When studying classical mechanics, work is defined as: $W_M=\int F_{tot} \hspace{2 mm} dx$.
However, for thermodynamics, work is defined as: $W_T=\int -F_{ext} \hspace{2 mm} dx$.
I'm having trouble reconciling both definitions. I have found the mathematical relationship between both of them (see below), but I wouldn't know how to interpret either this or the other thermodynamical functions in the light of this. For example, $\Delta U$ is usually related to the conservation of energy, but since the definition of $W$ is different, it isn't that direct a relationship now (for me at least).
$$ W_M=\int F_{tot} \hspace{3pt} dx \hspace{3pt} =\int F_{int}-F_{ext} \hspace{3pt} dx $$ $$ W_M=\int F_{int}\hspace{3pt} dx+W_T \hspace{3pt} $$
The books I've seen usually just throw the definition and start talking about the state functions, but do not spend any time reflecting on this definition or analyzing why is it different from the classical one. I found this source which tries to deal with this same issue: it is very interesting, but it gets a little confusing towards the end when he tries to relate his mechanical w and q with his thermodynamical W and Q (I don't understand how he combines equations 1 and 3 or 1 and 5 to get his results). Does anyone understand that last step?
Has anyone seen or thought about this issue before? Does anyone know of any other source or book where this issue is dealt with more detail?
EXAMPLE PROBLEM (to motivate the question): Consider a very big cylinder with a mobile piston in the middle separating two sections filled with gas (A and B). Both the cylinder walls and the piston are adiabatic, so there is no exchange of heat with the exterior or between sections. The pressure of the gas in A is higher than the pressure of the gas in B. The piston is originally being hold, but we let go of it for just a second and then hold it again (we assume the change in volume was small enough so as to not produce a change of pressures).
In this situation, I could consider the gas of section A as my system and claim it gave $p_B \Delta V$ of work to its exterior (the gas in section B, since it is the only exterior with which it interacts). But I could also consider the gas in section B as my system, and claim it received $p_A \Delta V$ of work from its exterior (gas in section A). But since $\Delta V$ is equal in both cases (only changes the sign, which is what determines is work is given or received) but the $P_i$ aren't, there is an energy difference I cannot account for.
My intuition tells me that the energy difference must come from the potential energy stored in the original pressure difference, but since thermodynamics (or the way I was taught thermodynamics) starts with a different definition of work and does not explain its relationship with the mechanical one, it is not that trivial to me how to incorporate this into a thorough thermodynamical analysis of this process.
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