Question

Explain qualitatively how \(\Delta G\) changes for each of the following reactions as the partial pressure of \(\mathrm{O}_{2}\) is increased: (a) \(2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)\) (b) \(2 \mathrm{H}_{2} \mathrm{O}_{2}(l) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{O}_{2}(g)\) (c) \(2 \mathrm{KClO}_{3}(s) \longrightarrow 2 \mathrm{KCl}(s)+3 \mathrm{O}_{2}(g)\)

Short answer

For each reaction: (a) As the partial pressure of \(O_{2}\) increases, the reaction will shift towards the products, making it more spontaneous and causing \(\Delta G\) to decrease. (b) With an increased partial pressure of \(O_{2}\), the reaction will shift towards the reactants, resulting in a decrease in spontaneity and an increase in \(\Delta G\). (c) Similar to reaction (b), increasing the partial pressure of \(O_{2}\) will shift the reaction towards the reactants, leading to a decrease in spontaneity and an increase in \(\Delta G\).

Step-by-step solution

ΔG is the Gibbs free energy change, which can be used to determine the spontaneity of a reaction. Mathematically, ΔG is related to the equilibrium constant (K) and the reaction quotient (Q) by the equation: \[ ΔG = ΔG^⦵ + RT\ln(Q) \] where ΔG^⦵ is the standard free energy change, R is the gas constant, and T is the temperature in Kelvin. As the partial pressure of O₂ changes, so will Q, causing ΔG to vary. We will apply Le Chatelier's principle, which states that when a system at equilibrium is subjected to a change in pressure, temperature, or concentration, it will shift to counteract the change and restore equilibrium. In this case, we will focus on the effect of increased partial pressure of O₂ on each reaction. #Step 2: Analyze reaction (a) with increased partial pressure of O₂#

The first reaction is given by: \[ 2CO(g) + O₂(g) \longrightarrow 2CO₂(g) \] In this reaction, as we increase the partial pressure of O₂ (a reactant), the reaction will try to counteract the change and restore equilibrium, according to Le Chatelier's principle. This leads to a shift in the reaction towards the products (CO₂). As a result, this reaction will become more spontaneous, meaning that ΔG will decrease. #Step 3: Analyze reaction (b) with increased partial pressure of O₂#

The second reaction is given by: \[ 2H₂O₂(l) \longrightarrow 2H₂O(l) + O₂(g) \] In this reaction, increasing the partial pressure of O₂ (a product) leads to a shift in the reaction in order to counteract the change. According to Le Chatelier's principle, the reaction will shift towards the reactants (H₂O₂) in order to restore equilibrium. Consequently, this reaction will become less spontaneous, meaning that ΔG will increase. #Step 4: Analyze reaction (c) with increased partial pressure of O₂#

The third reaction is given by: \[ 2KClO₃(s) \longrightarrow 2KCl(s) + 3O₂(g) \] In this reaction, increasing the partial pressure of O₂ (a product) also causes a shift back towards the reactants (KClO₃), as predicted by Le Chatelier's principle. Therefore, the reaction will become less spontaneous, implying an increase in ΔG.

Key concepts

Le Chatelier's Principle

Le Chatelier's Principle is essential in understanding how a system at equilibrium reacts to changes in conditions, such as pressure, concentration, or temperature. The principle states that if an external change is applied to a system in equilibrium, the system adjusts to partially counteract the change, ultimately restoring equilibrium. This results in a shift in the position of the equilibrium:

• If a reactant is added, or its partial pressure is increased, the system shifts towards the products to reduce the change.
• If a product is added or its pressure is increased, the equilibrium shifts towards the reactants.
• The principle also applies to changes in concentration and temperature, following similar logic.

In the context of the original exercise, as the partial pressure of oxygen (\(\mathrm{O}_{2}\)) is increased, Le Chatelier's Principle helps predict the direction that the reactions will shift to balance the change. Understanding this principle allows us to anticipate whether a reaction becomes more or less spontaneous when pressure conditions are altered.

Equilibrium Constant

The Equilibrium Constant (denoted as \(K\)) is a numerical value that describes the ratio of concentrations of products to reactants at equilibrium for a reversible reaction. It's an intrinsic property of the reaction at a given temperature:

• If \(K > 1\), the equilibrium position favors the products.
• If \(K < 1\), the equilibrium position favors the reactants.

Mathematically, the equilibrium constant can be expressed in terms of partial pressures (for gases) or concentrations (for solutions). However, it is crucial to note that changing the conditions of a reaction (such as pressure or temperature) does not change \(K\) unless temperature is altered.
The original exercise requires us to understand how the shift in equilibrium affects the Gibbs free energy (\(\Delta G\)), yet this balance itself remains constant unless external temperature changes are introduced.

Reaction Quotient

The Reaction Quotient (\(Q\)) is a measure that helps us understand the current state of a reaction in comparison to its equilibrium state. \(Q\) is calculated in a similar way to the equilibrium constant (\(K\)), but it uses the initial concentrations or partial pressures.

• If \(Q < K\), the reaction will proceed in the forward direction, towards the products, to reach equilibrium.
• If \(Q > K\), the reaction tends to proceed in the reverse direction, towards the reactants.
• If \(Q = K\), the system is already at equilibrium.

In the exercise, as the partial pressure of \(\mathrm{O}_{2}\) changes, \(Q\) will shift accordingly. This shift is indicative of how the reaction moves towards or away from equilibrium, which directly influences the Gibbs free energy (\(\Delta G\)), making the reaction more or less spontaneous depending on the direction of the shift.