The Relationship Between $ \Delta G^\circ $ and $ K $
Standard Gibbs Free Energy Change ($ \Delta G^\circ $) and Its Significance
- As discussed in Reactivity 1.4, the Gibbs free energy change ($ \Delta G $) is a thermodynamic quantity that determines whether a reaction is spontaneous.
- A spontaneous reaction proceeds without requiring external energy input.
- $ \Delta G< 0 $: The reaction is spontaneous in the forward direction.
- $ \Delta G >0 $: The reaction is non-spontaneous in the forward direction (but spontaneous in reverse).
- $ \Delta G = 0 $:The system is at equilibrium.
When we discuss $ \Delta G^\circ $, we refer to the Gibbs free energy change under standard conditions: 1 atm pressure for gases, 1 $mol \, dm^{-3}$ concentration for solutions, and a specified temperature (usually 298 K unless otherwise stated).
But how does $ \Delta G^\circ $ relate to the equilibrium constant ($ K $)?
The Equation Relating $ \Delta G^\circ $ and $ K $
- The connection between $ \Delta G^\circ $ and $ K $ is expressed mathematically as: $$
\Delta G^\circ = -RT \ln K
$$ where:- $ \Delta G^\circ $: Standard Gibbs free energy change (in $\text{J mol}^{-1}$).
- $ R $: Gas constant ($ 8.31 \, \mathrm{J \, mol^{-1} \, K^{-1}} $).
- $ T $: Absolute temperature in Kelvin.
- $ K $: Equilibrium constant (unitless).
- This equation links thermodynamics ($ \Delta G^\circ $) to equilibrium ($ K $) and provides critical insights into reaction behavior:
- When $ \Delta G^\circ < 0 $:
- The reaction is product-favored.
- $ K > 1 $, indicating that the equilibrium lies toward the products.
- When $ \Delta G^\circ > 0 $:
- The reaction is reactant-favored.
- $ K < 1 $, indicating that the equilibrium lies toward the reactants.
- When $ \Delta G^\circ = 0 $:
- The system is at equilibrium.
