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

2.5.15 · Chemistry › Thermodynamics (Chemical)

HUM KISI BAARE MEIN BAAT KAR RAHE HAIN?

Sabse important relation: $\boxed{\Delta G=\Delta G^{\circ }+RT\ln Q}$ aur iska equilibrium special case: $\boxed{\Delta G^{\circ }=-RT\ln K}$

AISA KYUN HONA CHAHIYE? (Scratch se Derivation)

Step 1 — Ek substance ki free energy uske pressure par depend karti hai. 1 mole ideal gas ke liye constant $T$ par, $dG=VdP$ (from $dG=VdP-SdT$, with $dT=0$). $V=RT/P$ use karte hue: $dG=\frac{RT}{P}dP\Rightarrow {\int }_{P^{\circ }}^{P}dG=RT{\int }_{P^{\circ }}^P\frac{dP}{P}$ $G(P)=G^{\circ }+RT\ln \frac{P}{P^{\circ }}$ Yeh step kyun? Chemical potential pressure ke saath logarithmically badhti hai — gas ko squeeze karne mein free energy lagti hai, aur woh cost hai $RT\ln (\text{ratio})$. Likho $\mu ={\mu }^{\circ }+RT\ln a$ jahan $a=P/P^{\circ }$ activity hai.

Step 2 — Ek reaction ke liye $\Delta G$ banao. $aA+bB\rightleftharpoons cC+dD$ ke liye: $\Delta G=\sum {\nu }_i{\mu }_i=(c{\mu }_C+d{\mu }_D)-(a{\mu }_A+b{\mu }_B)$ Har ${\mu }_i={\mu }_i^{\circ }+RT\ln a_i$ substitute karo: $\Delta G=\underset{\Delta G^{\circ }}{\underbrace{(c{\mu }_C^{\circ }+d{\mu }_D^{\circ }-a{\mu }_A^{\circ }-b{\mu }_B^{\circ })}}+RT(c\ln a_C+d\ln a_D-a\ln a_A-b\ln a_B)$ Log terms $\ln x^n=n\ln x$ aur $\ln x+\ln y=\ln xy$ use karke collapse ho jaate hain: $\Delta G=\Delta G^{\circ }+RT\ln \underset{Q}{\underbrace{\frac{a_C^c{a}_D^d}{a_A^a{a}_B^b}}}$ Yeh step kyun? Yeh sirf algebra hai jo saare logs ko ek quotient $Q$ mein collect karta hai — bilkul equilibrium expression ki shape.

Step 3 — Equilibrium impose karo. Equilibrium par system mein change ki koi tendency nahi hoti, isliye $\Delta G=0$ aur $Q=K$: $0=\Delta G^{\circ }+RT\ln K\Rightarrow \Delta G^{\circ }=-RT\ln K$ Yeh step kyun? $\Delta G$, $G$ ka slope hai reaction extent ke relative. $G$ ke minimum (equilibrium) par slope zero hota hai. Yahi ek physical fact $\Delta G^{\circ }$ ko $K$ se pin karta hai.

FORMULA KO KAISE PADHEIN

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

• Gibbs Free Energy G = H - TS — $\Delta G$ kahaan se aata hai.
• Reaction Quotient Q and Equilibrium — $\Delta G=\Delta G^{\circ }+RT\ln Q$.
• Chemical Potential μ — $\mu ={\mu }^{\circ }+RT\ln a$ ki backbone.
• Van't Hoff Equation — $\Delta G^{\circ }=\Delta H^{\circ }-T\Delta S^{\circ }$ se $K$ ki temperature dependence.
• Spontaneity and the Second Law — $\Delta G<0$ ka matlab "hoga" kyun hota hai.
• Le Chatelier's Principle — $Q$ vs $K$ ka qualitative version.

Concept Map