Consider this reaction occurring at 298 K: BaCO₃(s) ⇌ BaO(s) + CO₂(g) c. Can the reaction be made more spontaneous by an increase or decrease in temperature? If so, at what temperature is the partial pressure of carbon dioxide 1.0 atm?

Verified step by step guidance

1. insert step 1> Calculate the standard Gibbs free energy change (\( \Delta G^\circ \)) for the reaction at 298 K using the equation \( \Delta G^\circ = \Delta H^\circ - T\Delta S^\circ \), where \( \Delta H^\circ \) is the standard enthalpy change and \( \Delta S^\circ \) is the standard entropy change.
2. insert step 2> Determine the sign of \( \Delta G^\circ \) to assess the spontaneity of the reaction at 298 K. If \( \Delta G^\circ < 0 \), the reaction is spontaneous; if \( \Delta G^\circ > 0 \), it is non-spontaneous.
3. insert step 3> To find the temperature at which the partial pressure of CO2 is 1.0 atm, use the equation \( \Delta G = \Delta G^\circ + RT \ln Q \), where \( Q \) is the reaction quotient. Set \( Q = 1 \) for \( P_{CO2} = 1.0 \) atm.
4. insert step 4> Rearrange the equation to solve for the temperature \( T \) when \( \Delta G = 0 \) (equilibrium condition), which gives \( T = \frac{\Delta H^\circ}{\Delta S^\circ} \).
5. insert step 5> Analyze how changes in temperature affect the spontaneity of the reaction by considering the signs of \( \Delta H^\circ \) and \( \Delta S^\circ \). If \( \Delta H^\circ > 0 \) and \( \Delta S^\circ > 0 \), increasing temperature makes the reaction more spontaneous.

Key Concepts

Le Chatelier's Principle

Le Chatelier's Principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change. In the context of this reaction, altering the temperature can affect the equilibrium position, favoring either the reactants or products depending on whether the reaction is exothermic or endothermic.

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. A reaction is spontaneous when the change in Gibbs Free Energy (ΔG) is negative. The relationship between ΔG, enthalpy (ΔH), and entropy (ΔS) is given by the equation ΔG = ΔH - TΔS, which is crucial for determining the spontaneity of the reaction at different temperatures.

Equilibrium Constant and Partial Pressure

The equilibrium constant (K) for a reaction at a given temperature is a measure of the ratio of the concentrations of products to reactants at equilibrium. For the reaction BaCO3(s) ⇌ BaO(s) + CO2(g), the partial pressure of CO2 is directly related to the equilibrium constant. At 1.0 atm of CO2, the temperature can be adjusted to find the specific conditions under which the reaction favors the formation of products, thus influencing the spontaneity of the reaction.