Inexact differential defined

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

We must recall that the process or processes that we have been discussing have not been completely defined that is, we have not stated whether the process is adiabatic or isothermal, or whether any specific quantity of heat has been added to or removed from the system during the process. Although we have essentially defined the initial state of the system, we have not defined its final state, neither have we defined the path that we choose to connect the two states. When we do so, we find by experience that the quantity of work done by the system depends upon the path and, therefore, the differential quantity of work, dlT, is an inexact differential quantity. 

In thermodynamics, the quantity fV is defined as the work done by the system on its surroundings in a specified change of state along a specified path. The path must be specified, since dW is an inexact differential form. Physically, the line integral depends upon the path because of the inclusion of dissipative forces. Rigorously, the definition of W demands that the initial and the final states of the system be equilibrium states. The work done by the system when the only external force is a uniform normal pressure is given by... 

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