Gibbs Free Energy ΔG = ΔH − TΔS — Spontaneity Explained
Why does ice melt above 0 °C but not below? Why does limestone need a furnace to decompose, while rusting happens quietly on its own? Gibbs free energy answers all of these with one equation. It is the single most examined idea in Class 11–12 thermodynamics, and it carries straight through into IIT-JAM, GATE and CSIR-NET physical chemistry.
The equation
ΔG < 0 → the process is spontaneous (feasible) in the forward direction
ΔG = 0 → the system is at equilibrium
ΔG > 0 → not spontaneous forward; the reverse direction is spontaneous
What each term means
| Symbol | Meaning | Usual unit |
|---|---|---|
| ΔG | Change in Gibbs free energy — the useful work obtainable at constant P and T | kJ mol⁻¹ |
| ΔH | Enthalpy change; negative = exothermic, positive = endothermic | kJ mol⁻¹ |
| T | Absolute temperature | kelvin, always |
| ΔS | Entropy change; positive = more disorder | J K⁻¹ mol⁻¹ — note joules, not kilojoules |
That unit mismatch in the last row is the number one source of wrong answers in this whole chapter. ΔH is tabulated in kilojoules and ΔS in joules. Divide ΔS by 1000 before multiplying by T, or multiply ΔH by 1000 — but never mix them.
The four sign combinations
| ΔH | ΔS | Spontaneous when | Example |
|---|---|---|---|
| − (exothermic) | + (disorder rises) | at all temperatures | Combustion of petrol |
| + (endothermic) | − (disorder falls) | never | Reverse of a combustion |
| − (exothermic) | − (disorder falls) | only at low T | Water freezing; ammonia synthesis |
| + (endothermic) | + (disorder rises) | only at high T | Limestone decomposing; ice melting |
Read it from the equation rather than memorising the table. The TΔS term grows with temperature, so whichever way ΔS points, heating the system makes entropy the dominant partner.
Worked example 1 — Ammonia synthesis at room temperature
For N₂(g) + 3H₂(g) → 2NH₃(g): ΔH° = −92.4 kJ mol⁻¹ and ΔS° = −198.3 J K⁻¹ mol⁻¹. Find ΔG° at 298 K.
First fix the units: ΔS° = −198.3 J K⁻¹ mol⁻¹ = −0.1983 kJ K⁻¹ mol⁻¹
TΔS = 298 × (−0.1983) = −59.09 kJ mol⁻¹
ΔG° = ΔH° − TΔS = (−92.4) − (−59.09) = −92.4 + 59.09 = −33.3 kJ mol⁻¹
Negative, so ammonia formation is spontaneous at 298 K. Watch the double negative in the subtraction — that is where careless working goes wrong.
Worked example 2 — The crossover temperature
At equilibrium ΔG = 0, so ΔH = TΔS. Rearranging gives the temperature at which spontaneity flips:
Above what temperature does ammonia synthesis stop being spontaneous?
T = (−92.4) ÷ (−0.1983) = 466 K (about 193 °C)
Below 466 K, ΔG is negative and the reaction is feasible. Above it, the TΔS penalty grows larger than the enthalpy gain and ΔG turns positive. This is exactly the industrial dilemma of the Haber process: high temperature helps the rate but hurts the yield, so the plant runs at a compromise temperature with a catalyst and very high pressure.
Worked example 3 — Decomposition of limestone
For CaCO₃(s) → CaO(s) + CO₂(g): ΔH° = +178.3 kJ mol⁻¹ and ΔS° = +160.6 J K⁻¹ mol⁻¹. Find ΔG° at 298 K, and the minimum temperature for decomposition.
ΔS° = +0.1606 kJ K⁻¹ mol⁻¹
At 298 K: TΔS = 298 × 0.1606 = 47.86 kJ mol⁻¹
ΔG° = 178.3 − 47.86 = +130.4 kJ mol⁻¹ → strongly non-spontaneous. Limestone
is quite safe sitting on a shelf.
Crossover: T = 178.3 ÷ 0.1606 = 1110 K (about 837 °C)
That is why a lime kiln has to be that hot. The entropy gain comes from releasing CO₂ gas from a solid — a large positive ΔS — but it only wins above 1110 K.
Worked example 4 — From ΔG° to the equilibrium constant
Gibbs free energy is not only a yes/no test. It tells you how far a reaction goes:
with R = 8.314 J K⁻¹ mol⁻¹ (so ΔG° must be in joules here)
Find K for ammonia synthesis at 298 K, given ΔG° = −33.3 kJ mol⁻¹.
ΔG° = −33300 J mol⁻¹
ln K = −ΔG° / (RT) = 33300 ÷ (8.314 × 298) = 33300 ÷ 2477.6 = 13.44
K = e13.44 ≈ 6.9 × 10⁵
A large K, confirming that the position of equilibrium lies well to the product side at room temperature. Quick mental checks worth keeping: ΔG° negative → K > 1; ΔG° = 0 → K = 1; ΔG° positive → K < 1.
ΔG versus ΔG° — a distinction that carries marks
ΔG° is the standard free-energy change, measured with every species at its standard state (1 bar for gases, 1 M for solutions). ΔG is the actual value under the conditions in your flask, and it changes as the reaction proceeds:
At equilibrium Q = K and ΔG = 0 — which is precisely how ΔG° = −RT ln K is derived. A reaction with a positive ΔG° can still run forward if Q is small enough, which is how many biochemical steps proceed inside a cell.
Common mistakes that cost marks
- Mixing kJ and J. If ΔH is in kJ, convert ΔS from J K⁻¹ to kJ K⁻¹ by dividing by 1000. Skipping this makes the TΔS term a thousand times too large and usually flips the sign of the answer.
- Using Celsius for T. Kelvin only. At 25 °C use 298 K, and TΔS at "25" would be wrong by a factor of about twelve.
- Losing the double negative. ΔH − TΔS where both are negative means subtracting a negative — the result gets larger, not smaller.
- Reading "spontaneous" as "fast". ΔG is thermodynamics, not kinetics. Diamond turning into graphite has a negative ΔG and still takes geological time. Rate is governed by activation energy, not by ΔG.
- Confusing ΔG with ΔG°. Only ΔG° goes into −RT ln K. ΔG is zero at equilibrium; ΔG° is generally not.
- Forgetting R changes with the unit. Use 8.314 J K⁻¹ mol⁻¹ for the ln K relation and convert ΔG° into joules.
Where Gibbs free energy appears in exams
| Exam | Typical use |
|---|---|
| CBSE/ICSE Class 11 | Thermodynamics: spontaneity, entropy, sign analysis of ΔH and ΔS |
| CBSE/ICSE Class 12 | Equilibrium and electrochemistry: ΔG° = −nFE°cell and ΔG° = −RT ln K |
| JEE / NEET | Crossover temperature, sign-combination reasoning, ΔG from ΔG° and Q |
| IIT-JAM / CUET-PG | Maxwell relations, chemical potential, phase equilibria |
| GATE / CSIR-NET | Ellingham diagrams in metallurgy, coupled reactions, bioenergetics |
Never lose a mark to the kJ/J trap again. The Gibbs Free Energy calculator takes ΔH, ΔS and T, handles the unit conversion internally, and reports ΔG with the spontaneity verdict — so you can verify a full thermodynamics question in seconds.
Open the Gibbs Free Energy (ΔG) Calculator →Thermodynamics rewards clear reasoning more than memory. ABC Chemistry teaches Class 11–12 chemistry at the Gurugram coaching centre and through online classes across India, with home tuition available in Delhi-NCR — see abcchemistry.in.