Question

The Gibbs free energy change ($\Delta G^\circ$) and equilibrium constant (K) for a chemical reaction are related by

• A. \(\Delta G^\circ = RT \ln K\)
• B. \(\Delta G^\circ = T \ln K\)
• C. \(\Delta G^\circ = RTK\)
• D. \(\Delta G^\circ = -RT \ln K\)

Hint:

Remember the negative sign in the relationship \(\Delta G^\circ = -RT \ln K\). This signifies that a spontaneous reaction (\(\Delta G^\circ < 0\)) is associated with an equilibrium constant greater than 1 (\(\ln K > 0\)), favoring product formation.

Answer 1

The Correct Option is D

Step 1: Understand the concepts of Gibbs free energy change and equilibrium constant.
Gibbs Free Energy Change (\(\Delta G^\circ\)): This thermodynamic potential measures the maximum (or reversible) work that may be performed by a thermodynamic system at a constant temperature and pressure. The superscript \(\circ\) indicates standard conditions. A negative \(\Delta G^\circ\) indicates a spontaneous reaction under standard conditions, a positive \(\Delta G^\circ\) indicates a non-spontaneous reaction, and \(\Delta G^\circ = 0\) indicates that the reaction is at equilibrium under standard conditions. Equilibrium Constant (K): This constant expresses the ratio of products to reactants at equilibrium for a reversible chemical reaction at a given temperature. It indicates the extent to which a reaction will proceed to completion. A large \(K\) indicates that the equilibrium lies to the right (favoring products), a small \(K\) indicates that the equilibrium lies to the left (favoring reactants), and \(K = 1\) indicates that the concentrations of reactants and products are roughly equal at equilibrium.
Step 2: Recall the relationship between \(\Delta G^\circ\) and \(K\).
The standard Gibbs free energy change (\(\Delta G^\circ\)) and the equilibrium constant (\(K\)) are thermodynamically related by the following equation: $$\Delta G^\circ = -RT \ln K$$ Where:
\(\Delta G^\circ\) is the standard Gibbs free energy change.
\(R\) is the ideal gas constant (approximately 8.314 J/(mol·K)).
\(T\) is the absolute temperature in Kelvin.
\(\ln K\) is the natural logarithm of the equilibrium constant.

Step 3: Compare the recalled relationship with the given options.
By comparing the derived relationship with the given options, we can see that option (4) matches the correct thermodynamic equation. Option (1) has a positive sign, which is incorrect.
Option (2) is missing the ideal gas constant \(R\), making it dimensionally incorrect.
Option (3) uses \(K\) instead of \(\ln K\), which is incorrect.
Option (4) \(\Delta G^\circ = -RT \ln K\) is the correct relationship.