ΔG° and equilibrium constant - ΔG° = −RT ln K

WHAT are we talking about?

The single most important relation: $\boxed{\;\Delta G = \Delta G^\circ + RT\ln Q\;}$ and its equilibrium special case: $\boxed{\;\Delta G^\circ = -RT\ln K\;}$

WHY should this be true? (Derivation from scratch)

Step 1 — Free energy of one substance depends on its pressure. For 1 mole of an ideal gas at constant $T$, $dG = V\,dP$ (from $dG = V\,dP - S\,dT$, with $dT=0$). Using $V = RT/P$: $dG = \frac{RT}{P}\,dP \;\Rightarrow\; \int_{P^\circ}^{P} dG = RT\int_{P^\circ}^{P}\frac{dP}{P}$ $G(P) = G^\circ + RT\ln\frac{P}{P^\circ}$ Why this step? Chemical potential grows logarithmically with pressure — squeezing gas costs free energy, and the cost is $RT\ln(\text{ratio})$. Write $\mu = \mu^\circ + RT\ln a$ where $a=P/P^\circ$ is the activity.

Step 2 — Build $\Delta G$ for a reaction. For $aA + bB \rightleftharpoons cC + dD$: $\Delta G = \sum \nu_i \mu_i = \big(c\mu_C + d\mu_D\big) - \big(a\mu_A + b\mu_B\big)$ Substitute each $\mu_i = \mu_i^\circ + RT\ln a_i$: $\Delta G = \underbrace{\Big(c\mu_C^\circ+d\mu_D^\circ - a\mu_A^\circ - b\mu_B^\circ\Big)}_{\Delta G^\circ} + RT\Big(c\ln a_C + d\ln a_D - a\ln a_A - b\ln a_B\Big)$ The log terms collapse using $\ln x^n = n\ln x$ and $\ln x+\ln y=\ln xy$: $\Delta G = \Delta G^\circ + RT\ln\underbrace{\frac{a_C^{\,c}\,a_D^{\,d}}{a_A^{\,a}\,a_B^{\,b}}}_{Q}$ Why this step? This is just algebra collecting all the logs into one quotient $Q$ — the exact shape of the equilibrium expression.

Step 3 — Impose equilibrium. At equilibrium the system has no tendency to change, so $\Delta G = 0$ and $Q = K$: $0 = \Delta G^\circ + RT\ln K \;\Rightarrow\; \Delta G^\circ = -RT\ln K$ Why this step? $\Delta G$ is the slope of $G$ vs extent of reaction. At the minimum of $G$ (equilibrium) the slope is zero. That single physical fact is what pins $\Delta G^\circ$ to $K$.

HOW to read the formula

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

• Gibbs Free Energy G = H - TS — where $\Delta G$ comes from.
• Reaction Quotient Q and Equilibrium — $\Delta G=\Delta G^\circ+RT\ln Q$.
• Chemical Potential μ — the $\mu=\mu^\circ+RT\ln a$ backbone.
• Van't Hoff Equation — temperature dependence of $K$ from $\Delta G^\circ=\Delta H^\circ-T\Delta S^\circ$.
• Spontaneity and the Second Law — why $\Delta G<0$ means "goes."
• Le Chatelier's Principle — qualitative version of $Q$ vs $K$.

Concept Map

Hinglish (regional understanding)