Helmholtz free energy

This page develops the Helmholtz free energy from the point of view of thermal and statistical physics. For applications to chemistry, see work content.

The Helmholtz free energy is a thermodynamic potential and is therefore a state function of a thermodynamic system. It is sometimes known as the "work content". For a simple system, with a fixed number of particles, the Helmholtz free energy is equal to the maximum amount of work extractable from a thermodynamic process in which the initial and final states are at the same temperature.

The Helmholtz free energy is denoted by the letter A  (from the german "arbeit" or work), or the letter F . The letter A  is preferred by IUPAC and will be used here.

The Helmholtz free energy is defined as:

$A=U-TS\,$

where

• A  is the Helmholtz free energy (SI: joules, CGS: ergs),
• U  is the internal energy of the system (SI: joules, CGS: ergs),
• T  is the absolute temperature (kelvins),
• S  is the entropy (SI: joules per kelvin, CGS: ergs per kelvin).

From the first law of thermodynamics we have for a reversible process:

$dU=\delta Q-\delta W\,$

where $U$ is the internal energy, $\delta Q=TdS$ is the energy added by heating and $\delta W=PdV$ is the work done by the system. Differentiating the expression for A  we have:

$dA=dU-(TdS+SdT)\,$

$=(TdS-pdV)-TdS-SdT\,$

$=-pdV-SdT\,$

For a process which is not reversible, the entropy will be smaller than its equilibrium value so we may say that, in general,

$dA\geq -pdV-SdT\,$

It is seen that if a thermodynamic process is isothermal (i.e. occurs at constant temperature), then dT = 0  and thus

$dA\geq -\delta W\,$

The Helmholtz free energy is the maximum work attainable from the system in an isothermal process. In more mathematical terms, the integral of dA over any isotherm in state space is the maximum work attainable from the system. However, the Helmholtz free energy is a state function, which means that dA is an exact differential. It follows that the integral over any path will give the same value for the maximum work, a value which is only dependent upon the location of the beginning and end points in state space. It follows that, for a simple two dimensional system, the Helmholtz free energy is the maximum work attainable from any thermodynamic process in which the initial and final states are at the same temperature.

In a more general form, the first law describes the internal energy with additional terms involving the chemical potential and the number of particles of various types. The differential statement for dA is then:

$dA\geq -pdV-SdT+\sum _{i}\mu _{i}dN_{i}\,$

where $\mu _{i}$ is the chemical potential for an i-type particle, and $N_{i}$ is the number of such particles. It is seen that not only must the SdT term be set to zero by requiring the temperatures of the initial and final states to be the same, but the $\mu _{i}dN_{i}$ terms must be zero as well, by requiring that the particle numbers remain unchanged. Any further generalization will add even more terms whose extensive differential term must be set to zero in order for the interpretation of the Helmholtz free energy to hold.

References

• Atkins' Physical Chemistry, 7th edition, by Peter Atkins and Julio de Paula, Oxford university press