Question

Calculate the free energy change for this reaction at \(25^{\circ} \mathrm{C}\). Is the reaction spontaneous? (Assume that all reactants and products are in their standard states.) $$ \begin{array}{c} 2 \mathrm{Ca}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CaO}(s) \\ \Delta H_{\mathrm{rxn}}^{\circ}=-1269.8 \mathrm{~kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-364.6 \mathrm{~J} / \mathrm{K} \end{array} $$

Short answer

The free energy change for the reaction at \(25^\text{o}C\) is \(-1161.15 \text{ kJ}.\) The reaction is spontaneous as indicated by the negative \(\triangle G_{\text{rxn}}^\text{o}\).

Step-by-step solution

Convert temperature to Kelvin

First, convert the temperature from Celsius to Kelvin using the formula: \( T(K) = T(^\text{o}C) + 273.15 \). For a temperature of \(25^\text{o}C\), the Kelvin temperature is \(25 + 273.15 = 298.15 K\).

Convert the entropy change to kJ/K

Convert the entropy change \( \triangle S_{\text{rxn}}^\text{o} \) from J/K to kJ/K by dividing by 1000. This converts \(-364.6 J/K\) to \(-0.3646 kJ/K\).

Calculate the free energy change

Use the Gibbs free energy equation: \( \triangle G_{\text{rxn}}^\text{o} = \triangle H_{\text{rxn}}^\text{o} - T\triangle S_{\text{rxn}}^\text{o} \). Plug in the values to get: \( \triangle G_{\text{rxn}}^\text{o} = (-1269.8 \text{ kJ}) - (298.15 \text{ K})(-0.3646 \text{ kJ/K}) \).

Perform the calculation

Now, perform the arithmetic to find the free energy change: \( \triangle G_{\text{rxn}}^\text{o} = -1269.8 \text{ kJ} + (298.15 \text{ K})(0.3646 \text{ kJ/K}) = -1269.8 \text{ kJ} + 108.65 \text{ kJ} = -1161.15 \text{ kJ} \).

Determine the spontaneity of the reaction

A negative value for \( \triangle G_{\text{rxn}}^\text{o} \) indicates that the reaction is spontaneous under standard conditions.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, energy, and work. It is particularly concerned with how energy is transferred within systems and how those energy transformations can do work. One of the key components of thermodynamics is the study of the laws of energy conservation and how they pertain to thermal processes.

The field of thermodynamics lays the groundwork for understanding how energy conversions can inform whether a chemical reaction will occur spontaneously. For instance, thermodynamics tells us that in an isolated system (one that does not exchange matter or energy with its surroundings), energy can neither be created nor destroyed. This means the total energy before and after a reaction must be the same, leading to calculations of energy changes such as enthalpy and entropy changes during reactions.

Spontaneity of Reactions

The spontaneity of a reaction refers to the natural tendency of a reaction to occur without being driven by an external force. A spontaneous reaction can proceed on its own once it has been initiated. To determine whether a reaction is spontaneous, scientists use a thermodynamic quantity called Gibbs free energy, represented by the symbol \( G \).

The change in Gibbs free energy (\( \Delta G \)) during a reaction indicates the spontaneity of the process. If \(-\Delta G > 0\), the reaction is exergonic and occurs spontaneously under the given conditions. If \(-\Delta G < 0\), the reaction is endergonic and is not spontaneous. The conditions typically refer to standard temperature and pressure, but the concept can be applied to any set of environmental conditions.

Standard Enthalpy Change

In thermodynamics, the standard enthalpy change (\( \Delta H^\circ \) of a reaction represents the heat absorbed or released when a reaction occurs at standard conditions, which typically means 1 bar pressure and a specified temperature, usually 25 degrees Celsius or 298.15 K. It is an intrinsic property of the reaction and does not depend on the pathway taken.

For a reaction such as the formation of calcium oxide from calcium and oxygen, the standard enthalpy change is negative (\( \Delta H_{\text{rxn}}^\circ = -1269.8 \text{ kJ} \)), reflecting that the process is exothermic – it releases heat to the surroundings. This release of energy is one component that is considered in evaluating whether a reaction will be spontaneous.

Standard Entropy Change

Entropy is a measure of the disorder or randomness of a system. The standard entropy change (\( \Delta S^\circ \)) for a reaction indicates how the disorder changes as reactants are converted into products under standard conditions. An increase in entropy means the system has become more disordered, while a decrease means it has become more ordered.

During the formation of calcium oxide from calcium and oxygen, as illustrated in the given exercise, the standard entropy change is negative, signifying a decrease in disorder (\( \Delta S_{\text{rxn}}^\circ = -364.6 \text{ J/K} \)). Despite the decrease in disorder, the reaction's standard enthalpy change is significantly exothermic. This signifies that the energy released due to bond formation outweighs the decrease in entropy, favoring the reaction's spontaneity under standard conditions.