Question

The following reaction occurs in pure water: $${\mathrm{H}}_2\mathrm{O}(l)+{\mathrm{H}}_2\mathrm{O}(l)\rightleftharpoons {\mathrm{H}}_3{\mathrm{O}}^{+}(aq)+{\mathrm{OH}}^{-}(aq)$$ which is often abbreviated as $${\mathrm{H}}_2\mathrm{O}(l)\rightleftharpoons {\mathrm{H}}^{+}(aq)+{\mathrm{OH}}^{-}(aq)$$ For this reaction, $\mathrm{\Delta }{G}^{\circ }=79.9\mathrm{kJ}/\mathrm{mol}$ at ${25}^{\circ }\mathrm{C}$ . Calculate the value of $\mathrm{\Delta }G$ for this reaction at ${25}^{\circ }\mathrm{C}$ when $\left[{\mathrm{OH}}^{-}\right]=0.15M$ and $\left[{\mathrm{H}}^{+}\right]=0.71M.$

Short answer

The value of ΔG for the reaction at 25°C with the given concentrations of OH⁻ and H⁺ is approximately 74370.3 J/mol.

Step-by-step solution

Identify the given values

We are given that: 1. ΔG° = 79.9 kJ/mol 2. T = 25°C = 298.15 K (converted to Kelvin) 3. [OH⁻] = 0.15 M 4. [H⁺] = 0.71 M

Calculate the reaction quotient (Q)

For the reaction: H2O(l) ⮌ H⁺(aq) + OH⁻(aq), the reaction quotient Q is given by: Q = [H⁺][OH⁻], Now, substituting the given values of [H⁺] and [OH⁻], Q = (0.71 M)(0.15 M) = 0.1065

Calculate ΔG using ΔG°, RT, and ln(Q)

We have the equation: ΔG = ΔG° + RT ln(Q), We have already found the values for ΔG° (= 79.9 kJ/mol) and Q (= 0.1065). Let's convert ΔG° from kJ/mol to J/mol, which is 79.9 × 1000 = 79,900 J/mol. The gas constant R = 8.314 J/(mol·K). Now we can calculate ΔG: ΔG = 79900 J/mol + (8.314 J/(mol·K))(298.15 K)ln(0.1065) ΔG ≈ 79900 J/mol + (8.314 J/(mol·K))(298.15 K)(-2.2398) ΔG ≈ 79900 J/mol - 5529.7 J/mol ΔG ≈ 74370.3 J/mol So, the value of ΔG for the reaction at 25°C with the given concentrations of OH⁻ and H⁺ is approximately 74370.3 J/mol.

Key concepts

Equilibrium Constants

Equilibrium constants are crucial in understanding the balance point of chemical reactions. For any reversible reaction at equilibrium, there exists a constant at a given temperature, known as the equilibrium constant (K). This constant is the ratio of the concentration of products to reactants, each raised to the power of their stoichiometric coefficients. In the case of the water ionization reaction, it can be represented as:

• $K_w=[H^{+}][O{H}^{-}]$

Notice how this relationship shows the intrinsic nature of water to slightly ionize, even in its purest form.
The equilibrium constant is significant because it tells us about the extent to which a reaction can proceed under given conditions. Markedly, a large K indicates a reaction favoring products, while a small K suggests the opposite.
Understanding the equilibrium constant
is essential in predicting the position of equilibrium and calculating other important quantities like the reaction quotient.

Reaction Quotient

The reaction quotient, denoted as $Q$, is similar to the equilibrium constant but provides insight into the current state of the system before equilibrium is reached. It is calculated in the same way as K, using the actual concentrations of reactants and products at any given moment.

• $Q=[H^{+}][O{H}^{-}]$

If $Q=K$, the system is at equilibrium. If $Q<K$, the reaction will proceed in the forward direction to produce more products. Conversely, if $Q>K$, the reaction will shift in the reverse direction to form more reactants.
This concept helps predict the shift in the equilibrium based on concentration changes, effectively acting as a measure of how far the system deviates from equilibrium conditions.
Calculating Q can aid in determining how the system will react to various stresses, such as changes in concentration, pressure, or temperature.

Thermodynamics

Thermodynamics is the study of energy changes that accompany chemical reactions. Gibbs Free Energy ($\mathrm{\Delta }G$) is a thermodynamic function that describes the spontaneity of a reaction, which can predict whether a reaction will proceed forward or not.
The relationship between $\mathrm{\Delta }G$ and equilibrium is given by the equation:

• $\mathrm{\Delta }G=\mathrm{\Delta }{G}^{\circ }+RT\ln (Q)$

where $\mathrm{\Delta }{G}^{\circ }$ is the standard Gibbs Free Energy change, $R$ is the gas constant, $T$ is the temperature in Kelvin, and $Q$ is the reaction quotient.
If $\mathrm{\Delta }G<0$, the reaction is spontaneous. If $\mathrm{\Delta }G>0$, the reaction is non-spontaneous. At equilibrium, $\mathrm{\Delta }G=0$. This relationship highlights how reaction spontaneity is linked with the ratio of products to reactants at a given time.
This fundamental concept assists in comprehending how systems evolve over time to reach equilibrium, dictated by energy considerations.

Acid-Base Reactions

Acid-base reactions involve the transfer of protons (H⁺ ions) between substances. In the reaction given, water self-ionizes to produce hydronium (H₃O⁺) and hydroxide ions (OH⁻), which can be simplified to H⁺ and OH⁻ for ease of use in calculations.
This particular reaction is a classic example of an acid-base reaction where water acts both as an acid (donating a proton) and as a base (accepting a proton). The ability of water to self-ionize is expressed by the equilibrium constant for water, $K_w$, which at 25°C has a value of $1.0\times {10}^{-14}$.
Understanding acid-base reactions is essential in predicting the pH of a solution and its behavior in various chemical environments. Such reactions are omnipresent, from biochemical processes in the body to industrial chemical processes.
Grasping the nuances of these reactions helps in calculating concentrations, predicting reaction directions, and understanding the fundamental nature of chemical equilibria.