Gibbs free energy ΔG = ΔH − TΔS; spontaneity criteria: Visual walkthrough

Before we start, three plain words we will need. Nothing here is assumed; each gets a picture below.

Step 1 — The one rule nature obeys

WHAT. The Second Law of Thermodynamics (Second Law of Thermodynamics) says: a change happens by itself (is spontaneous) only if the total mess of the universe increases. "Mess" has a proper name — entropy (Entropy and ΔS), written $S$ . More spread-out, more ways to arrange things = more entropy.

$change in total mess Δ S_{u ni v} > 0 \Rightarrow happens on its own$

WHY this and not something else. You might guess "things happen if they release energy." That's often true but not always — ice melts in your warm hand while absorbing energy. The only rule that never fails is the entropy-of-the-universe rule. So we start there.

PICTURE. Two boxes side by side — the system (a beaker) and the surroundings (the room). An arrow labelled "spontaneous" points from an ordered arrangement to a messier one, and the scoreboard $Δ S_{u ni v}$ ticks upward.

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

Step 2 — Splitting the universe into two pieces

WHAT. The universe is just system + surroundings, so its entropy change is the sum of the two:

$Δ S_{u ni v} = mess made inside beaker Δ S_{sy s} + mess made in the room Δ S_{s u r r}$

WHY. Entropy is additive — like counting people in two rooms, you just add the totals. This split is the whole trick: the room term is annoying to measure directly, so our plan is to rewrite it using only beaker quantities. That is the entire point of Gibbs free energy.

PICTURE. The universe scoreboard splits into two smaller scoreboards, one under the beaker, one under the room, joined by a big "+".

Step 3 — Heat leaves the beaker, enters the room

WHAT. When the reaction runs, it either releases heat (feels hot) or absorbs heat (feels cold). At constant pressure — the normal open-beaker situation — the heat the system gives out is exactly its enthalpy change (Enthalpy and ΔH):

$q_{sy s} = Δ H_{sy s} (at constant pressure)$

Here $q$ means "heat exchanged." Whatever heat leaves the system must enter the room, so the room receives the opposite:

$q_{s u r r} = - Δ H_{sy s}$

WHY the minus sign. Heat is not created or destroyed — it just moves. If the beaker loses $Δ H_{sy s}$ (exothermic, $Δ H_{sy s} < 0$ ), the room gains that amount, so its heat is $+ ∣Δ H_{sy s} ∣$ . The minus sign flips the beaker's bookkeeping into the room's bookkeeping.

WHY "constant pressure = enthalpy." Enthalpy $H$ is defined precisely so that $q_{P} = Δ H$ . It is the natural energy currency for reactions open to the atmosphere. That is why chemists use $Δ H$ and not the raw internal energy.

PICTURE. An arrow of heat leaving the beaker (labelled $- Δ H_{sy s}$ from the beaker's view) and the same arrow arriving at the room (labelled $+ q_{s u r r} = - Δ H_{sy s}$ ). Same arrow, opposite sign depending on who you ask.

Step 4 — Turning the room's heat into the room's entropy

WHAT. The room is enormous — a "heat reservoir" held at the constant temperature $T$ . Its entropy change for a heat input is heat divided by temperature:

$Δ S_{s u r r} = \frac{q _{s u r r}}{T} = \frac{- Δ H _{sy s}}{T}$

Each symbol: $q_{s u r r}$ = heat the room got (top of the fraction), $T$ = the constant temperature in kelvin (bottom), and the result is the room's mess change.

WHY divide by $T$ . The same lump of heat makes a big difference to a cold, quiet room but barely disturbs a hot, already-chaotic one. Dividing by temperature captures that: adding a bucket of water matters more to a small pond than to an ocean. This is the exact definition of entropy from reversible heat, $Δ S = q_{r e v} / T$ .

WHY dividing by a single $T$ is valid (the isothermal assumption). Because we fixed $T$ constant (see the conditions box), the temperature during the whole heat exchange is one number, not a changing one — so we can write a plain $q / T$ instead of adding up many tiny slices. On top of that, the room is so vast its own temperature never budges, so from the room's point of view the heat trickles in gently (reversibly) — making the formula exact.

PICTURE. A thermometer at the fixed temperature $T$ ; the incoming heat $- Δ H_{sy s}$ divided by the height of the thermometer gives the entropy bar for the room.

Step 5 — Substituting: everything now in beaker language

WHAT. Put the room result from Step 4 back into the split from Step 2:

$Δ S_{u ni v} = Δ S_{sy s} + from Step 4 (\frac{- Δ H _{sy s}}{T}) = Δ S_{sy s} - \frac{Δ H _{sy s}}{T}$

WHY this is a triumph. Look at the right side: $Δ S_{sy s}$ , $Δ H_{sy s}$ , $T$ — all three are properties of the beaker (or its temperature). The room has vanished! We can now judge spontaneity without ever measuring the surroundings. That was the whole mission.

PICTURE. The room scoreboard is crossed out and replaced by the expression $- Δ H_{sy s} / T$ , so the universe formula now lives entirely inside the beaker box.

Step 6 — One cosmetic move: multiply by $- T$

WHAT. Multiply both sides by $- T$ :

$- T Δ S_{u ni v} = Δ H_{sy s} - T Δ S_{sy s}$

Now name the right-hand side the Gibbs free energy change. From here on we drop the "sys" subscript: by convention $Δ G$ , $Δ H$ and $Δ S$ always mean the system's values (the universe's entropy has already been folded away). So $Δ H \equiv Δ H_{sy s}$ and $Δ S \equiv Δ S_{sy s}$ below.

Term by term in $Δ G = Δ H - T Δ S$ :

$Δ H$ — the energy pull of the system (nature likes to shed energy, $Δ H < 0$ is favourable).
$T Δ S$ — the mess pull of the system, weighted by temperature $T$ (nature likes disorder, $Δ S > 0$ is favourable).
The minus sign between them makes low $Δ G$ mean "both wants satisfied."

WHY multiply by $- T$ . Two reasons. (1) It clears the ugly fraction $Δ H / T$ into a clean $Δ H$ . (2) It converts an entropy statement (about the whole universe) into an energy statement (about the beaker) — energies are what chemists already tabulate. The price: because $- T$ is negative, multiplying flips the direction of an inequality (see next step).

PICTURE. The universe-entropy expression is scaled by $- T$ and rebadged with the boxed label $Δ G$ ; a small tag reminds us $Δ G = - T Δ S_{u ni v}$ .

Step 7 — Reading off the spontaneity rule

WHAT. Since temperature in kelvin is always positive ( $T > 0$ ), the sign of $Δ G$ is just the opposite of the sign of $Δ S_{u ni v}$ :

$Δ S_{u ni v} > 0 ⟺ - T Δ S_{u ni v} < 0 ⟺ Δ G < 0$

WHY the flip is safe. We multiplied by a negative number, and multiplying an inequality by a negative flips its arrow — that is exactly why "universe entropy up" becomes "Gibbs energy down." The two statements are perfectly equivalent.

PICTURE. A number line for $Δ G$ : left of zero shaded "spontaneous," the point zero marked "equilibrium," right of zero "non-spontaneous," with a mirror arrow showing $Δ S_{u ni v}$ pointing the opposite way.

Step 8 — The four sign scenarios (every quadrant, then the edge cases)

The boxed formula $Δ G = Δ H - T Δ S$ is a straight line in $T$ : intercept $Δ H$ , slope $- Δ S$ . So the behaviour depends only on the signs of $Δ H$ and $Δ S$ . There are exactly four combinations — enumerate them all:

The crossover, $Δ G = 0$ . For scenarios 3 and 4 the line actually crosses the zero axis. Setting $Δ H - T Δ S = 0$ : $T = \frac{Δ H}{Δ S}$ This is where the reaction switches between spontaneous and not — the melting/boiling point of a phase transition. Scenarios 1 and 2 never cross zero (their line stays on one side), so they have no switch temperature.

Now the truly degenerate inputs (the boundaries between the four scenarios):

$Δ S = 0$ (no mess change). Then $Δ G = Δ H$ : a horizontal line. Spontaneity is decided purely by energy, independent of $T$ .
$Δ H = 0$ (no energy change). Then $Δ G = - T Δ S$ : a line through the origin. Only disorder matters; $Δ S > 0$ is spontaneous at any $T > 0$ (e.g. two gases mixing).

PICTURE — how five bullet points map to four lines. The plot draws one line per sign scenario (four lines total): teal = scenario 1, orange = scenario 2, plum = scenario 3, olive = scenario 4. The two degenerate cases ( $Δ S = 0$ , $Δ H = 0$ ) are not separate lines — they are the limiting boundaries the four coloured lines approach (a flat line, and a line through the origin), so they are described in text rather than plotted as extra curves. The crossover dots sit where the plum and olive lines cut the dashed $Δ G = 0$ axis.

The one-picture summary

The whole chain, compressed: universe rule → split → heat swap → divide by $T$ → substitute → scale by $- T$ → read off $Δ G$ .

Recall Feynman: the walkthrough in plain words

Nature keeps one score: how messy the whole universe is, and it only lets changes happen that make that score go up. But measuring the whole universe is silly, so we cheated. The messiness of the room outside is just the heat the beaker dumps into it, divided by how hot the (steady-temperature) room is. We plugged that in, and suddenly the "room" disappeared — everything was written in beaker-only quantities: the system's energy change $Δ H$ , its own mess change $Δ S$ , and the temperature $T$ . Then we did one tidy multiplication by $- T$ to turn the messiness score into an energy score, and called it $Δ G$ . Because we multiplied by a negative, the arrow flipped: "universe gets messier" turned into " $Δ G$ goes down." So the final rule is dead simple — if $Δ G$ goes down, the change happens. Cold days let energy decide; hot days let mess decide, because temperature is the weight sitting on the mess term.

Connections

Second Law of Thermodynamics — Step 1, the master rule we started from.
Enthalpy and ΔH — Step 3, the constant-pressure heat term.
Entropy and ΔS — Steps 1–2, the "mess" being counted.
Chemical Equilibrium and K — the $Δ G = 0$ crossover links to $Δ G^{\circ} = - R T ln K$ .
Phase Transitions — the $Δ G = 0$ crossover, melting/boiling.
Electrochemistry ΔG = −nFE — free energy read from a cell voltage.
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