# What are Helmholtz free energy and Gibbs free energy?

Jun 6, 2017

Both Helmholtz and Gibbs free energies are important thermodynamic functions known as Thermodynamic potentials.

The Helmholtz free energy is defined as,

$A = U - T S$

Where, $U$ is the internal energy, $T$ is the absolute temperature and $S$ is the entropy.

The above definition may be obtained from the internal energy function by means of one Legendre transform.

The Helmholtz free energy has $\left(T , V\right)$ as the natural pair of variables.

Differentiating the expression for $A$,

$\mathrm{dA} = \mathrm{dU} - T \mathrm{dS} - S \mathrm{dT}$

Using the combined mathematical form of first and second laws of thermodynamics, $T \mathrm{dS} = \mathrm{dU} + p \mathrm{dV}$,

$\implies \mathrm{dA} = - p \mathrm{dV} - S \mathrm{dT}$

Thus, $A = A \left(V , T\right)$
That is why the Helmholtz free energy is known as thermodynamic potential at constant volume.
It stays constant during any isothermal-isochoric change.

For such a system, the Helmholtz free energy tends to minimize as the system tends to equilibrium.

Now coming to Gibbs free energy, the expression is,

$G = U + p V - T S$ where symbols have their usual meaning.

The above relation may be derived from the internal energy function by means of Legendre's transformations to change variables.

It may also be cast in the form,

$G = H - T S$ where, $H = U + p V$ is the the enthalpy.

Now, Differentiating $G$,

$\mathrm{dG} = \mathrm{dU} + p \mathrm{dV} + V \mathrm{dp} - S \mathrm{dT} - T \mathrm{dS}$

Again using the combined mathematical form of the first and second law of thermodynamics (for reversible transformations),

$\mathrm{dG} = V \mathrm{dp} - S \mathrm{dT}$

Thus, $G = G \left(T , p\right)$
The Gibbs function is also called thermodynamic potential at constant pressure.

For an Isothermal-isobaric transformation, $G$ is constant.
Such a system tending to equilibrium requires $G$ to be minimum.

It may also be of some interest to mention that the specific heats at constant volume and pressure are related respectively to $A$ and $G$ as -

${C}_{v} = - T {\left(\frac{{\partial}^{2} A}{\partial {T}^{2}}\right)}_{v}$

And

${C}_{p} = - T {\left(\frac{{\partial}^{2} G}{\partial {T}^{2}}\right)}_{p}$

Jun 6, 2017

The Gibbs' free energy is the energy available to do non-PV work in a thermodynamically-closed system at constant pressure and temperature.

The Helmholtz free energy is the maximum amount of "useful" (non-PV) work that can be extracted from a thermodynamically-closed system at constant volume and temperature.

The following are derivations that prove these definitions at the undergraduate level. It will involve some minor multi-variable calculus.

($\delta$ will indicate path functions (inexact differentials), meaning those that are dependent on the path taken, and are not state functions.)

GIBBS FREE ENERGY

The Gibbs' free energy is a thermodynamic function of the natural variables, temperature $T$ and pressure $P$.

At constant $T$ and $P$, though, in a thermodynamically-closed system, the following formulas are useful:

$\mathrm{dG} = \mathrm{dH} - T \mathrm{dS} - {\overbrace{\cancel{S \mathrm{dT}}}}^{\Delta T = 0 \text{ assumed}}$ $\text{ "" } \boldsymbol{\left(1\right)}$

(recall $\Delta G = \Delta H - T \Delta S$ from general chemistry.)

$\mathrm{dH} = \mathrm{dU} + P \mathrm{dV} + {\overbrace{\cancel{V \mathrm{dP}}}}^{\Delta P = 0 \text{ assumed}}$ $\text{ "" } \boldsymbol{\left(2\right)}$

(which may be familiar from general chemistry as $\Delta H = \Delta U + P \Delta V$ at constant pressure.)

By plugging $\left(2\right)$ into $\left(1\right)$, we obtain:

$\mathrm{dG} = \mathrm{dU} + P \mathrm{dV} - T \mathrm{dS}$

If we examine only the non-PV work, i.e. the mechanical work that does not relate to expansions/compressions, we ignore PV work ($\Delta V = 0$ too).

$\implies \mathrm{dG} = \mathrm{dU} - T \mathrm{dS}$$\text{ "" } \boldsymbol{\left(3\right)}$,

From the first law of thermodynamics, the change in internal energy $U$ as a function of reversible heat flow ${q}_{r e v}$ and reversible work ${w}_{r e v}$ (with respect to the surroundings) is:

$\mathrm{dU} = \delta {q}_{r e v} + {\cancel{\delta {w}_{r e v , P V}}}^{\text{assumed 0}} + \delta {w}_{r e v , n o n - P V}$$\text{ "" } \boldsymbol{\left(4\right)}$,

where $\delta {w}_{r e v , P V} = - P \mathrm{dV}$.

As a result, from utilizing $\left(4\right)$ in $\left(3\right)$:

$\mathrm{dG} = \delta {q}_{r e v} + \delta {w}_{r e v , n o n - P V} - T \mathrm{dS}$

Lastly, entropy can be defined as the influence of the reversible heat flow at a given temperature:

$\mathrm{dS} = \frac{\delta {q}_{r e v}}{T}$,

So for reversible heat flow:

$\mathrm{dG} = \cancel{T \mathrm{dS}} + \delta {w}_{r e v , n o n - P V} - \cancel{T \mathrm{dS}}$

$\textcolor{b l u e}{\mathrm{dG} = \delta {w}_{r e v , n o n - P V}}$ $\text{ }$($\text{const. T and P}$)

So, in a reversible process in a thermodynamically-closed system, the Gibbs' free energy is the maximum amount of non-PV work with respect to the surroundings that can be accomplished at constant temperature and pressure.

HELMHOLTZ FREE ENERGY

We define $A = A \left(T , V\right)$, the Helmholtz free energy. The analogous relation to $\left(1\right)$ at constant temperature and volume in a thermodynamically-closed system is:

$\mathrm{dA} = \mathrm{dU} - T \mathrm{dS}$ $\text{ "" } \boldsymbol{\left(5\right)}$

(Note that even though $\left(3\right)$ seems to imply that $\mathrm{dA} = \mathrm{dG}$, these are different conditions. These are constant $T$ and $V$, not $T$ and $P$.)

From $\left(4\right)$ again, and the definition of entropy, we have plugging into $\left(5\right)$:

$\mathrm{dA} = \cancel{T \mathrm{dS}} + {\overbrace{\cancel{\delta {w}_{r e v , P V}}}}^{\text{0 at const. V}} + \delta {w}_{r e v , n o n - P V} - \cancel{T \mathrm{dS}}$

As a result,

$\textcolor{b l u e}{\mathrm{dA} = \delta {w}_{r e v , n o n - P V}}$ $\text{ }$($\text{const. T and V}$)

and the Helmholtz free energy for a reversible process in a thermodynamically-closed system is analogously the maximum available non-PV (useful) work with respect to the surroundings at constant temperature and volume.