TΔS

Work-transformed-into-Heat
Heat-transformed-into-Work
Couple arguingman at work
mechanical equivalent of heat
The Joule paddle wheel experiment (1843) representative of the work transformed into heat aspect of the mechanical equivalent of heat:

 W = JQ \,

which states that W can convert into Q.
The Hero aeolipile (50AD) representative of the heat transforming into work aspect of the mechanical equivalent of heat:

 Q = \frac{W}{J} \,

which states that Q can convert into W.
In equations, the function:

 T \Delta S \,

or TdS, in differential form, is called the bound energy, or transformation content energy change, or energy associated with the equivalence-value of all uncompensated transformations, of a process or reaction occurring in a system, where T is the absolute temperature of the system, and dS is the differential entropy change and ΔS is the entropy change on going from an initial state entropy to a final state entropy. In expanded form:

 T \Delta S = T (S_f - S_i) \,

where Sf is the entropy measure of the system in its final state and Si is the entropy of he system in its initial state.

History
The function was first derived in the work of Rudolf Clausius (1850-1865) and expanded on by Willard Gibbs in 1876.

The labeling of the function TdS as "bound energy", in contrast to "free energy" (internal energy less bound energy) was done in 1882 by German physicist Hermann Helmholtz in his “On the Thermodynamics of Chemical Processes”. [1]

References
1. Helmholtz, Hermann. (1882). “On the Thermodynamics of Chemical Processes”, in: Physical Memoirs Selected and Translated from Foreign Sources, 1: 43-97. Physical Society of London, Taylor and Francis, 1888.

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