Problem 104 - Chemistry: An Atoms-Focused Approach

Question

The equation \(\Delta G^{\circ}=-R T\) ln \(K\) relates the valuc of \(K_{\mathrm{p}},\) not \(K_{\mathrm{c}},\) to the change in standard free energy for a reaction in the gas phasc. Explain why.

Step-by-Step Solution

Answer

The equation relates the value of \(K_p\) rather than \(K_c\) because in a gaseous reaction, the change in standard free energy is closely linked to pressure changes within the system. \(K_p\) is an equilibrium constant expressed in terms of partial pressures, which accounts for the pressure effects of gases on the change in standard free energy.

Step 1: Understand the equation

The equation \(\Delta G^{\circ}=-RT\) ln \(K\) connects the change in the standard free energy of a reaction (\(\Delta G^{\circ}\)) to its equilibrium constant (K). The R is the gas constant, and T is the temperature in Kelvin.

Step 2: Difference between \(K_p\) and \(K_c\)

\(K_p\) and \(K_c\) are both equilibrium constants but are expressed in different units. \(K_p\) is the equilibrium constant expressed in terms of partial pressures of the gaseous reactants and products, while \(K_c\) is the equilibrium constant expressed in terms of molar concentrations of the reactants and products.

Step 3: Convert between \(K_p\) and \(K_c\) in a gaseous reaction

In a gaseous reaction, we can convert between \(K_p\) and \(K_c\) using the relation \(K_p = K_c(RT)^{\Delta n}\), where \(\Delta n\) is the difference in the number of moles of gaseous products and reactants in the balanced chemical equation.

Step 4: Connecting the equation to \(K_p\) instead of \(K_c\)

The equation \(\Delta G^{\circ}=-RT\) ln \(K\) is applicable for reactions in the gas phase because of the dependence on pressure (and thereby the partial pressures of the reactants and products). In a gaseous reaction, the change in free energy is closely linked to pressure changes within the system. Hence, \(K_p\) (which uses partial pressures) is the appropriate equilibrium constant to use in this equation as it accounts for the pressure effects of gases on the change in standard free energy. In conclusion, the equation \(\Delta G^{\circ}=-RT\) ln \(K\) relates the value of \(K_p\), not \(K_c\), to the change in standard free energy for a reaction in the gas phase because it accounts for the pressure effects of gases on the change in standard free energy.

Key Concepts

Equilibrium Constant

The equilibrium constant of a reaction tells us about the concentration or pressure balance of reactants and products once the reaction reaches equilibrium. This balance can occur in either a gaseous or aqueous phase, and the equilibrium constant can be expressed in different ways, depending on the circumstances.

• **Equilibrium Constant (\(K\) notation):** In the context of gas phase reactions, the equilibrium constant often comes in two forms: \(K_p\) and \(K_c\).
• **\(K_c\) - Concentration Based:** Used for reactions where substances are measured in terms of their concentrations (moles per liter). Appropriate for reactions occurring in liquids or solutions.
• **\(K_p\) - Pressure Based:** Used when dealing with gases, as this format considers the partial pressures of each gas involved, making it more suitable for interpreting gas-phase equilibria.

Each of these constants serves different types of reactions and conditions, helping chemists understand how systems behave when they have reached equilibrium.

Gas Phase Reactions

Gas phase reactions are fascinating because they mainly involve substances in their gaseous state, which allows molecules to move freely. This freedom leads to a unique set of behaviors and calculations when predicting reaction outcomes.
Gas phase reactions often involve the balancing of partial pressures rather than concentrations.

• **Molecular Freedom:** Because gases fill the container they occupy, distributing molecules equally throughout. This means their interactions are highly dependent on temperature and pressure.
• **Pressure Dependent:** Given that gases can expand and compress, their behavior under different pressures must be considered. This is why \(K_p\) is preferred over \(K_c\) for equilibrium expressions involving gases.

These characteristics make calculations involving gas phase reactions fundamentally different, emphasizing the importance of pressure in understanding these systems.

Pressure Effects

Pressure significantly affects reactions that occur in the gaseous phase because gases respond swiftly to pressure changes. This is crucial in understanding why \(K_p\) is used instead of \(K_c\) when calculating the Gibbs Free Energy change for gas phase reactions.
Pressure is especially important because:

• **Partial Pressure Influence:** In gas reactions, molecules are constantly moving and colliding. The partial pressure of a gas affects how molecules interact and react, often influencing the rate and yield of a reaction.
• **Pressure Changes and Equilibrium:** As the pressure changes, it can shift the position of equilibrium, meaning the amount of product or reactant may increase or decrease.

When predicting how a reaction will behave under different conditions, pressure is a key factor that modifies the Gibbs Free Energy of the system, thus highlighting why \(K_p\) is applied to ensure the calculations incorporate these pressure dynamics.