For processes occurring at constant temperature and pressure, the entropy criterion for spontaneity can be expressed using properties of the system alone. Define the Gibbs free energy

$$\boxed{G=H-TS},$$

where $H$ is enthalpy, $T$ is absolute temperature and $S$ is entropy.

At constant temperature,

$$\boxed{\Delta G=\Delta H-T\Delta S}.$$

To connect this with the second law, consider a system exchanging heat with surroundings at the same constant temperature and pressure. The surroundings receive

$$q_{\mathrm{surr}}=-q_p=-\Delta H,$$

so

$$\Delta S_{\mathrm{surr}}=-\frac{\Delta H}{T}.$$

Therefore

$$\Delta S_{\mathrm{univ}}=\Delta S-\frac{\Delta H}{T},$$

and multiplying by $-T$ gives

$$\boxed{\Delta G=-T\Delta S_{\mathrm{univ}}}.$$

Because $T>0$, the signs are opposite:

• $\Delta G<0$: the process is spontaneous in the stated direction;
• $\Delta G>0$: the stated direction is nonspontaneous and the reverse direction is favored;
• $\Delta G=0$: the system is at equilibrium.

These criteria apply under the stated constant-$T$, constant-$p$ conditions. A negative $\Delta G$ says that change is thermodynamically favorable; it does not say that it happens rapidly.

Competition between enthalpy and entropy

The expression

$$\Delta G=\Delta H-T\Delta S$$

shows why temperature can change spontaneity.

• If $\Delta H<0$ and $\Delta S>0$, both terms favor spontaneity at every temperature.
• If $\Delta H>0$ and $\Delta S<0$, neither favors spontaneity.
• If $\Delta H$ and $\Delta S$ have the same sign, temperature can determine which term dominates.

Example

Suppose a process has

$$\Delta H=-40,\mathrm{kJ/mol},\qquad \Delta S=-100,\mathrm{J,mol^{-1},K^{-1}}.$$

Convert entropy to kilojoules:

$$\Delta S=-0.100,\mathrm{kJ,mol^{-1},K^{-1}}.$$

Then

$$\Delta G=-40+0.100T.$$

The crossover occurs when $\Delta G=0$:

$$T=\frac{40}{0.100}=400,\mathrm K.$$

Below $400,\mathrm K$, $\Delta G<0$; above it, $\Delta G>0$, assuming $\Delta H$ and $\Delta S$ remain approximately constant. Gibbs free energy therefore packages the system-and-surroundings entropy criterion into a practical state function for common chemical conditions.