Foundations — Gibbs free energy ΔG = ΔH − TΔS; spontaneity criteria

This page is the toolbox. Before you can read $\Delta G=\Delta H-T\Delta S$ and believe it, every letter in it must mean something you can picture. We build them one at a time, each resting on the one before.

0. What "system" and "surroundings" even mean

Everything starts with drawing a box.

Picture: a small blue box (system) sitting inside a huge grey region (surroundings). Heat can cross the boundary between them, shown by the orange arrow.

Why the topic needs it: the second law is a statement about the whole universe, but we only get to measure the little blue box. The entire point of Gibbs energy is to smuggle information about the grey region into quantities we measure in the blue box.

1. Temperature $T$ — the "how jiggly" number

Picture: two boxes of dots — a cold box with short motion-arrows, a hot box with long ones. More arrow = more $T$.

Why we need it: $T$ appears as a multiplier on the disorder term. Because it can never be negative, we are always allowed to multiply or divide by it and (when needed) flip an inequality knowing the sign for certain. That single fact makes the whole derivation clean.

2. Heat $q$ — energy that flows because of a temperature difference

Picture: the orange arrow on the box diagram. Point it inward → $q>0$; outward → $q<0$.

Why we need it: heat is the currency traded between system and surroundings. Whatever heat the system loses, the surroundings gain — this exact bookkeeping ($q_{surr}=-q_{sys}$) is the bridge that lets us describe the grey region using the blue box.

3. Enthalpy $H$ and $\Delta H$ — the energy scoreboard

Picture: an energy hill. Products sitting lower than reactants → energy released → $\Delta H<0$ (exothermic). Products higher → $\Delta H>0$ (endothermic). The vertical drop/rise is $\Delta H$.

Why the topic needs it: $\Delta H$ is the "energy wants to go low" half of the scoreboard, and through $q_P=\Delta H$ it is also exactly the heat handed to the surroundings.

See Enthalpy and ΔH for the full build.

4. Entropy $S$ and $\Delta S$ — the "how spread out" number

Picture: left panel — particles neatly stacked (low $S$); right panel — the same particles scattered across a bigger space (high $S$). The arrow from left to right is $\Delta S>0$.

Why we need it: $\Delta S$ is the "nature wants to spread out" half of the scoreboard. The formula $\Delta S=q/T$ is the exact tool used in the parent derivation to describe the surroundings: $\Delta S_{surr}=-\Delta H/T$.

See Entropy and ΔS and the parent principle in Second Law of Thermodynamics.

5. The second-law criterion $\Delta S_{univ}>0$

Picture: return to the box-in-a-box. Add up the disorder change of the blue box and the grey region; if the total goes up, the process runs forward.

Why we need it: this is the true law. Gibbs free energy is just a re-write of this inequality using only blue-box quantities — that is what the parent note derives.

6. Reading $\Delta G=\Delta H-T\Delta S$

Now every symbol is earned. Line by line:

Why $T$ sits on $\Delta S$ and not on $\Delta H$: $T$ is the exchange rate between "disorder units" and "energy units". Hot conditions make each unit of disorder worth more energy, so the $-T\Delta S$ term grows — which is exactly how temperature becomes the tie-breaker in the four-sign table.

7. The Greek and math shorthand, decoded

Why we need it: the parent note uses $\Delta G^{\circ }$, $\ln K$, and $⟺$ without pause. You should never meet a symbol there for the first time.

Prerequisite map

Equipment checklist

Answer each before moving to the parent note.