Free Energy and Equilibrium — AP Chemistry
1. The Core Relationship Between $\Delta G^\circ$ and $K$ ★★☆☆☆ ⏱ 5 min
At equilibrium, the total free energy of the system is at its minimum, so there is no net driving force for forward or reverse reaction, meaning $\Delta G = 0$. We start with the general expression for free energy under any conditions to relate standard free energy change to the equilibrium constant.
\Delta G = \Delta G^\circ + RT \ln Q
At equilibrium, $\Delta G = 0$ and $Q = K$, so substituting gives the core relationship connecting thermodynamics and equilibrium:
\Delta G^\circ = -RT \ln K
- If $\Delta G^\circ < 0$, $\ln K > 0$ so $K > 1$: products are favored at equilibrium
- If $\Delta G^\circ > 0$, $\ln K < 0$ so $K < 1$: reactants are favored at equilibrium
- If $\Delta G^\circ = 0$, $\ln K = 0$ so $K = 1$: reactants and products are equally favored
Exam tip: Always convert $R$ to match the energy units of $\Delta G^\circ$: use 0.008314 kJ/(mol·K) when $\Delta G^\circ$ is in kJ/mol, and 8.314 J/(mol·K) when $\Delta G^\circ$ is in J/mol to avoid 1000× errors in $K$.
2. Non-Standard $\Delta G$ and Spontaneity Prediction ★★★☆☆ ⏱ 6 min
Most reactions do not occur under standard conditions (1 M concentration, 1 atm pressure, pure solids/liquids). To predict the direction of spontaneous change under non-standard conditions, we use the general free energy formula:
\Delta G = \Delta G^\circ + RT \ln Q
Where $Q$ is the reaction quotient calculated from current non-standard concentrations or partial pressures. The sign of $\Delta G$ directly indicates the direction of spontaneity:
- $\Delta G < 0$: forward reaction is spontaneous
- $\Delta G > 0$: reverse reaction is spontaneous
- $\Delta G = 0$: the reaction is at equilibrium
We can derive a useful shortcut by substituting $\Delta G^\circ = -RT \ln K$ into the non-standard $\Delta G$ formula:
\Delta G = -RT \ln K + RT \ln Q = RT \ln\left(\frac{Q}{K}\right)
Since $R$ and $T$ are always positive, the sign of $\Delta G$ matches the sign of $\ln\left(\frac{Q}{K}\right)$, so we can predict spontaneity directly from comparing $Q$ and $K$ without calculating $\Delta G$.
Exam tip: Do not confuse $\Delta G^\circ$ and $\Delta G$: $\Delta G^\circ$ tells you about the equilibrium position (whether $K$ is greater than or less than 1), while $\Delta G$ tells you the direction of spontaneity under your specific non-standard conditions.
3. Temperature Dependence of $K$ and the van't Hoff Equation ★★★★☆ ⏱ 7 min
Equilibrium constants change with temperature. We can derive the relationship by combining two expressions for $\Delta G^\circ$: $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$ and $\Delta G^\circ = -RT \ln K$. Setting these equal and rearranging gives the linear form of the van't Hoff equation:
\ln K = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R}
To calculate $K$ at a new temperature when $K$ is known at an initial temperature, we use the two-point form of the van't Hoff equation:
\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)
This relationship confirms Le Chatelier's principle for temperature changes:
- Endothermic reactions ($\Delta H^\circ > 0$): increasing temperature increases $K$, shifting equilibrium right
- Exothermic reactions ($\Delta H^\circ < 0$): increasing temperature decreases $K$, shifting equilibrium left
Exam tip: After calculating $K$ at a new temperature, always cross-check against Le Chatelier's principle. If your result contradicts Le Chatelier, you have a sign error in the van't Hoff equation.
4. Exam-Style Concept Check ★★★☆☆ ⏱ 4 min
Common Pitfalls
Why: Most students memorize $R$ as 8.314 but forget $\Delta G$ is usually reported in kJ, leading to a 1000× error in $\ln K$ and an incorrect $K$ by many orders of magnitude.
Why: Students confuse $\Delta G^\circ$ (standard conditions) with $\Delta G$ (non-standard conditions).
Why: Students mix up logarithm rules: $\ln 1 = 0$, not $\ln 0 = 0$.
Why: Sign errors in the van't Hoff equation are common, and students do not cross-check their result.
Why: Students mix up what $\Delta G$ vs $\Delta G^\circ$ tells you about $K$.