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At its basis, the energy equation equates system energy change to the causes of energy change. The causes or mechanisms are energies of mass transfers (in or out), the sum of relevant work modes and the and the sum of heats and friction. We find energy equations written in three mathematical forms which relate the aspects of time. The forms are differential, increment, and rate.
These mathematical forms are not unique - each can be transformed into either of the other two. The differential form is of use with the mathematics of equation development and modification. The increment form applies to the many events having a "start-stop" or "before-after" nature.
One might ask, "There being three equation forms, each equivalent to the other two, is any of the forms superior to the others? If so, which form and how so?"
In a physical sense, the rate form is superior. Time proceeds. Everything that exists does so as the rate of time goes on. The increment form addresses an era, part of the past. The differential form applies to time in its differential limit - vanishingly small. The rate form includes time properly, as proceeding.
Equations (1), being "symbolic;" omit detail. Our goal is a "rate form" for typical "normal" heating, cooling or phase change occurance. To proceed we identify energy, in Eqn-1, as "extrinsic and intrinsic" and work as having extrinsic and intrinsic components. When a "summation sign" is written, it means "sum over all instances."
Most substances are classed as "simple" meaning "has only one appreciable work mode." That work mode all substance have is the "compression work." The energy of all substances can be changed by compression. Also with every event there is friction and often more than one heat interaction.
The compression work mode and its "rate," are:
For the cases we will address now, pressure is constant. This permits the compression work rate to be simplified (takes a step or so):
Algebra of the compression work rate term.
Now place this in Eqn-3 and do a little more algebra:
So with a bit of manipulation we come out with something simple.