Q.ΔG is net energy available to do useful work and is thus a measure of "free energy". Show mathematically that ΔG is a measure of free energy. Find the unit of ΔG. If a reaction has positive enthalpy change and positive entropy change, under what condition will the reaction be spontaneous?

The key idea is that Gibbs free energy ($G$) represents the maximum useful work obtainable from a closed system at constant temperature and pressure, minus the work done by expansion.

Step 1: Relate $\Delta G$ to work. From the first and second laws, for a reversible process at constant $T$ and $P$:

Since $dU = T\,dS - P\,dV + dW_{\text{non-expansion}}$, substitution gives:

Thus $\Delta G$ equals the maximum non-expansion work (useful work) the system can do.

Step 2: Unit of $\Delta G$. Since $\Delta G = \Delta H - T\Delta S$, and $\Delta H$ and $T\Delta S$ are in energy units:

In SI, it is $\text{J mol}^{-1}$ if per mole of reaction. …

Why $\Delta G$ is a measure of free energy

The term "free energy" refers to energy that is available to do useful work — as opposed to energy that is inevitably lost as heat or tied up in entropy. The second law of thermodynamics tells us that for any process, the total entropy of the universe (system + surroundings) must increase. But we want a criterion that depends only on the system itself, not on the surroundings. That is exactly what Gibbs free energy provides.

The derivation connects the system's properties to the maximum work it can deliver.

1. Start with the second law for the universe. For a spontaneous process at constant temperature and pressure, the total entropy change is:

2. Express $\Delta S_{\text{surroundings}}$ in terms of the system's heat. The surroundings exchange heat with the system at constant pressure. The heat absorbed by the surroundings is $-\Delta H_{\text{system}}$ (since heat lost by the system is gained by the surroundings). At constant temperature:

3. Combine to get a system-only condition. Substituting into the inequality:

Multiply through by $T$ (positive):

Rearranging:

The quantity in parentheses is defined as the Gibbs free energy change:

So spontaneity means $\Delta G < 0$.

4. Connect $\Delta G$ to useful work. From the first law, the total work done by a system is $w = w_{\text{useful}} + w_{\text{expansion}}$, where $w_{\text{expansion}} = -P\Delta V$ (work done against constant pressure). At constant pressure, $\Delta H = q_P$ (heat at constant pressure). The change in internal energy is $\Delta U = q + w = q_P + w_{\text{useful}} - P\Delta V$. But $\Delta H = \Delta U + P\Delta V$, so:

The maximum work occurs for a reversible process, where $q_P = T\Delta S$. Thus:

Rearranging:

Therefore, $\Delta G$ equals the maximum useful work (other than expansion work) that the system can perform. This is why it is called "free" energy — it is the energy free to do useful work.

Unit of $\Delta G$ …