Animated Solution for Chemistry - Chemical Thermodynamics: The correct relationship between free energy change in a reaction and the corresponding equilibrium constant $K_c$ is

Imagine a ball rolling down a rugged hill. It will keep rolling until it finds the lowest possible point in the valley, where it finally comes to rest. In the world of chemistry, a reaction behaves much like that ball. It proceeds spontaneously as long as it can lower its free energy. The "valley" where it finally rests is what we call chemical equilibrium.

To understand this mathematically, we use the Reaction Isotherm Equation, which acts as the master bridge connecting the thermodynamic properties of a system to its current state. The equation is:

Here, $\Delta G$ is the Gibbs free energy change at any given moment, $\Delta G^{\circ }$ is the standard Gibbs free energy change (a fixed value for a specific reaction at a given temperature), $R$ is the universal gas constant, $T$ is the absolute temperature, and $Q$ is the reaction quotient.

The Equilibrium State

As the reaction proceeds, the concentrations of reactants and products change, which means $Q$ is constantly changing. Consequently, $\Delta G$ is also changing.

When the reaction finally reaches equilibrium (the bottom of the valley), two critical things happen: 1. The system can no longer do any useful work. The driving force is gone, meaning $\Delta G=0$. 2. The reaction quotient $Q$ reaches a constant value known as the equilibrium constant, $K_c$.

Let's substitute these equilibrium conditions back into our isotherm equation. Replacing $\Delta G$ with $0$ and $Q$ with $K_c$, we get:

Now, it's just a matter of simple algebra. We want to isolate the terms to match the standard conventions used in chemistry. By moving $\Delta G^{\circ }$ to the other side of the equation, we arrive at the beautiful and profound relationship:

This equation is incredibly powerful. It tells us that if we know the standard free energy change of a reaction (which we can calculate from tables of thermodynamic data), we can predict exactly where the equilibrium will lie! If $\Delta G^{\circ }$ is highly negative, $K_c$ will be very large, meaning the reaction strongly favors the products. Conversely, a positive $\Delta G^{\circ }$ means the reaction barely proceeds at all.