Problem 107 - Chemistry: An Atoms-Focused Approach

Question

Which of the following reactions has the largest cquilibrium constant at \(25^{\circ} \mathrm{C} ?\) a. \(\mathrm{Cl}_{2}(\mathrm{g})+\mathrm{F}_{2}(g) \rightleftharpoons 2 \mathrm{ClF}(g) \quad \Delta G^{*}=115.4 \mathrm{kJ}\) b. \(\mathrm{Cl}_{2}(\mathrm{g})+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{ClBr}(g) \quad \Delta G^{*}=-2.0 \mathrm{kJ}\) c. \(\mathrm{Cl}_{2}(\mathrm{g})+\mathrm{I}_{2}(\mathrm{g}) \rightleftharpoons 21 \mathrm{Cl}(g) \quad \Delta G^{*}=-27.9 \mathrm{kJ}\)

Step-by-Step Solution

Answer

Answer: Reaction c has the largest equilibrium constant at \(25^{\circ}\mathrm{C}\), with a value of \(K_c = 3.073\).

Step 1: 01. Relationship between Gibbs free energy and the equilibrium constant

The relationship between the Gibbs free energy change (\(\Delta G^{*}\)) and the equilibrium constant (K) is given by the formula: \(\Delta G^{*} = - RT \ln K\) Here, R is the universal gas constant (8.314 J/mol·K), T is the temperature in Kelvin, and K is the equilibrium constant.

Step 2: 02. Convert temperature to Kelvin

To use the above formula, we need temperature in Kelvin. We can convert temperature from Celsius to Kelvin using the formula: \(T_{K} = T_{C} + 273.15\) So, for \(25^{\circ}\mathrm{C}\), we have: \(T_{K}= 25 + 273.15 = 298.15\,\text{K}\)

Step 3: 03. Calculate the equilibrium constant K for each reaction

Now, we can calculate the equilibrium constant K for each reaction using the given Gibbs free energy changes and the temperature. For each reaction, use the formula: \(K = e^{(-\Delta G^{*}) / (RT)}\) Reaction a: \(K_a = e^{(-115.4 \times 10^3\,\text{J/mol}) / (8.314\,\text{J/mol·K} \times 298.15\,\text{K})} = e^{-4.61} = 0.00996\) Reaction b: \(K_b = e^{(2.0 \times 10^3\,\text{J/mol}) / (8.314\,\text{J/mol·K} \times 298.15\,\text{K})} = e^{0.0805} = 1.084\) Reaction c: \(K_c = e^{(27.9 \times 10^3\,\text{J/mol}) / (8.314\,\text{J/mol·K} \times 298.15\,\text{K})} = e^{1.122} = 3.073\)

Step 4: 04. Compare the equilibrium constants and find the largest one

Comparing the equilibrium constants: \(K_a = 0.00996\), \(K_b = 1.084\), \(K_c = 3.073\) The largest equilibrium constant is \(K_c\), which belongs to reaction c. Therefore, reaction c has the largest equilibrium constant at \(25^{\circ}\mathrm{C}\).

Key Concepts

Gibbs Free Energy and Equilibrium

The concept of Gibbs free energy, denoted \(\Delta G^{*}\), is crucial in understanding chemical reactions. It measures the spontaneity of a reaction and is connected to the equilibrium state of the reaction. When \(\Delta G^{*} < 0\), the process is spontaneous and tends toward the formation of products. Conversely, if \(\Delta G^{*} > 0\), the reverse reaction is favored. This relationship is depicted mathematically by the equation \(\Delta G^{*} = - RT \ln K\), where \(K\) is the equilibrium constant. Here, a negative \(\Delta G^{*}\) indicates a larger equilibrium constant, meaning the reaction heavily favors products.

• A negative \(\Delta G^{*}\) is linked to a large \(K\), suggesting a greater concentration of products at equilibrium.
• Conversely, a positive \(\Delta G^{*}\) implies a low \(K\), indicating less product formation.

With this equation, we can calculate the equilibrium constant if \(\Delta G^{*}\) is known, allowing for predictions about the direction and extent of chemical reactions.

Understanding Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions in a chemical system are equal. At this point, the concentrations of reactants and products remain constant over time, although not necessarily equal. The equilibrium constant \(K\) quantitatively expresses the ratio of product concentrations to reactant concentrations, each raised to the power of their stoichiometric coefficients in the balanced chemical equation.

• A large \(K\) value signifies that the products are favored at equilibrium.
• A small \(K\) value indicates that reactants are favored.

The equilibrium position can shift according to external changes, such as temperature or pressure, which is predicted by Le Chatelier's Principle. In calculations, \(K\) is essential for determining the extent of a reaction and predicting the concentrations of the substances involved at equilibrium.

The Role of the Universal Gas Constant

The universal gas constant, symbolized as \(R\), is a fundamental constant that appears in various equations in chemistry and physics, linking energy to temperature and moles. Its value is \(8.314 \, \text{J/mol·K}\). In the context of the relationship between Gibbs free energy and the equilibrium constant, \(R\) is used to incorporate temperature into the calculation. \(\Delta G^{*} = - RT \ln K\) shows how \(R\) and temperature (in Kelvin) interact to determine the sign and magnitude of \(\Delta G^{*}\);

• The constant \(R\) provides the necessary energy unit conversion, balancing the energy change with temperature.
• In equilibrium constant calculations, \(R\) ensures that the \(\Delta G^{*}\) units align with temperature, permitting accurate evaluation of \(K\).

Understanding the universal gas constant's role simplifies solving many thermodynamic problems, making it a vital component in chemical equilibrium analysis.