9.) Consider the simplest system, where the energy state function has only
one x variable (eg., this might be a gas with fixed mole number and x
being the gas volume). The energy function is then some E(S,x) and
dE(S,x) = TdS + Ydx, where T(temperature) and Y are the partial
derivatives (for the gas, Y would be the pressure P). This E(S,x)
references only one non-thermal mode of energy exchange with the
environment (in the "canonical" form dW = Y*dx.
Of course there is also an equation of state (relating say T, Y and x)
which can be used to express E in terms of other state variables. It
might also be noted that the function E(S,x) can be inverted to make
explicit the entropy function S(E,x), which can also be written in terms
of other state variables. In what follows we assume that we explicitly
know the energy function E(S,x) and the equation of state, say Y(x,T) for
the system under consideration.
10.) In principle, dE = TdS + Ydx can be integrated over any path which is
a continuum of equilibrium states (ie., in which all state variables are
well defined). (In practice this integration might be done in T/x , Y/x ,
etc space.) Of course this integration will simply result in the energy
function change (E2 - E1).
11.) In a typical practical problem. one knows only the initial (1)
state - one does not know a priori the final (2) state, and furthermore
the process is irreversible, so that the intermediate states/variables are
not even definable throughout the real process. In general, only the
initial state and the (unknown) final state are defined equilibrium states
with defined state variables.
12.) What one does know in the typical problem under consideration is a
means of calculating the energy transfers to/from the system during the
stated process. One can calculate the RHS of Delta(E) = Q + W, where the
RHS terms are simply quantities of transferred energy evaluated from an
appropriate model of the system/process; they are not the change in any
state function.
13.) For example, suppose the process consisted in
a) holding x fixed,
b) injecting a measured amount of microwave energy into the (otherwise
thermally insulated) system, and
c) dissipating into the system a measured amount of kinetic energy from a
rotating paddle wheel.
The RHS of Delta(E) = Q + W is known through the behavior of the external
environment, and the problem is solved. People may argue about what is Q
and what is W, but that is of no consequence to any testable conclusion.
It matters not whether you tell yourself "this is Q" or "this is W" as you
write down the numbers; that concerns only a personal, intermediate
accounting convention adopted for calculational/conceptual convenience.
The numerical value of the transferred energy is un-arguable, and that is
all that matters. Once the total energy increment is known, the new E, S,
etc can be calculated. If you have a model for the "internal" or
"thermal" energy of this system, it can now be evaluated in terms of the
new values of the state variables.
14.) Think of the simple gas system and suppose we also allow x (volume)
to vary by allowing a known weight to fall, pushing a piston and
compressing the gas a measurable amount (irreversibly). Another term must
be added to account for the additional energy transferred to the system.
This is easily evaluated as the gravitational potential energy lost by the
falling weight. Note that this term cannot be evaluated as Y*dx, using the
system state variables - they are not even defined during this process.
Again, the energy transferred is evaluated by looking at the environment -
especially when the system state is undefined during the process. And
again, as before, once the final state is identified - and only then - one
can evaluate the new E, S, etc values and even say how much of the new
system energy is now "thermal", "internal", etc according to some pet
model which one finds useful.
Excuse the typo boo-boos (they're always with me).