Chemistry · Ch 6 — Equilibrium

Relationship between Equilibrium Constant K, Reaction Quotient Q and Gibbs Energy G

6.7

Relationship between Equilibrium Constant K, Reaction Quotient Q and Gibbs Energy G

The Thermodynamic Link Between $K$ , $Q$ , and $\Delta G$

The equilibrium constant $K_c$ for a reaction is a number that tells you where equilibrium lies, but it is not a kinetic quantity — it does not depend on how fast the reaction gets there. Instead, $K_c$ is rooted in thermodynamics, specifically in the Gibbs energy change $\Delta G$ of the reaction. This connection is what lets us predict the direction of a reaction from a single measurement of concentrations.

The sign of $\Delta G$ tells you the story of the reaction's spontaneity:

If $\Delta G$ is negative, the forward reaction is spontaneous — it will proceed on its own.
If $\Delta G$ is positive, the forward reaction is non-spontaneous. But the reverse reaction then has a negative $\Delta G$ , so the products will convert back into reactants.
If $\Delta G$ is zero, the system is at equilibrium. No net free energy remains to drive the reaction in either direction.

This thermodynamic view is captured in a single equation that relates $\Delta G$ to the reaction quotient $Q$ :

\Delta G = \Delta G^\circ + RT \ln Q \qquad(6.21)

Here $\Delta G^\circ$ is the standard Gibbs energy change (when all reactants and products are in their standard states, typically 1 bar pressure for gases and 1 M concentration for solutions), $R$ is the gas constant ( $8.314\ \text{J mol}^{-1}\text{K}^{-1}$ ), and $T$ is the absolute temperature in Kelvin.

The Equilibrium Condition: Deriving $\Delta G^\circ = -RT \ln K$

At equilibrium, the system has no net driving force — $\Delta G = 0$ . At that same point, the reaction quotient $Q$ equals the equilibrium constant $K$ (whether $K_c$ or $K_p$ , depending on the units). Substituting these two conditions into equation (6.21) gives:

0 = \Delta G^\circ + RT \ln K

Rearranging:

\Delta G^\circ = -RT \ln K \qquad(6.22)

This is the master equation that links thermodynamics and equilibrium. It can also be written as:

\ln K = -\frac{\Delta G^\circ}{RT}

Or, exponentiating both sides:

K = e^{-\Delta G^\circ / RT}

Important

Equation (6.22) is the bridge between two worlds: the thermodynamic quantity $\Delta G^\circ$ (which you can look up in tables of standard Gibbs energies of formation) and the equilibrium constant $K$ (which you measure experimentally). A single value of $\Delta G^\circ$ determines $K$ uniquely at a given temperature.

What This Equation Tells Us About $K$

The relationship $K = e^{-\Delta G^\circ / RT}$ reveals three clear regimes:

Sign of $\Delta G^\circ$	Value of $K$	Meaning
Negative ( $\Delta G^\circ < 0$ )	$K > 1$	Products are favoured at equilibrium; the equilibrium lies to the right.
Zero ( $\Delta G^\circ = 0$ )	$K = 1$	Reactants and products are equally favoured at equilibrium.
Positive ( $\Delta G^\circ > 0$ )	$0 < K < 1$	Reactants are favoured at equilibrium; the equilibrium lies to the left.

Watch out

Do not confuse $\Delta G$ with $\Delta G^\circ$ . $\Delta G^\circ$ is a fixed value for a given reaction at a given temperature — it determines $K$ . $\Delta G$ depends on the actual concentrations (or pressures) of reactants and products through $Q$ , and it tells you whether the reaction will proceed forward or backward from those conditions. A reaction with $\Delta G^\circ > 0$ (and therefore $K < 1$ ) can still have $\Delta G < 0$ if the initial concentrations are far from equilibrium — for example, if you start with no products at all.

Using $Q$ and $K$ to Predict Direction

The equation $\Delta G = \Delta G^\circ + RT \ln Q$ can be combined with $\Delta G^\circ = -RT \ln K$ to give a direct comparison: …

Start your 14-day free trial to unlock the full solution →

← Previous · 6.6.3Problem 6.9 — 3.00 mol of PCl 5 kept in 1L closed reaction vessel was allowed to att…Next · 6.7 →Problem 6.10 — The value of ∆G ° for the phosphorylation of glucose in glycolysis is …

Relationship between Equilibrium Constant K, Reaction Quotient Q and Gibbs Energy G