Gibbs Energy and Spontaneity

Gibbs Energy and Spontaneity

We have seen that it is the total entropy change, $\Delta S_{\text{total}}$, that decides the spontaneity of a process. But most chemical reactions occur in closed or open systems, not isolated ones. For such systems, there are changes in both enthalpy and entropy. Neither a decrease in enthalpy nor an increase in entropy alone can determine the direction of spontaneous change.

For this purpose, we define a new thermodynamic function: the Gibbs energy (or Gibbs function), $G$:

Gibbs energy $G$ is an extensive property and a state function. The change in Gibbs energy for the system, $\Delta G_{\text{sys}}$, can be written as:

At constant temperature, $\Delta T = 0$, so:

Usually, we drop the subscript "system" and write simply:

This is the Gibbs equation, one of the most important equations in chemistry. It combines both energy (through $\Delta H$) and entropy (through $\Delta S$) into a single criterion for spontaneity.

Dimensionally, $\Delta G$ has units of energy because both $\Delta H$ and $T\Delta S$ are energy terms: $T\Delta S = (\text{K})(\text{J/K}) = \text{J}$.

Relating $\Delta G$ to Spontaneity

We know that $\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}}$.

If the system is in thermal equilibrium with the surroundings, the temperature of the surroundings is the same as that of the system. Also, the increase in enthalpy of the surroundings equals the decrease in enthalpy of the system. Therefore:

So:

Rearranging:

For a spontaneous process, $\Delta S_{\text{total}} > 0$, so:

Or:

Using equation (5.21), this becomes:

Thus, the criterion for spontaneity at constant pressure and temperature is:

• If $\Delta G < 0$ (negative), the process is spontaneous.
• If $\Delta G > 0$ (positive), the process is non-spontaneous.
• If $\Delta G = 0$, the system is at equilibrium.

$\Delta H_{\text{sys}}$ is the enthalpy change of the reaction, and $T\Delta S_{\text{sys}}$ is the energy that is not available to do useful work. So $\Delta G$ represents the net energy available to do useful work — it is a measure of the "free energy."