Problem 77 - Chemistry: The Central Science

Question

Indicate whether \(\Delta G\) increases, decreases, or stays the same 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) \(2 \mathrm{KClO}_{3}(s) \longrightarrow 2 \mathrm{KCl}(s)+3 \mathrm{O}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

For the given reactions, increasing the partial pressure of \(O_{2}\) has the following effect on the Gibbs free energy change, \(\Delta G\): (a) \(\Delta G\) increases (b) \(\Delta G\) decreases (c) \(\Delta G\) decreases

Step 1: (Step 1: Review Gibbs free energy change and equilibrium constant relationship)

Recall that the relationship between the Gibbs free energy change, reaction quotient (Q), and the equilibrium constant (K) is given by: \[\Delta G = \Delta G^{\circ} + RT \ln Q\] Where: - \(\Delta G\) is the Gibbs free energy change - \(\Delta G^{\circ}\) is the standard Gibbs free energy change - R is the gas constant - T is the absolute temperature - Q is the reaction quotient At equilibrium, \(\Delta G = 0\) , and \(Q = K\) (the equilibrium constant).

Step 2: (Step 2: Effect of partial pressure changes on reaction quotient Q)

The reaction quotient, Q, is calculated using partial pressures of products and reactants. Thus, changes in partial pressure of any reactant or product can alter Q, which in turn can affect \(\Delta G\). To find the effect of increasing the partial pressure of \(\mathrm{O}_{2}\), we need to see how Q changes for each reaction and determine the effect of that change.

Step 3: (Step 3: Reaction (a))

For reaction (a): \(2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)\) The reaction quotient, Q, can be expressed as: \[Q = \frac{[\mathrm{CO}_{2}]^{2}}{[\mathrm{CO}]^{2}[\mathrm{O}_{2}]}\] When the partial pressure of \(\mathrm{O}_{2}\) is increased, the denominator of Q increases, resulting in a decrease in Q. Since Q is decreased, \(\Delta G\) will increase (using the relationship mentioned in Step 1).

Step 4: (Step 4: Reaction (b))

For reaction (b): \(2 \mathrm{H}_{2} \mathrm{O}_{2}(l) \longrightarrow 2 \mathrm{H}_{2}\mathrm{O}(l)+\mathrm {O}_{2}(g)\) The reaction quotient, Q, can be expressed as: \[Q = \frac{[\mathrm{O}_{2}]}{[\mathrm{H}_{2} \mathrm{O}_{2}]^{2}}\] When the partial pressure of \(\mathrm{O}_{2}\) is increased, the numerator of Q increases, resulting in an increase in Q. Since Q is increased, \(\Delta G\) will decrease.

Step 5: (Step 5: Reaction (c))

For reaction (c): \(2 \mathrm{KClO}_{3}(s) \longrightarrow 2 \mathrm{KCl}(s)+3\mathrm{O}_{2}(g)\) The reaction quotient, Q, can be expressed as: \[Q = \frac{[\mathrm{O}_{2}]^{3}}{1}\] When the partial pressure of \(\mathrm{O}_{2}\) is increased, the numerator of Q increases, resulting in an increase in Q. Since Q is increased, \(\Delta G\) will decrease.

Step 6: (Conclusion)

The effect of increasing partial pressure of \(\mathrm{O}_{2}\) on \(\Delta G\) for each reaction is as follows: (a) \(\Delta G\) increases (b) \(\Delta G\) decreases (c) \(\Delta G\) decreases

Key Concepts

Reaction Quotient

The reaction quotient, often denoted as \( Q \), is a numerical value that mirrors the relative concentrations of products and reactants for a given reaction at any state. It is calculated in a similar manner to the equilibrium constant \( K \), but applicable at any point during the reaction process. The formula for \( Q \) involves the partial pressures of the gases involved:

• For a reaction \( aA + bB \rightarrow cC + dD \), the reaction quotient is expressed as: \[ Q = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
• The values for \( [C] \), \( [D] \), \( [A] \), and \( [B] \) are the partial pressures or concentrations.

When \( Q \) is compared with \( K \), we get insight into the direction a reaction must proceed to reach equilibrium:

• If \( Q < K \), the reaction proceeds in the forward direction to produce more products.
• If \( Q > K \), the reaction moves in the reverse direction to form more reactants.
• If \( Q = K \), the system is at equilibrium, and no net change occurs.

Understanding \( Q \) plays a crucial role when analyzing the impact of changing conditions on a reaction and predicting if \( \Delta G \) will shift.

Equilibrium Constant

The equilibrium constant, represented as \( K \), provides insight into the ratio of concentrations of products to reactants at equilibrium for a reversible reaction. It is a constant value at a particular temperature and helps in predicting the extent of a reaction. In mathematical terms, for the reaction \( aA + bB \rightleftharpoons cC + dD \):

• \( K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \)

Unlike the reaction quotient \( Q \), \( K \) is always constant at a given temperature, thus exclusively represents the reaction at equilibrium. Different reactions have different \( K \) values, indicating their favorability:

• A large \( K >> 1 \) signifies a reaction where products are heavily favored at equilibrium.
• A small \( K << 1 \) indicates that reactants are favored.
• \( K = 1 \) implies appreciable amounts of both reactants and products.

Changes in temperature can shift \( K \), altering the balance of products and reactants at equilibrium. However, changes in pressure or concentrations impact \( Q \) and the position of equilibrium but not \( K \).

Partial Pressure

Partial pressure refers to the pressure exerted by an individual gas component in a mixture. In the context of chemical reactions, partial pressures are crucial when gaseous reactants or products are involved. Each gas in a mixture contributes to the total pressure based on its proportion in the mixture.

• Dalton's Law of Partial Pressures states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of all the individual gases.
• The partial pressure \( P_i \) of a gas \( i \) in a mixture is given by: \[ P_i = \frac{n_i}{n_{total}} \times P_{total} \]
• Where \( n_i \) is the number of moles of gas \( i \), and \( n_{total} \) is the total number of moles of all gases present.

Partial pressure changes have a direct impact on the reaction quotient \( Q \). For instance, increasing the partial pressure of a reactant in the denominator of \( Q \) will decrease \( Q \), affecting the Gibbs free energy \( \Delta G \). Hence, understanding partial pressures is essential for predicting the direction and extent of a reaction under non-equilibrium conditions.

Standard Gibbs Free Energy Change

The standard Gibbs free energy change, symbolized as \( \Delta G^{\circ} \), is a critical concept in thermodynamics. It indicates the available energy to do work under standard conditions, typically 1 atm pressure and 298 K temperature. This variable helps predict whether a reaction will occur spontaneously.

• A negative \( \Delta G^{\circ} \) signals a spontaneous reaction under standard conditions.
• If \( \Delta G^{\circ} \) is positive, the reaction is non-spontaneous.
• \( \Delta G^{\circ} = 0 \) means the system is at equilibrium.

The relationship between \( \Delta G \), the standard Gibbs free energy, the reaction quotient \( Q \), and the absolute temperature \( T \) is given by the equation: \[ \Delta G = \Delta G^{\circ} + RT \ln Q \] This expression highlights that the actual \( \Delta G \) changes based on the reaction conditions (\( Q \)). Here, \( R \) is the gas constant, and \( T \) is the absolute temperature. Thus, changes in \( Q \) due to varying partial pressures or concentrations directly affect the Gibbs free energy change from its standard state, allowing us to understand energetic feasibility beyond ideal conditions.