Principle of maximum work
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In thermodynamics, the principle of maximum work was a postulate put forward in 1875 by the French chemist Marcellin Berthelot which stated that chemical reactions will tend to yield the maximum amount of chemical energy in the form of work as the reaction progresses.
This is in fact a particular case of a more general statement:
| For all thermodynamic processes between the same initial and final state, the delivery of work is a maximum (and the delivery of heat is a minimum) for a reversible process. |
The principle of maximum work was a precursor to the development of the thermodynamic concept of free energy.
[edit] History
In thermodynamics, the Gibbs free energy or Helmholtz free energy is essentially the energy of a chemical reaction "free" or available to do external work. Historically, the "free energy" it is a more advanced and accurate replacement for the term “affinity” used by chemists, of olden days, to describe the “force” that caused chemical reactions. The term dates back to at least the time of Albertus Magnus in 1250.
From the 1998 textbook Modern Thermodynamics by Nobelist and chemical engineering professor Ilya Prigogine’s we find: “as motion was explained by the Newtonian concept of force, chemists wanted a similar concept of ‘driving force’ for chemical change? Why do chemical reactions occur, and why do they stop at certain points? Chemists called the ‘force’ that caused chemical reactions affinity, but it lacked a clear definition.
During the entire 18th century, the dominate view in regards to heat and light was that put forward by Isaac Newton, called the “Newtonian hypothesis”, which stated that light and heat are forms of matter attracted or repelled by other forms of matter, with forces analogous to gravitation or to chemical affinity.
In the 19th century, the French chemist Marcellin Berthelot and the Danish chemist Julius Thomsen had attempted to quantify chemical affinity using heats of reaction. In 1875, after quantifying the heats of reaction for a large number of compounds, Berthelot proposed the “principle of maximum work” in which all chemical changes occurring without intervention of outside energy tend toward the production of bodies or of a system of bodies which liberate heat.
[edit] Mathematical derivation
Suppose we have a general thermodynamic system, called the "primary" system. Suppose also that we mechanically connect it to a "reversible work source". A reversible work source is a system which, when it does work, or has work done to it, does not change its entropy. It is therefore not a heat engine and does not suffer dissipation due to friction or heat exchanges. A simple example would be a frictionless spring, or a weight on a pully in a gravitational field. Suppose further, that we thermally connect the primary system to a third system, a "reversible heat source". A reversible heat source may be thought of as a heat source in which all transformations are reversible. For such a source, the heat energy δQ added will be equal to the temperature of the source (T) times the increase in its entropy. (If it were an irreversible heat source, the entropy increase would be larger than δQ/T)
Define:
-
The loss of internal energy by the primary system 
The gain in entropy of the primary system 
The gain in internal energy of the reversible work source 
The gain in entropy of the reversible work source 
The gain in internal energy of the reversible heat source 
The gain in entropy of the reversible heat source 
The temperature of the reversible heat source
We may now make the following statements
-
(First law of thermodynamics) 
(Second law of thermodynamics) 
(Reversible work source) 
(Reversible heat source)
Eliminating dSw, δQ, and dSh gives the following equation:
When the primary system is reversible, the equality will hold and the amount of work delivered will be a maximum. Note that this will hold for any reversible system which has the same values of dU and dS . Note also that the maximum amount of work recoverable is -dA where A is the Helmholtz free energy of the primary system.
