The equilibrium constant for the reaction given below is \(2.0 \times 10^{-7}\) at \(300 \mathrm{~K}\). Calculate the standard free energy change for the reaction; $$ \mathrm{PCl}_{5(\mathrm{~g})} \rightleftharpoons \mathrm{PCl}_{3(\mathrm{~g})}+\mathrm{Cl}_{2(\mathrm{~g})} $$ Also, calculate the standard entropy change if \(\Delta H^{\circ}=28.40 \mathrm{~kJ} \mathrm{~mol}^{-1}\).

Thermodynamics is a field of physics that deals with the relationships and conversions between heat and other forms of energy. Specifically, it focuses on how thermal energy is transformed and how it affects matter. The three main laws of thermodynamics describe how energy moves and changes form, whether in a steam engine, a living cell, or a chemical reaction.

One of the central concepts in thermodynamics is that of the system, which refers to the specific portion of the universe under consideration. Everything outside of the system is called the surroundings. In studying chemical reactions, we consider the system to be the reactants and products, with the surroundings being everything else, including the container holding the reactants.

In thermodynamics, we are often interested in how energy changes during a reaction and what these changes tell us about the reaction's tendency to occur or its spontaneity. A reaction's spontaneity is influenced by changes in enthalpy (heat content), entropy, and temperature—concepts that are all intertwined with the Gibbs free energy.

Entropy is a measure of the degree of disorder or randomness in a system. The second law of thermodynamics states that in any spontaneous process, the total entropy of the universe increases. This law implies that entropy can serve as an indicator of the spontaneity of a process.

The standard entropy change (\(\Delta S^\circ\)) of a reaction is the entropy change that occurs when all reactants and products are in their standard states. A standard state is typically defined as the pure form of a substance at 1 atmosphere of pressure and a specified temperature, usually 298.15 K.

The calculation of standard entropy change in a chemical reaction involves considering the entropy of the products and reactants. If the products are more disordered than the reactants, the entropy change (\(\Delta S^\circ\)) will be positive, signifying an increase in disorder, which often corresponds to a spontaneous process at a given temperature if not opposed by enthalpy changes.

The equilibrium constant (\(K\)) for a chemical reaction at a given temperature is a measure of how far the reaction proceeds before reaching a state of balance or equilibrium. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time.

For a general reaction, the equilibrium constant is defined as the product of the concentrations of the products raised to the power of their stoichiometric coefficients divided by the product of the concentrations of the reactants raised to the power of their coefficients. This ratio shows the relative proportions of products to reactants at equilibrium.

Calculating the equilibrium constant can tell us a lot about the reaction's properties. A large equilibrium constant (significantly greater than 1) indicates that the reaction heavily favors the production of products, suggesting a higher product concentration at equilibrium. In contrast, a small equilibrium constant (significantly less than 1) indicates that reactants are favored.

Gibbs free energy (\(G\)), named after Josiah Willard Gibbs, is a thermodynamic quantity that helps predict the spontaneity of a process at constant pressure and temperature. It combines the concepts of enthalpy (\(H\)) and entropy (\(S\)) into a single value (\(G = H - TS\)), where \(T\) is the temperature in Kelvins. If the change in free energy (\(\Delta G\)) for a process is negative, the process is spontaneous; if positive, it is non-spontaneous; and if zero, the system is at equilibrium.

The calculation of Gibbs free energy change (\(\Delta G^\circ\)) for a reaction using the equilibrium constant is a powerful tool. It allows us to predict the direction of a chemical reaction under standard conditions. A negative \(\Delta G^\circ\) value indicates a process that can occur spontaneously under standard conditions, while a positive value would require an input of energy to proceed.

Understanding Gibbs free energy is crucial for various applications in chemical thermodynamics, including the calculation of battery potentials, predicting reaction spontaneity, and understanding biological energy transformations.