Thermodynamic Criterion for Equilibrium
The equilibrium constant \(K\) and standard Gibbs energy change \(\Delta G^\circ\) are quantitatively linked by \(\Delta G^\circ = -RT\ln K\). Dynamic equilibrium is reached at the minimum of Gibbs energy where \(\Delta G = 0\).
The Key Equations
At any point during a reaction (not necessarily at equilibrium):
At equilibrium, ΔG = 0 and Q = K, so:
What Does the Sign of ΔG° Tell Us?
ΔG°, K, and Spontaneity
Worked Example: Calculating K from ΔG°
Calculating Equilibrium Constant K from ΔG°
Problem: At 25 °C, ΔG° = −4.38 kJ mol⁻¹ for a reaction. Calculate K.
Step 1: Convert T to Kelvin: T = 25 + 273.15 = 298.15 K
Step 2: Convert ΔG° to J: ΔG° = −4380 J mol⁻¹
Step 3: Rearrange: \(\ln K = \frac{-\Delta G^\circ}{RT}\)
Step 4: Substitute: \(\ln K = \frac{-(-4380)}{8.314 \times 298.15} = \frac{4380}{2478.8} = 1.767\)
Step 5: Solve: \(K = e^{1.767} = \textbf{5.85}\)
Conclusion: K > 1 and ΔG° is negative, so the equilibrium lies to the right (products favoured). ✅
Gas Constant R and ΔG° Unit Matching
Critical Unit Trap: \(\Delta G^\circ\) is typically provided in \(\text{kJ mol}^{-1}\), while \(R = 8.314\text{ J K}^{-1}\text{mol}^{-1}\) uses Joules. You must multiply \(\Delta G^\circ\) by 1000 to convert to Joules before evaluating \(\ln K = -\dfrac{\Delta G^\circ}{RT}\).
Thermodynamic Feasibility vs Equilibrium Extent
A positive \(\Delta G^\circ\) means \(K < 1\), indicating that reactants predominate at equilibrium. It does not mean zero product forms; even a reaction with \(\Delta G^\circ = +20\text{ kJ mol}^{-1}\) produces detectable equilibrium product concentrations (\(K \approx 3 \times 10^{-4}\)).
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