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The work done is given by the dot product of the two vectors, where the result is a scalar. When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by: = If the force is variable, then work is given by the line integral:
For a proof, imagine two paths 1 and 2, both going from point A to point B. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B.
For a quasi-static adiabatic process, the change in internal energy is equal to minus the integral amount of work done by the system, so the work also depends only on the initial and final states of the process and is one and the same for every intermediate path. As a result, the work done by the system also depends on the initial and final states.
For conservative forces, path independence can be interpreted to mean that the work done in going from a point to a point is independent of the moving path chosen (dependent on only the points and ), and that the work done in going around a simple closed loop is :
The dependence of work on the path of the thermodynamic process is also unrelated to reversibility, since expansion work, which can be visualized on a pressure–volume diagram as the area beneath the equilibrium curve, is different for different reversible expansion processes (e.g. adiabatic, then isothermal; vs. isothermal, then adiabatic ...
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Energy gained by the system, through work done on it, is lost to the bath, so that its temperature remains constant. An adiabatic process is a process in which there is no matter or heat transfer, because a thermally insulating wall separates the system from its surroundings. For the process to be natural, either (a) work must be done on the ...