the natural logarithm of the equilibrium constant:\[ \ln (1.0 \times 10^{-14}) = -32.236\]

Calculate ΔG°

Substitute the value from Step 4 into the calculation for \( \Delta G^{\circ} \):\[ \Delta G^{\circ} = -(8.314)(298.15)(-32.236)\]Calculate the result:\[ \Delta G^{\circ} = (8.314)(298.15)(32.236) \]\[ \Delta G^{\circ} = 1.84 \times 10^5 \, \text{J/mol} \, (or \, 184 \, \text{kJ/mol})\]

Interpret the Result

The calculated \(\Delta G^{\circ}\) value is positive, indicating that the autoionization of water is non-spontaneous under standard conditions.

Key concepts

Equilibrium Constant

The equilibrium constant, often represented by the letter \( K \), is central to understanding reactions that have reached a state of balance. It quantifies the ratio of the concentrations of products to reactants at equilibrium for a given reaction. In the case of water autoionization, the equilibrium constant \( K_w \) is \( 1.0 \times 10^{-14} \) at \( 25^{\circ}C \).

This small value indicates that the concentration of ions formed from water is very low, hence the reaction strongly favors the reactants in this condition.

When calculating Gibbs free energy change, the equilibrium constant is crucial because it relates to how far the reaction lies towards the products or reactants, influencing the spontaneity of the process.

Gibbs Free Energy Change

Gibbs free energy change, represented as \( \Delta G^{\circ} \), tells us whether a reaction is spontaneous under standard conditions. This energy change can be calculated using the formula:
\[ \Delta G^{\circ} = -RT \ln K \]where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. If \( \Delta G^{\circ} \) is negative, the reaction is spontaneous. If positive, like in water's autoionization, the reaction is non-spontaneous.

The magnitude of \( \Delta G^{\circ} \) indicates the amount of energy required to drive the reaction. In our exercise, a calculated \( \Delta G^{\circ} \) of 184 kJ/mol reflects significant non-spontaneity under standard conditions.

Kelvin Temperature Conversion

Temperature plays a key role in calculations involving thermodynamics. Often, temperatures are provided in Celsius but need to be converted to Kelvin for accurate thermodynamic calculations, since Kelvin is the SI unit for temperature.

Conversion is simple: add 273.15 to the Celsius temperature to get Kelvin.
For example, \( 25^{\circ}C = 25 + 273.15 = 298.15 \, K \).

This unit is essential as it ensures a uniform standard that is widely accepted, leading to accurate and reliable thermodynamic results.

Universal Gas Constant

The universal gas constant, symbolized as \( R \), is a key fundamental constant in chemistry and appears in many equations, including those involving energy calculations like Gibbs free energy.

It relates energy scale in chemistry to temperature and amounts of substance.

The value used in thermodynamics is \( 8.314 \, J/mol\cdot K \).

When calculating \( \Delta G^{\circ} \), this constant acts as a bridge between the temperature in Kelvin, the equilibrium constant, and Gibbs free energy, thereby integrating different aspects of the reaction dynamics into one coherent calculation.