At constant temperature, the Helmholtz free energy is minimized at equilibrium.
It is also frequently used to define fundamental equations of state of pure substances.
The concept of free energy was developed by Hermann von Helmholtz, a German physicist, and first presented in 1882 in a lecture called "On the thermodynamics of chemical processes".
[1] From the German word Arbeit (work), the International Union of Pure and Applied Chemistry (IUPAC) recommends the symbol A and the name Helmholtz energy.
[2] In physics, the symbol F is also used in reference to free energy or Helmholtz function.
(ignoring electrical and other non-PV work) and so: Applying the product rule for differentiation to
allows us to rewrite this as Because F is a thermodynamic function of state, this relation is also valid for a process (without electrical work or composition change) that is not reversible.
The laws of thermodynamics are only directly applicable to systems in thermal equilibrium.
If we wish to describe phenomena like chemical reactions, then the best we can do is to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium.
If the system is kept at fixed volume and is in contact with a heat bath at some constant temperature, then we can reason as follows.
Therefore, the entropy change of the heat bath is The total entropy change is thus given by Since the system is in thermal equilibrium with the heat bath in the initial and the final states, T is also the temperature of the system in these states.
The fact that the system's temperature does not change allows us to express the numerator as the free energy change of the system: Since the total change in entropy must always be larger or equal to zero, we obtain the inequality We see that the total amount of work that can be extracted in an isothermal process is limited by the free-energy decrease, and that increasing the free energy in a reversible process requires work to be done on the system.
If no work is extracted from the system, then and thus for a system kept at constant temperature and volume and not capable of performing electrical or other non-PV work, the total free energy during a spontaneous change can only decrease.
In reality there is no contradiction: In a simple one-component system, to which the validity of the equation dF = −S dT − P dV is restricted, no process can occur at constant T and V, since there is a unique P(T, V) relation, and thus T, V, and P are all fixed.
To allow for spontaneous processes at constant T and V, one needs to enlarge the thermodynamical state space of the system.
In case of a chemical reaction, one must allow for changes in the numbers Nj of particles of each type j.
A system kept at constant volume, temperature, and particle number is described by the canonical ensemble.
The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values.
In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.
Combining the definition of Helmholtz free energy along with the fundamental thermodynamic relation
one can find expressions for entropy, pressure and chemical potential:[4] These three equations, along with the free energy in terms of the partition function, allow an efficient way of calculating thermodynamic variables of interest given the partition function and are often used in density of state calculations.
For example, for a system with a magnetic field or potential, it is true that Computing the free energy is an intractable problem for all but the simplest models in statistical physics.
, which has different interactions and may depend on extra parameters that are not present in the original model.
We can easily generalize this proof to the case of quantum mechanical models.
We denote the diagonal components of the density matrices for the canonical distributions for
we can replace: On the right-hand side we can use the inequality where we have introduced the notation for the expectation value of the operator Y in the state r. See here for a proof.
In the case of linear elastic materials that obey Hooke's law, the stress is related to the strain by where we are now using Einstein notation for the tensors, in which repeated indices in a product are summed.
to obtain the Helmholtz energy: The Helmholtz free energy function for a pure substance (together with its partial derivatives) can be used to determine all other thermodynamic properties for the substance.
Hinton and Zemel[6] "derive an objective function for training auto-encoder based on the minimum description length (MDL) principle".
The true expected combined cost is "which has exactly the form of Helmholtz free energy".