What is Wrong with Eqn (3)?

Again: Eqn (1) and/or (2) are path independant exact differentials while the work as defined in terms of the compressed gas is not, i.e eqn (3).  Eqn (3) is similar to eqn (10) in our expanding system blog, in that it is for isothermal compression and is based upon the ideal gas law (PV=NkT), i.e. quai-static pressure increase i.e. the addition of one small pebble at a time in Fig 1. Again this is because any heat generated is allowed to radiate out of the compressed system into the surrounding atmosphere (heat sink). And for infinitesimal processes eqn (3) approximates eqn (1).

 Now reconsider the first law: Qin=dEsys+Wdone. Since the compression is isothermal, then there is no energy change to the isothermally compressed gas, hence dEsys = 0. If the isothermal system's energy is constant, then how can one claim that work is done onto the gas is often traditionally done. Again the answer lay in not understanding that when one writes in terms of the compressed gas's parameters you are Not writing work in terms that can be readily applied to the first law! see parameters

 Conversely, if the compression is rapid then the gas's temperature will increase, in which case eqn (3) no longer applies. Again given sufficient time then enough heat (thermal energy) can radiate out through the piston-cylinder's walls, in which case the compressed system would be in thermal equilibrium with the surrounding atmosphere and eqn (3) would again be valid but only in terms of the compressed gas's parameters, which again is Not first law applicable. see parameters

 And if the piston-cylinder is insulated then compression would result in both  its pressure and temperature increasing and eqn (3) becomes a rough approximation at best. Moreover as the temperature increases then eqn (3)'s validity decreases!

  And if the compressed gas is then allowed to expand then the irreversible work (AKA Lost work) it does onto the atmosphere would be defined by eqn (2). And if the compressed gas does other work like move man and/or machine then that work is added to the lost work as defined by eqn (2).

  Most importantly: Just as was the case for expanding gaseous systems, eqn (3) would only be valid in traditionally accepted equations if the internal energy change included all energy changes other than those associated with the mechanical parameters P and/or V! see parameters

 Atmosphere and Compression

 What happens to the atmosphere during compression? During compression  no work is done onto the atmosphere. However any decrease in volume of the atmosphere will result a change of some of the atmosphere’s potential energy into kinetic energy. So although the atmosphere’s total energy remains the same during a system's compression, the atmosphere's experiences an infintesimal temperature increase, which is a direct result of its kinetic energy increase. It should be said by a decrease in the atmophere's volume, we actually are considering the compressed system as being part of the atmosphere's total volume! 

It must be stated that in terms of the atmosphere's actual temperature that such infintesimal temperature increases will in general NOT be measureable by a thermometer. However after an infinite/massive number of such compressions then the temperature increase could become measureable. Remember that the atmosphere is massive i.e. a heat sink, which means that real systems that give energy into the surrounding atmosphere will generally remain immeasurable unless the amount of energy given is massive. 

 And if the compressed piston-cylinder is allowed to expand  it will then increase the potential and/or thermal energy of the atmosphere as defined by eqn (3) i.e. W=PdV. Again infinitesimal arguments apply, in that it would take a massive number (infinite) of such expansions in order to become measureable.

 Never forget that the atmosphere is a heat sink/bath hence it takes vast amounts of energy for change to be readily observable. see infinitesimal change

Energy for Compression

 One must realize that the atmosphere did not cause the energy changes associated with compression. It was whatever was doing the compressing that forced the changes to the atmosphere's natural state, for example the addition of the pebbles in Fig 1. Moreover the compression, not only increased the potential of the gas within the piston-cylinder to do work, it also caused some of the atmosphere’s potential energy to transform into kinetic energy. And  it is this change of potential into kinetic energy that the reader may have the most difficulty in comprehending.

Work into a System via Compression

 I find it interesting that many will talk about work being done onto the gas during its compression. This is again making the mistakes that are beheld by eqn (3)  such as the internal energy change (dE) including all energy change except those assocaited with the mechanical parameters (P,V). see parameters

 To empahsize: Now consider that an ideal monatomic gas is compressed isothermally. Since its temperature remains constant then that compressed gas's kinetic energy also remains constant, i.e. there is no energy increase to the gas. 

 Now ask: Is work really done onto the gas? It depends upon how one envisions it. Although the isothermal gas's energy remains constant, certainly the gas's potential to do work increases as its pressure increases. Again this potential is really an increase in the work that the gas can do onto the surrounding atmosphere and be used to move man and/or his machines, when that compressed gas is allowed to expand.

 It would certainly be dangeous to try and apply entropy based conceptualizations to this. On the other hand one could consider that isothermal compression of a gas is a concentration of the gas's energy while its isothermal expansion of the same gas is a dispersal of its energy.