Question

Consider the reaction \(6 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{P}_{4}(g) \longrightarrow 4 \mathrm{PH}_{3}(g)\). (a) Using data from Appendix C, calculate \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\). (b) Calculate \(\Delta G\) at \(298 \mathrm{~K}\) if the reaction mixture consists of \(8.0 \mathrm{~atm}\) of \(\mathrm{H}_{2}, 0.050 \mathrm{~atm}\) of \(\mathrm{P}_{4}\), and \(0.22 \mathrm{~atm}\) of \(\mathrm{PH}_{3}\).

Short answer

The standard Gibbs free energy change (ΔG°) for the reaction is -51.2 kJ/mol. And under the given conditions of partial pressures at 298 K, the actual Gibbs free energy change (ΔG) is -48.24 kJ/mol.

Step-by-step solution

Calculate Standard Gibbs Free Energy Change (ΔG°)

Using the standard Gibbs free energies of formation from Appendix C, we can find the values for H2(g), P4(g), and PH3(g). According to the values given: ΔGf°(H2) = 0 kJ/mol (since H2 is in its most stable elemental form) ΔGf°(P4) = 0 kJ/mol (since P4 is in its most stable elemental form) ΔGf°(PH3) = -12.8 kJ/mol Now, we can apply the equation for ΔG°: ΔG° = Σ(ν_products * ΔGf°_products) - Σ(ν_reactants * ΔGf°_reactants) ΔG° = 4(-12.8) - (6(0) + 0) ΔG° = -51.2 kJ/mol

Calculate the Reaction Quotient (Q)

The reaction quotient, Q, is given by the ratio of the product of the concentrations (or partial pressures) of the products to the product of the concentrations (or partial pressures) of the reactants, each raised to the power of their stoichiometric coefficients. For the given reaction, Q = (P_PH3)^4 / ((P_H2)^6 * (P_P4)^1) Given partial pressures are: P_H2 = 8.0 atm P_P4 = 0.050 atm P_PH3 = 0.22 atm Plugging these values into the equation for Q, we get: Q = (0.22)^4 / ((8.0)^6 * (0.050)^1) = 2.3465 × 10^(-6)

Calculate the actual Gibbs Free Energy Change (ΔG)

Finally, we can find the actual Gibbs free energy change (ΔG) using the relationship between ΔG and ΔG°: ΔG = ΔG° + RTln(Q) Where R = 8.314 J/(mol*K), T = 298 K, and Q = 2.3465 × 10^(-6) ΔG = -51,200 J/mol + (8.314 J/(mol*K))(298 K)ln(2.3465 × 10^(-6)) ΔG = -51,200 J/mol + 2960 J/mol ΔG = -48,240 J/mol So, the Gibbs free energy change at 298 K under the given conditions is -48,240 J/mol or -48.24 kJ/mol.

Key concepts

Standard Gibbs Free Energy Change

Understanding the concept of Standard Gibbs Free Energy Change (G\textdegree) is crucial in the study of chemical reactions and their spontaneity. It's a thermodynamic property that indicates the maximum amount of work obtained from a chemical process under standard conditions of temperature and pressure, which usually means 298 K and 1 atmosphere of pressure for each substance.

G\textdegree is calculated based on the energies of formation of the products and the reactants. The formula G\textdegree = \(G%\textdegree_{products} -G%\textdegree_{reactants}\) incorporates the standard Gibbs free energies of formation (Gf\textdegree) for each substance involved in the reaction. Remember, elements in their most stable form, like H2 and P4 in our exercise, have a Gf\textdegree of zero. The significance of a negative G\textdegree is that the reaction is spontaneous under standard conditions, as it was in our exercise with G\textdegree = -51.2 kJ/mol.

Reaction Quotient

The Reaction Quotient (Q) helps predict the direction of a chemical reaction at any moment in time. Q is a dimensionless number calculated from the ratio of the concentrations or partial pressures of the products to those of the reactants, each raised to their stoichiometric coefficients. In terms of partial pressures, the formula is given by:\[ Q = \frac{(P_{\text{products}})^{\text{stoichiometric coefficient}}}{(P_{\text{reactants}})^{\text{stoichiometric coefficient}}} \]

By comparing Q to the equilibrium constant (K), we can gauge whether the reaction will proceed forward or reverse to reach equilibrium. If Q < K, the reaction tends to form more products; if Q > K, it will form more reactants. In the given solution, Q was calculated using the partial pressures of the gases H2, P4, and PH3, indicating the state of the reaction at a specific instance, which can be helpful in calculating the actual Gibbs Free Energy Change.

Chemical Thermodynamics

The field of Chemical Thermodynamics involves the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. One of the most important concepts within this field is Gibbs Free Energy (G), a function that combines the system's enthalpy (H), temperature (T), and entropy (S) to predict the spontaneity of processes: \[ G = H - TS \]

Any chemical process that decreases the Gibbs Free Energy of a system is thermodynamically favorable and may occur spontaneously. Thermal equilibrium, phase changes, and chemical reaction direction all fall within the purview of chemical thermodynamics. As we've seen with the reaction involving H2 and P4 creating PH3, thermodynamic concepts like G\textdegree and G provide insights into the feasibility and extent of chemical reactions.

Partial Pressure

The term Partial Pressure refers to the pressure exerted by a single gas in a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. Mathematically, the partial pressure can be expressed as:\[ P_i = X_i \times P_{\text{total}} \]

where P_i is the partial pressure of the gas i, X_i is the mole fraction of the gas, and P_{\text{total}} is the total pressure of the mixture. This concept is important when dealing with gas-phase reactions and is pivotal for calculating the Reaction Quotient (Q). In the given exercise, partial pressures are used directly to calculate Q and assess the status and progress of the reaction relative to its equilibrium state.