Helmholtz free energy

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The Helmholtz free energy is a thermodynamic potential which measures the "useful" work obtainable from constant temperature, constant volume thermodynamic systems. It is sometimes known as the "work content". For a simple system, with a fixed number of particles, the negative of the difference in the Helmholtz free energy is equal to the maximum amount of work extractable from a thermodynamic process in which temperature is held constant.

The Helmholtz free energy was developed by Hermann von Helmholtz and 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:

<math>A=U-TS\,</math>

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).

Mathematical development

From the first law of thermodynamics we have:

<math>dU = \delta Q - \delta W\,</math>

where <math>U</math> is the internal energy, <math>\delta Q</math> is the energy added by heating and <math>\delta W=PdV</math> is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that <math>\delta Q=TdS</math>. Differentiating the expression for A  we have:

<math>dA = dU - (TdS + SdT)\,</math>

<math>= (TdS - pdV) - TdS - SdT\,</math>

<math>= - pdV - SdT\,</math>

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

<math>dA \le - pdV - SdT\,</math>

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

<math>dA \le -\delta W\,</math>

The negative of the change in 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.

If, in addition the volume is held constant as well, the above equation becomes:

<math>dA \le 0\,</math>

with the equality holding at equilibrium. It is seen that the Helmholtz free energy for a general system in which the temperature and volume are held constant will continuously decrease to its minimum value, which it maintains at equilbrium.

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:

<math>dA \le - pdV - SdT + \sum_i \mu_i dN_i\,</math>

where <math>\mu_i</math> is the chemical potential for an i-type particle, and <math>N_i</math> is the number of such particles. With this definition, we may say that the negative of the Helmholtz free energy is the maximum amount of work energy available from a system in which the initial and final states have the same temperature and number of particles. Further generalizations 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.

See also

• work content - for applications to chemistry
• Statistical Mechanics

[this page details the Helmholtz free energy from the point of view of thermal and statistical physics.]

References

• Atkins' Physical Chemistry, 7th edition, by Peter Atkins and Julio de Paula, Oxford University Pressde:Helmholtz-Energie

it:Energia libera di Helmholtz ja:ヘルムホルツエネルギー pl:Energia swobodna sv:Helmholtz fria energi zh:亥姆霍茨自由能