8.1 Helmholtz and Gibbs free energies

Helmholtz and Gibbs Free Energies

Helmholtz and Gibbs free energies are thermodynamic potentials that tell you how much energy in a system is actually available to do useful work. The difference between them comes down to the constraints: Helmholtz applies at constant temperature and volume, while Gibbs applies at constant temperature and pressure.

These potentials are essential for predicting whether a process will happen spontaneously, identifying equilibrium states, and calculating maximum work output. They show up everywhere from chemical reaction analysis to battery design, and they form the foundation for the Maxwell relations you'll derive later in this unit.

Helmholtz Free Energy

Helmholtz Free Energy Definition

The Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed system at constant temperature and volume. It's defined as:

$A = U - TS$

where:

• $U$ is the internal energy
• $T$ is the absolute temperature (K)
• $S$ is the entropy (J/K)

The change in Helmholtz free energy, $\Delta A$, equals the maximum work extractable from a closed system during an isothermal process.

Physical Significance of Helmholtz Energy

Think of the internal energy $U$ as split into two parts. The Helmholtz energy $A$ is the portion available for useful work at constant $T$ and $V$. The remaining portion, $TS$, is energy that's "locked up" by the system's entropy and can't be harnessed.

• A decrease in $A$ during an isothermal, constant-volume process means the system can perform work on its surroundings. Examples include a battery discharging or a gas expanding against a piston in an isothermal setup.
• At equilibrium, $A$ is minimized for a system held at constant $T$ and $V$. This gives you a criterion for spontaneity: if a proposed change would lower $A$, it can happen spontaneously under those constraints.

Gibbs Free Energy

Gibbs Free Energy Definition

The Gibbs free energy measures the maximum reversible non-expansion work a system can perform at constant temperature and pressure. It's defined as:

$G = H - TS$

where:

• $H$ is the enthalpy (J)
• $T$ is the absolute temperature (K)
• $S$ is the entropy (J/K)

The change $\Delta G$ equals the maximum non-expansion work (also called "useful work") extractable during an isothermal, isobaric process. The distinction matters: because pressure is held constant, any $PdV$ expansion work is already accounted for in the enthalpy $H$, so $\Delta G$ captures only the work beyond that.

Physical Significance of Gibbs Energy

Gibbs energy plays the same role for constant $T$ and $P$ systems that Helmholtz energy plays for constant $T$ and $V$ systems. The enthalpy $H$ splits into a usable portion $G$ and an unavailable portion $TS$.

• A decrease in $G$ during an isobaric, isothermal process means the system can perform non-expansion work on its surroundings. This is the relevant quantity for electrochemical cells, fuel cells, and biochemical reactions, all of which operate near constant $T$ and $P$.
• At equilibrium, $G$ is minimized for a system at constant $T$ and $P$. This is why $\Delta G < 0$ is the standard spontaneity criterion for chemical reactions at fixed temperature and pressure.

Differential Forms of the Free Energies

The differential forms connect these potentials to measurable quantities and lead directly to the Maxwell relations.

Helmholtz Free Energy

Starting from $A = U - TS$, take the total differential:

1. $dA = dU - TdS - SdT$

2. Substitute the combined first and second law expression $dU = TdS - PdV$

3. $dA = TdS - PdV - TdS - SdT$

4. The $TdS$ terms cancel, giving:

$dA = -SdT - PdV$

This tells you that the natural variables of $A$ are $T$ and $V$. From this form you can read off:

• $\left(\frac{\partial A}{\partial T}\right)_V = -S$
• $\left(\frac{\partial A}{\partial V}\right)_T = -P$

Gibbs Free Energy

Starting from $G = H - TS$, take the total differential:

1. $dG = dH - TdS - SdT$

2. Substitute $dH = TdS + VdP$

3. $dG = TdS + VdP - TdS - SdT$

4. The $TdS$ terms cancel, giving:

$dG = -SdT + VdP$

The natural variables of $G$ are $T$ and $P$. From this form:

• $\left(\frac{\partial G}{\partial T}\right)_P = -S$
• $\left(\frac{\partial G}{\partial P}\right)_T = V$

These partial derivative relationships are the starting point for deriving the Maxwell relations in the next section of this unit.