Consider that we are extracting energy out of a system. The first law of thermodynamics is traditionally written^7,8 in terms extracted energy (Qout), internal energy change (dE) and work done (W):

      Qout = dE(internal) – W               eqn 3a)

   Herein Q is the total energy into the system and W is the total work done external to the system. If we consider that the internal energy change (dE) is the summation of all of the energy changes within a system (dEtot). Then a less confusing way to rewrite eqn 3) is:  

      Qout = dE(tot) – W                     eqn 3b)

Sure eqn 3a) and 3b) are similar, but eqn 3b) gives clarity in that it states the total (whole) system’s energy changes.

Or yet another way of writing

    Now consider that energy is being put into a system, then in terms of thermal energy into  a system (Ein), the first law is written:

No matter how we choose to write it, the first law states that “0energy is conserved”.

In writing the first law W is the total work done by the system.

The first law should never be written in terms of work do onto the system because that allows for unnecessary confusion, yet often you may find it written that way! Think about it, if work is done onto a system in question, then that work may, or may not, result in an energy change to that system! And any energy change to that system is part of system’s total energy change, whether you choose to write it i.e. dE, dE(internal) or dE(tot)  

Never forget that if work is limited to the displacement of our atmosphere by the some expanding system, then that work is lost work:

    W = PdV = P(atm)dV       (4)

Never forget: The atmosphere can only do work onto a system if it’s pressure is significantly greater than the pressure within that system. And in real life this only occurs when negative work was done in creating that system in the first place.  

   If an expanding system also moves an object with mass, does work onto a car, then the total work done is always:

   W = PdV + W(car) = P(atm)dV + W(car)     (5)

If you include the work done onto the car (or anything else) in the PdV term then you are only asking for undue confusion, as can be to often witnessed in tradiational writings of thermodynamics.

In some ways we have done nothing but cut hairs as often the total energy change of a system may arguably be equal to the change of the system’s internal but this may depend upon one’s definition of internal energy. So why not keep things simple and write the first law in terms of the system’s total energy change. It avoids confusion and it tells us that the work done is done to system’s surroundings and/or devices attached to the system’s exterior.

One needs to realize that a gaseous system’s thermal energy and work are never equal. Remember is work defined by the mechanical parameter P & V. While a system’s energy is defined by its temperature See work vs energy.

    Furthermore work is most often extracted from gases. Specifically work can be extracted from a gas’s translational and rotational energy but not necessarily its vibrational energy.  See kinetic theory