Entropy is a starting point for many advanced theoretical statistical
mechanics books but beyond it just being negative or positive, what
does it actually tell you? Specifically, what does the magnitude of
it's change tell you?
For one thing, it tells you if a process is irreversible the entropy generated due to the irreversible process results in less energy available to produce net work compared to a reversible process where no entropy is generated. The greater the magnitude of the entropy generated the less energy available to produce net work.
For the 1st law, a calculation of energy change can result in us
knowing the work and heat which are very practical things because it's
how I can set things in motion.
Correct, but the first law is only a statement of conservation of energy. It does not tell us the direction that processes can take. For example, the first law (conservation of energy) does not preclude the possibility of heat transferring naturally from a cold temperature body to a high temperature body, though that has never been observed in nature. Only the second law precludes that possibility.
But an entropy calculation only tells me about the reversible heat
absorbed from a reservoir at temperature T, so unless I know the
actual temperature then how is this a useful parameter for a
calculation?
With respect to heat transfer, the entropy calculation always involves the temperature at the boundary between the system and the surroundings. In the case of the surroundings being a thermal reservoir, that temperature is the temperature of the thermal reservoir.
My suspicion is that entropy change is useful for an engineer to know
how much maximum/minimum work can be produced from a cyclic process
but I'm not entirely sure how to show this with the thermodynamic
equations so I would be interested to see how this can be shown.
In most instances, where it is possible for a reversible and irreversible path to connect the same to equilibrium states, your suspicion is correct. An example I provided in one of your previous posts comparing a reversible isothermal path to an irreversible isothermal path between the same two equilibrium end states illustrates this. See the Figure below.
Clearly, the work done for the reversible isothermal process (area under the PV=constant curve) is greater than the work done for the irreversible isothermal process (area under the constant external pressure curve).
Hope this helps.
