Find out whether it is possible to reduce \(\mathrm{MgO}\) using carbon at \(298 \mathrm{~K}\). If not, at what temperature it becomes spontaneous. For reaction, \(\mathrm{MgO}(s)+\mathrm{C}(s) \longrightarrow \mathrm{Mg}(s)+\mathrm{CO}(g), \Delta H^{\circ}=+491.18 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(\Delta S^{0}=197.67 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\)

Understanding the spontaneity of chemical reactions is crucial in predicting whether a reaction will occur under a set of conditions. A reaction is spontaneous if it occurs without being driven by some external force. The Gibbs Free Energy (abla G) formula provides insight into this spontaneity. For a reaction at constant temperature and pressure, if abla G is less than zero (abla G < 0), the process is considered spontaneous. On the other hand, a positive abla G (> 0) indicates a non-spontaneous reaction, implying it needs external energy to proceed. At equilibrium, abla G equals zero, and no net reaction occurs; the forward and reverse reactions occur at the same rate.

The exercise aimed at determining the spontaneity of reducing magnesium oxide (MgO) with carbon. The calculation of Gibbs Free Energy at standard conditions (298 K) with the given enthalpy (H^{0}) and entropy (S^{0}) values allowed us to assess whether the reaction is spontaneous at this temperature. By resolving the Gibbs formula with abla G = abla H - Tabla S, we can infer the spontaneity based on the sign of the resultant abla G.

In thermodynamics, entropy (abla S) and enthalpy (abla H) are fundamental concepts that explain the energy changes in chemical reactions. Entropy relates to the degree of disorder or randomness in a system; it is a measure of the number of specific ways a system can be arranged. Generally, natural processes tend to increase the entropy of the universe.

Enthalpy, on the other hand, is a measure of the total energy of a thermodynamic system, including both internal energy and the energy required to displace its environment to make room for its volume. A positive change in enthalpy (abla H > 0) indicates that a reaction is endothermic, absorbing heat from its surroundings. Conversely, a negative abla H (< 0) signifies an exothermic reaction, where energy is released as heat to the surroundings.

The given exercise provides the values of both abla H and abla S for the reaction between magnesium oxide and carbon. By considering these values within the Gibbs Free Energy equation, we were able to calculate how they influence the reaction's spontaneity.

Chemical equilibrium is a state in a reversible chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no overall change in the concentrations of reactants and products over time. At equilibrium, the Gibbs Free Energy of the system is at its minimum, and thus the free energy change (abla G) for the reaction is zero.

The magnitude of the equilibrium constant (K_{eq}) provides insight into the position of equilibrium. A large K_{eq} value indicates a reaction that favors the formation of products, while a small K_{eq} value suggests a reaction favoring reactants. In the context of our exercise, by setting the Gibbs Free Energy equation to zero and solving for the temperature (T), we effectively find the temperature at which the system would reach equilibrium for the given reaction if it could achieve equilibrium at all.

The principles of thermodynamics are the foundation of physical chemistry, explaining the energy changes and exchanges that occur during chemical reactions and phase transitions. Among these, the first and second laws of thermodynamics govern the concepts of energy conservation and entropy, respectively.

In our exercise, thermodynamics helps predict the conditions under which the reduction of MgO by carbon would spontaneously occur. By applying the equations and concepts of thermodynamics, such as the calculation of Gibbs Free Energy, we determined not only the spontaneous nature of the reaction at a given temperature but also the theoretical temperature at which the reaction would become spontaneous, based on the enthalpy and entropy values provided.

The exercise guides students to leverage these thermodynamics concepts to assess reaction spontaneity and equilibrium, enhancing their understanding of how energy and matter interact in chemical processes.