1. Thermodynamics was originally created to understand the interaction between heat and work in steam engines. The internal energy U of the steam, presumed to exist in a tube with mobile plug, is said to be given by:
dU = where Q is heat, W work and stands for so called 'inexact differential', which means that only the sum of heat and work counts. Individually they can have any value and their value is path dependant, if one or the other comes first. This is true even if the work is applied reversibly.
When no work is applied dQ = T dS where T is temperature and S entropy. When the steam is being heated with no work transfer, the last entropy added required TdS heat. Earlier entropies required less heat because the temperature was then lower. In heat transfer, temperature is the intensive variable and entropy the extensive variable. We can plot temperature versus entropy. At zero temperature the entropy is zero. Temperature is the slope of the curve.
When work is applied adiabatically dW = FdX, where F is force and X the column length. X is the extensive and F the intensive variable. At X = infinite the steam can no longer transfer work and the compressive energy of the steam is zero. We can plot energy versus column length.
The two plots allow us to generate a 3D plot, energy versus entropy and column length, with energy as a surface and temperature and force as directional slopes. At zero entropy and infinite column length the surface is at zero. The surface is continuous.
According to Maxwell, the slopes of the surface are related as:
which can be given as integral as:
F =
or: F dX =
This means for steam compression: Work input means the increase of the heat requirements of all the entropy differentials of the steam but no entropy increase.
Heat input increases the entropy at the last heat requirement but does not alter the heat requirements of pre-existing entropies.
The last two equations can not be evaluated over its whole width since dT can only be measured while the steam is stable and we do not know the absolute value of entropy.
The question is if other forces, force against external force fields or altering the centre-of-mass velocity, have the same influence on the internal conditions of the system.

2.

3. Congratulations, you arrived in 18th century. Sadly your version of "Maxwell relation" doesnt make any sense and has different units on other sides. Your integrals doesnt make any sense either.

5. If the surface is continuous, it is a standard mathematical formula that two directional slopes , in this case T and F, are related with the extensive variables as
. The formula is one of several formula developed by Maxwell, a friend of Newton's, (18th century).

6. Originally Posted by Gere
Congratulations, you arrived in 18th century. Sadly your version of "Maxwell relation" doesnt make any sense and has different units on other sides. Your integrals doesnt make any sense either.
Let us dimensionally analyse the tree equations: Dimensions of F =
of entropy S =
of temperature T =
1.

2. F =

3. F dX=

Obviously the three equations have passed the dimensional analysis.

7. I am rather disappointment by the unfairness of reviews. If you do not understand an equation, why not ask for an explanation. I was looking forward to a further development of the ideas. If you do not know Maxwell relationships that is understandable. I could have used another method to develop the equation. It is quite clear that, according to equation 2, the force pressing on the plug of the tube containing the steam is related to temperature increases of each entropy differential of the steam when the volume or column length is is altered. That can be checked in any steam table. I have not been able to find out how I can put a drawing into a reply. That might have made it clearer.