Problem 78 - Chemistry: The Central Science in SI Units, Global Edition

Question

Consider the reaction \(\mathrm{CH}_{4}(\mathrm{~g})+4 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{CCl}_{4}(g)+\) \(4 \mathrm{HCl}(g) .\). (a) Using data from Appendix C, calculate \(\Delta G^{\circ}\) at \(298 \mathrm{~K} .(\mathbf{b})\) Calculate \(\Delta G\) at \(298 \mathrm{~K}\) if the reaction mixture consists of \(50.7 \mathrm{kPa}\) of \(\mathrm{CH}_{4}(g), 25.3 \mathrm{kPa}\) of \(\mathrm{Cl}_{2}(g), 10.13 \mathrm{kPa}\) of \(\mathrm{CCl}_{4}(\mathrm{~g})\) and \(15.2 \mathrm{kPa}\) of \(\mathrm{HCl}(\mathrm{g})\)

Step-by-Step Solution

Answer

(a) \(\Delta G^{\circ} = -191.5 \textrm{ kJ/mol}\) is obtained by calculating \(\Delta H^{\circ} - T\Delta S^{\circ}\) using the standard enthalpy and entropy change values. (b) \(\Delta G = -183.9 \textrm{ kJ/mol}\) is calculated by finding the reaction quotient (Q) using given partial pressures and substituting it along with the calculated \(\Delta G^{\circ}\) into the equation \(\Delta G = \Delta G^{\circ} + RT \ln(Q)\).

Step 1: Calculate the standard enthalpy and entropy changes

Using data from Appendix C, we determine the standard molar enthalpies (ΔH°) and standard molar entropies (ΔS°) of all species involved in the reaction. Then, we calculate the respective values for the entire reaction.

Step 2: Calculate the standard Gibbs free energy change

Now, we substitute the calculated values of ΔH° and ΔS° in the equation ΔG° = ΔH° - TΔS° and calculate ΔG° at the given temperature 298K.

Step 3: Calculate the reaction quotient (Q)

Based on the provided partial pressures of reactants and products, calculate the reaction quotient Q = [CCl₄][HCl]⁴ / ([CH₄][Cl₂]⁴).

Step 4: Calculate ΔG using the reaction quotient

Finally, use the calculated ΔG° and reaction quotient (Q) values to find ΔG at the given temperature using the equation, ΔG = ΔG° + RT ln(Q).

Key Concepts

Enthalpy

In chemistry, enthalpy (\(\Delta H\)) represents the heat content of a system at constant pressure. When considering a chemical reaction, the change in enthalpy (\(\Delta H^{\circ}\)) is the difference between the enthalpy of the products and the reactants.

For a reaction, you can find enthalpy changes using data from the Appendix C tables, which list standard molar enthalpies. The formula used is:\[\Delta H^{\circ} = \sum \Delta H^{\circ}_{\text{products}} - \sum \Delta H^{\circ}_{\text{reactants}}\]Here's a simple formula:

• Calculate the enthalpy for each compound.
• Multiply the calculated enthalpy by the number of moles in the reaction.
• Subtract the sum of the reactants' enthalpy from the products' enthalpy.

Enthalpy is critical in understanding how much energy is released or absorbed in a chemical change. A negative\(\Delta H\)indicates an exothermic reaction, meaning it releases energy.

Entropy

Entropy (\(\Delta S\)) is a measure of randomness or disorder within a system. It influences the spontaneity of a reaction, which is particularly important when paired with enthalpy to determine free energy.

In thermodynamics, an increase in entropy (\(\Delta S^{\circ} > 0\)) often corresponds to a system moving toward more chaos or disorder. To calculate the standard entropy change of a system, use:\[\Delta S^{\circ} = \sum \Delta S^{\circ}_{\text{products}} - \sum \Delta S^{\circ}_{\text{reactants}}\]Steps to compute entropy change:

• Look up standard entropy values for each compound present in the reaction.
• Calculate using the number of moles in the reaction.
• Determine the total entropy change by subtracting the entropy of reactants from that of products.

Entropy change and its sign help determine if a system will favor the formation of products, particularly in balance with enthalpy changes.

Reaction Quotient

The reaction quotient (\(Q\)) is a vital factor in assessing the progress of a reaction under specific conditions, different from the equilibrium state. It is defined by the ratio of the concentrations or pressures of the products to reactants, each raised to the power of their stoichiometric coefficients in the balanced chemical equation.

The general formula for gases, when pressures are provided, is:\[Q = \frac{P_{\text{products}}}{P_{\text{reactants}}}\]For the given reaction:

• Identify partial pressures for each gas involved.
• Apply the formula considering stoichiometry: \[Q = \frac{{[\mathrm{CCl}_4][\mathrm{HCl}]^4}}{[\mathrm{CH}_4][\mathrm{Cl}_2]^4}\]

The calculated\(Q\)helps in determining the Gibbs free energy (\(\Delta G\)) using:\[\Delta G = \Delta G^{\circ} + RT \ln(Q)\]Where\(R\) is the universal gas constant and \(T\) is the temperature in Kelvin. This equation allows chemists to predict the direction a reaction will naturally progress given current pressures, indicating if the reaction is moving towards equilibrium.