Helmholtz Free Energy

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: : $A=U-TS\,$ where ''A'' is the Helmholtz free energy (), ''U'' is the internal energy of the system (SI: ), ''T'' is the absolute temperature ( Kelvin s), ''S'' is the per Kelvin ). NON-VISCOUS FLUIDS From the First Law Of Thermodynamics we have: : $dU = \delta Q - \delta W\,$ where $U$ is the internal energy, $\delta Q$ is the energy added by heating and $\delta W=PdV$ is the work done by the system. From the Second Law Of Thermodynamics , for a Reversible Process we may say that $\delta Q=TdS$ . 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, : $dA \le - pdV - SdT\,$ It is seen that if a thermodynamic process is isothermal (i.e. occurs at constant temperature), then ''dT = 0'' and thus : $dA \le -\delta W\,$ 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: : $dA \le 0\,$ 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: : $dA \le - pdV - SdT + \sum_i \mu_i dN_i\,$ where $\mu_i$ is the chemical potential for an i-type particle, and $N_i$ is the number of such particles. 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. GENERALIZED HELMHOLTZ FREE ENERGY In the more general case, the mechanical term ( $pdV$ ) must be replaced by the product of the volume times the Stress times an infinitesimal Strain : : $dA \le V\sum_{ij}\sigma_{ij}darepsilon_{ij} - SdT + \sum_i \mu_i dN_i\,$ where $\sigma_{ij}$ is the stress tensor, and $arepsilon_{ij}$ is the strain tensor. In the case of linear Elastic materials which obey Hooke's Law , the stress is related to the strain by: : $\sigma_{ij}=C_{ijkl}arepsilon_{kl}$ where we are now using Einstein Notation for the tensors, in which repeated indices in a product are summed. We may integrate the expression for $dA$ to obtain the Helmholtz free energy: : $A = rac{1}{2}VC_{ijkl}arepsilon_{kl}^2 - ST + \sum_i \mu_i N_i\,$ : $= rac{1}{2}V\sigma_{ij}arepsilon_{ij} - ST + \sum_i \mu_i N_i\,$ SEE ALSO Work Content - for applications to Chemistry Statistical Mechanics : 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 Press
	Last	Landau
	First	L D
	Authorlink	Lev Landau
	Coauthors	Lifshitz, E M
	Languange	English
	Others	(Translated from Russian by JB Sykes and WH Reid)
	Year	1986
	Title	Theory of Elasticity (Course of Theoretical Physics Volume 7)
	Edition	Third ed
	Publisher	Butterworth Heinemann
	Location	Boston, MA
	Id	ISBN: 075062633X

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Helmholtz Free Energy