Derive the equation $$ \Delta G=R T \ln (Q / K) $$ where \(Q\) is the reaction quotient and describe how you would use it to predict the spontaneity of a reaction.

The reaction quotient, represented by the symbol Q, is a measure of the relative concentrations or partial pressures of the reactants and products at any point during a reaction. It's calculated using the same formula as the equilibrium constant K, but for a system that is not necessarily at equilibrium. The general form of the reaction quotient is given by:
\[ Q = \frac{\text{Products}}{\text{Reactants}} \]
The expression uses the molar concentrations (for solutions) or partial pressures (for gases) of the substances involved in the reaction, raised to the power corresponding to their coefficients in the balanced chemical equation. For a reaction at equilibrium, Q equals the equilibrium constant, K, indicating that the rates of the forward and reverse reactions are equal. However, if the system is not at equilibrium, Q can provide insights into predicting the direction in which the reaction will proceed to reach equilibrium.

The equilibrium constant, denoted as K, is a fundamental aspect of chemical thermodynamics that relates to the concentrations of reactants and products in a chemical reaction at the state of equilibrium. Unlike Q, the value of K is constant for a given reaction at a specified temperature, and its expression is:
\[ K = \frac{[\text{Products}]^{\text{coefficients}}}{[\text{Reactants}]^{\text{coefficients}}} \]
The equilibrium constant offers substantial information regarding the proportion of reactants and products. A larger value of K indicates that, at equilibrium, the reaction favors the formation of products, while a smaller K value suggests that reactants are more favored. Understanding the magnitude of K is crucial for predicting the predominant species in an equilibrium mixture, and it remains unchanged unless the temperature of the system changes.

Predicting the spontaneity of a chemical reaction is a significant utilization of the Gibbs free energy equation. Spontaneity indicates whether a reaction can occur without the input of external energy. The criterion for spontaneity is based on the sign of the Gibbs free energy change (ΔG):
- If ΔG is negative (ΔG < 0), the reaction is spontaneous, as it will proceed forward to form products.
- If ΔG is positive (ΔG > 0), the reaction is non-spontaneous, and will not proceed without external energy input.
- If ΔG is zero (ΔG = 0), the system is at equilibrium and the reaction is reversible.
The equation \(\Delta G = R T \ln (Q / K)\) helps in calculating ΔG, which is pivotal for predicting whether a reaction will spontaneously move towards equilibrium. By analyzing Q relative to K, we can determine the direction of spontaneity. When Q < K, the reaction will spontaneously proceed towards product formation to establish equilibrium (ΔG < 0). If Q > K, the reaction will proceed in the reverse direction, converting products back into reactants to achieve equilibrium (ΔG > 0). This assessment serves as a practical approach for chemists to anticipate the outcome of chemical reactions.