Question

Calculate \(\Delta G_{\mathrm{rmn}}^{\circ}\) for the reaction: $$\mathrm{CaCO}_{3}(s) \longrightarrow \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$$ Use the following reactions and given \(\Delta G_{\mathrm{m} \mathrm{v}}^{\circ}\) values: $$\mathrm{Ca}(s)+\mathrm{CO}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{CaCO}_{3}(s) \quad \Delta G_{\mathrm{mn}}^{\circ}=-734.4 \mathrm{kJ}$$ $$2 \mathrm{Ca}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CaO}(s) \quad \Delta G_{\mathrm{ron}}^{\circ}=-1206.6 \mathrm{kJ}$$

Short answer

ΔG° for the reaction CaCO3(s) → CaO(s) + CO2(g) is 131.1 kJ.

Step-by-step solution

Write the Given Reactions and Their Delta G Values

Two reactions are given with their standard Gibbs free energy change (ΔG°) values: 1) Ca(s) + CO2(g) + 1/2 O2(g) → CaCO3(s) with ΔG° = -734.4 kJ2) 2 Ca(s) + O2(g) → 2 CaO(s) with ΔG° = -1206.6 kJWe need to manipulate these reactions to find the ΔG° for the reaction: CaCO3(s) → CaO(s) + CO2(g).

Reverse Reaction 1 and Divide Reaction 2

To get to the desired reaction, we need to reverse reaction 1 and divide reaction 2 by half since we only need one mole of CaO. This changes their Delta G° signs and values:1) CaCO3(s) → Ca(s) + CO2(g) + 1/2 O2(g) with ΔG° = +734.4 kJ2) Ca(s) + 1/2 O2(g) → CaO(s) with ΔG° = -1206.6 kJ / 2 = -603.3 kJ

Add Reactions to Get the Desired Reaction

By adding the manipulated versions of reactions 1 and 2, we can cancel out Ca(s) and 1/2 O2(g), leaving us with the desired reaction:CaCO3(s) → CaO(s) + CO2(g)Now we can add the ΔG° values for these reactions to get the ΔG° for the desired reaction.

Calculate the Standard Gibbs Free Energy Change for the Desired Reaction

Adding the ΔG° of reversed reaction 1 and half of reaction 2 gives us:ΔG°(desired reaction) = ΔG°(reaction 1 reversed) + ΔG°(half of reaction 2)ΔG°(desired reaction) = (+734.4 kJ) + (-603.3 kJ)ΔG°(desired reaction) = 131.1 kJ

Key concepts

Chemical Thermodynamics

Understanding the flow of energy within chemical reactions is a fundamental aspect of chemical thermodynamics. At the heart of this branch of science is the study of how energy is transformed, especially in the form of heat and work during chemical and physical processes. A key component of this is the concept of Gibbs free energy, denoted as \( \Delta G \), which combines the system’s enthalpy (\( \Delta H \))—or heat content—with its entropy (\( \Delta S \))—a measure of disorder or randomness.

Gibbs free energy is particularly important because it helps predict the direction in which a chemical reaction will spontaneously proceed. The equation \( \Delta G = \Delta H - T\Delta S \) succinctly expresses the relationship between these quantities, where \( T \) stands for the temperature in Kelvin. A negative value of \( \Delta G \) indicates a reaction that can occur spontaneously, releasing energy, while a positive value shows that energy must be added for the reaction to take place.

Reaction Spontaneity

The spontaneity of a reaction is an essential concept when predicting whether a chemical reaction can occur by itself without the input of additional energy from its surroundings. Spontaneity is often assessed using the Gibbs free energy formula mentioned earlier. If \( \Delta G \) is negative, the reaction is spontaneous; if it's positive, the reaction is non-spontaneous.

Understanding \( \Delta G \) for Reaction Spontaneity

The value of \( \Delta G \) for a reaction is derived from standard Gibbs free energy values of reactants and products under standard conditions. By calculating \( \Delta G \) as shown in the exercise, we can predict whether the decomposition of \( \mathrm{CaCO}_3 \) to \( \mathrm{CaO} \) and \( \mathrm{CO}_2 \) is spontaneous. This calculation is critical in processes like the production of lime, where the spontaneity of \( \mathrm{CaCO}_3 \) decomposition is a key factor.

Enthalpy and Entropy

In the context of chemical reactions, enthalpy and entropy are pivotal in determining the energetic feasibility of a reaction.

Enthalpy (\( \Delta H \))

Enthalpy change is the heat absorbed or released under constant pressure during a reaction. It's a reflection of the energy content of a system. Reactions that give off heat, like combustion processes, have negative enthalpy changes and are termed exothermic. Conversely, those that absorb heat have positive enthalpy changes and are called endothermic.

Entropy (\( \Delta S \))

Entropy is a thermodynamic function that measures the degree of disorder in a system. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. Thus, a reaction is more likely to be spontaneous if it increases the entropy of the universe. This concept is exemplified in reactions where solid compounds break down into gases, typically resulting in an increase in entropy due to the higher disorder of the gaseous state.

Together, enthralpy and entropy can predict the direction of a chemical reaction—as seen with the \( \Delta G \) equation. Chemical thermodynamics intertwines these concepts to understand and predict the inherent energy changes and the feasibility of reactions in a broader sense.