What is Helmholtz Free Energy?

The Helmholtz free energy is a thermodynamic potential used to calculate the measure of useful work from a closed system. This system is kept at constant temperature and volume. German physicist Hermann Helmholtz introduced this concept in 1882.

In thermodynamics, free energy is a state property that has the dimension of energy. In other words, free energy is a state variable of a system at thermodynamic equilibrium. In thermodynamics, we express free energy in two forms. The first is Helmholtz free energy, and the second is Gibbs free energy. The direction of the spontaneity of reaction and the maximum useful work that the system can do can be calculated using the changes in free energy.

How to Calculate Helmholtz Free Energy?

The equation for Helmholtz free energy is:

F=U -TS

Here, F is the Helmholtz free energy, U is the internal energy, T is the absolute temperature, and S is the final entropy of the system.

We can think of internal energy (U) as the energy needed to create a system at constant volume and temperature. For creating this system in an environment with absolute temperature T, some part of the energy can be obtained via heat transfer from the system's environment. The amount of this energy transfer via heat is given as TS (can be seen in the formula of Helmholtz free energy), where S accounts for the final entropy of the system.

Relation Between the Thermodynamic Potentials

Schroeder suggested that a single mnemonic diagram can describe the relationship between all four thermodynamic potentials. In this diagram, as we move from left to right, the energy reduces as the environment term “TS”, and as we move from top to bottom, the energy adds as volume expansion work term “PV”. The diagram is shown below:

For example, to calculate Gibbs free energy of the system using enthalpy, subtract TS as Gibbs free energy is placed adjacent right of enthalpy.

G=H -TS

How Formula for Helmholtz Energy is Derived?

Consider that a closed thermodynamic system takes heat dQ from the surroundings while the temperature of the system remains constant.

Thus,

Entropy gain by the system = dS

Entropy lost by surrounding =  $\frac{dQ}{T}$

The net change in entropy must be positive by the second law of thermodynamics.

Using Clausius inequality,

$dS⩾\frac{dQ}{T}$

We can rewrite above equation as:

$TdS=dQ$T

Using first law of thermodynamics,

$TdS=dW+dU$

We can rewrite above equation as:

$TdS-dU=dW$

Integrating both sides,

${\int }_0^{W}dW⩽{\int }_{S_i}^{S_r}dS-{\int }_{U_i}^{U_r}dU$

Solving the above integrals,

$W⩽T(S_r-S_i)-(U_r-U_i)W⩽(U_i-T{S}_i)-(U_r-T{S}_r)$

Here, $(U_i-T{S}_i)$ is initial Helmholtz function and $(U_r-T{S}_r)$ is final Helmholtz function. The difference of both these functions is greater than or equal to the magnitude of work done.

Condition for Spontaneity of a Process

In an isothermal process, free energy decrease limits, the total amount of work done. This follows from the relation between free energy change and the work done. Moreover, an increase in the free energy of a reversible process will require work on the system. In case we do not extract work from the system,

$F⩽0$

Thus, for a feasible change to happen, net Helmholtz’s free energy will decrease in a system on which no work is extracted. Note that the system must be maintained at constant volume and temperature.

Gibbs Free Energy and its Comparison with Helmholtz Free Energy

Gibbs free energy is also a thermodynamic variable and it is given by the formula:

G=U-TS+PV

Change in Gibbs free energy is a useful parameter to understand thermodynamic processes. In a process, Gibbs free energy is the maximum amount of work that can be obtained. Similar to Helmholtz free energy, Gibbs energy also reaches its minimum value when the system is at equilibrium.

As discussed earlier, when the temperature of the surrounding environment is constant and equal to T, it contributes energy TS to the system. Thus, the total energy requirement reduces. This energy investment needed to create a system in the environment of temperature T, starting from the negligible initial volume, is called Gibbs free energy.

Relation between Gibbs Free Energy and Spontaneity

When temperature and pressure are constant, the change in Gibbs energy during a thermodynamic process is

$∆G=∆H-T∆S$

For a thermodynamic process to be spontaneous, change in Gibbs free energy must be negative, i.e., ∆G≤0. According to the equation of change in Gibbs free energy, as we increase the temperature, Gibbs free energy becomes more negative. Thus, the spontaneity of a process can depend on temperature.

For example, let us consider the case when the ice melts at a temperature of $0.01°\mathrm{C}$, and the solid and liquid phases are in equilibrium. We know that the latent heat of fusion of water is ∆H=79.7 $\mathrm{cal} {\mathrm{g}}^{-1}$.

By the definition of entropy,

$∆S=\frac{∆H}{T}=0.292 \mathrm{cal} {\mathrm{g}}^{-1} {\mathrm{K}}^{-1}$

By the definition of Gibb's free energy, ∆G=∆H-T∆S. Here, ∆G=0, which indicates that the two phases are in equilibrium and no work can be extracted from this transition. As we increase temperature, Gibbs free energy becomes more negative, indicating that the feasible direction of change is from ice to water.

Difference between Gibbs Free Energy and Helmholtz Free Energy

The following table lists some key differences between Gibbs and Helmholtz free energy.

How Helmholtz Free Energy is related to other Thermodynamic Variables?

In a reversible process, the negative change in Helmholtz energy is nothing but the maximum useful work that can be extracted from a system. This system must be kept at constant temperature and volume.

By the definition of Helmholtz free energy,

F=U-TS

The expression for entropy is:

$S=-\frac{\partial F}{\partial T}(V, N)$

The expression for pressure is:

$P=-\frac{\partial F}{\partial V}(T, N)$

The expression for chemical potential is:

$\mu =-\frac{\partial F}{\partial N}(T, V)$

Common Mistakes

Helmholtz free energy is many times confused with Gibbs free energy, as both are free energies and provide a measure of useful work. But it is important that students must know what are differences between both these free energies and where both these find applications.

Context and Applications

This topic finds its application in university courses related to thermodynamic systems and thermal physics.
It is significant for the following programs:

Bachelor of Science in physics

Master of Science in physics

Doctor of Philosophy in physics

Related Concepts

Gibbs free energy

Enthalpy

Grand potential

Statistical mechanics

Bennett acceptance ratio

Practice Problems

Q.1 Helmholtz free energy is?

(a) is a state variable

(b) is a path variable

(c) independent of temperature

(d) is always equal to the internal energy of the system

Correct option: (a)

Q.2 For a system at constant volume and temperature, it is given that no work is being extracted from the system, then for a feasible change, net Helmholtz free energy must?

(a) Decrease

(b) Increase

(c) Remain constant

(d) Cannot say

Correct option: (a)

Q.3 Correct expression for Helmholtz free energy is?

(a) F=U+TS

(b) F=U-TS

(c) F=U-ST

(d) F=H-TS

Correct option: (b)

Q.4 Relation between Helmholtz free energy (F) and Gibb’s free energy (G) is given as:

(a) G=F+PV

(b) F=G+PV

(c) F=G-TS

(d) F=G+TS

Correct option: (a)

Q.5 For a process to be spontaneous ∆G  (change in Gibb’s free energy) must be:

(a) Negative

(b) Positive

(c) Zero

(d) Cannot say

Correct option: (a)