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Free energy and equilibrium

Standard free energy and the equilibrium constant are two ways of saying the same thing. One equation links them, and it explains why ΔG° = 0 does not mean 'nothing happens.'

§1

ΔG° = −RT ln K.

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Standard free energy and the equilibrium constant are locked together by ΔG° = −RT ln K. The minus sign ties the two: a large K (> 1) forces a negative ΔG° (products favored), and a small K (< 1) forces a positive ΔG°.

The special case is ΔG° = 0, which gives K = 1 — a balanced mixture of reactants and products, not a reaction that fails to occur. Reaction still happens; it just settles with comparable amounts on both sides.

Note that ΔG° (standard) sets K, while the actual ΔG under given conditions tells you which way the reaction currently shifts. At equilibrium the actual ΔG = 0, but ΔG° need not be.

UNIT 9 TOPIC 9.5 • FREE ENERGY AND EQUILIBRIUM ΔG AND K The standard Gibbs free energy change ΔG° is tied to the equilibrium constant K at a given temperature. THE RELATIONSHIP ΔG° = −RT ln K K = e−ΔG°/RT VARIABLES ΔG° = std Gibbs free energy (J·mol⁻¹) R = gas constant (8.314 J·mol⁻¹·K⁻¹) T = temperature (K) K = equilibrium constant (unitless) UNITS CHECK R → J·mol⁻¹·K⁻¹ T → K R × T → J·mol⁻¹ ln K & e−ΔG°/RT unitless ΔG° must be in J·mol⁻¹. kJ·mol⁻¹ → J·mol⁻¹: multiply by 1000. The sign of ΔG° sets the size of K and which side is favored at equilibrium. ΔG° < 0 K > 1 (large) Products favored Equilibrium lies to the right A + B ⇌ C + D ΔG° = 0 K = 1 Neither side favored Equilibrium in the middle A + B ⇌ C + D ΔG° > 0 K < 1 (small) Reactants favored Equilibrium lies to the left A + B ⇌ C + D Large K (≫ 1) Small K (≪ 1) ΔG° negative spontaneous forward ΔG° = 0 at equilibrium (K = 1) ΔG° positive spontaneous reverse increasing ΔG° CED ANCHOR · KEY TAKEAWAY Use ΔG° = −RT ln K to link free energy to K. Negative ΔG° → products (large K); positive ΔG° → reactants (small K). REMEMBER ΔG° grows more negative as K grows larger. AP Chemistry · Unit 9 · Applications of Thermodynamics
Fig. 9.5.1 ΔG° = −RT ln K links standard free energy to the equilibrium constant. A large K (> 1) forces a negative ΔG°; a small K forces a positive one; ΔG° = 0 means K = 1 — a balanced mixture, not no reaction.
§2

Reading K from ΔG° and back.

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Let the minus sign do the work.

  1. Write the relation. ΔG° = −RT ln K, with R the gas constant and T in kelvin.
  2. Map size to sign. K > 1 gives ln K > 0, so ΔG° < 0 (favored); K < 1 gives ΔG° > 0.
  3. Handle the K = 1 case. ΔG° = 0 means K = 1 — comparable products and reactants, not no reaction.
  4. Separate ΔG° from ΔG. ΔG° sets K; the actual ΔG (with Q) says which way it shifts now.
§3

The pieces you'll meet.

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Free energy and the equilibrium constant.

ΔG°
Standard free energy
= −RT ln K.
K
Equilibrium constant
Large K favors products.
sign
Sign map
K > 1 → ΔG° < 0; K < 1 → ΔG° > 0.
K=1
ΔG° = 0
Means K = 1, a balanced mixture.
R,T
R and T
Gas constant and kelvin temperature.
not stop
Not 'no reaction'
ΔG° = 0 still means reaction occurs.
§4

Worked example: sign of ΔG° from K.

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Large K. A reaction has K = 1 × 10⁵. Since K > 1, ln K > 0, so ΔG° = −RT ln K is negative — products are favored.

Small K. Another has K = 1 × 10⁻⁴. Since K < 1, ln K < 0, so ΔG° is positive — reactants are favored.

K = 1. If ΔG° = 0, then ln K = 0 and K = 1 — a mixture with comparable products and reactants.

Key point. ΔG° = 0 does not mean 'no reaction' — it means the equilibrium lies with K = 1.

§5

Mistakes that cost real points.

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Pitfall · 01

"A large K goes with a positive ΔG°."

The minus sign in ΔG° = −RT ln K means a large K (ln K > 0) gives a negative ΔG°. Pairing a large K with a positive ΔG° is impossible. Large K and negative ΔG° both mean products are favored.

Fix. Use ΔG° = −RT ln K: K > 1 → negative ΔG°; K < 1 → positive ΔG°.

Pitfall · 02

"ΔG° = 0 means the reaction does not happen."

ΔG° = 0 corresponds to K = 1, a balanced equilibrium mixture — reaction still occurs and reaches a state with comparable products and reactants. It does not mean the reaction is frozen or absent.

Fix. Read ΔG° = 0 as K = 1 — significant amounts of both sides — not as 'no reaction.'

Pitfall · 03

"Drop the minus sign: ΔG° = RT ln K."

The relationship carries a negative sign: ΔG° = −RT ln K. Omitting it reverses the connection between the sign of ΔG° and the size of K, giving backwards conclusions about which side is favored.

Fix. Keep the minus sign in ΔG° = −RT ln K.

§6

Skill Check.

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Ten scenarios. Pick the chips that match your answer, then check. A scenario marks complete the first time every part is right. Progress saves on this device.

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