Integration of inexact differentials in Thermodynamics

[Sokolov]

In Thermodynamics, I have seen that some equations are expressed in terms of inexact differentials, $\delta$, instead of $d$. I understand that this concept is introduced to point out that these differential forms are path-dependent, although I am not clear how they can be handled.

So, are there any special rules to be taken into account when doing operations with them (such as integrals) or can they be handled just as ordinary differential forms?

For example, with respect to the second law of thermodynamics, $dS= \frac{\delta Q}{T}$ , if $T$ remains constant, can it be integrated as if $\delta$ were an ordinary differential? And what about the first law, $dU=\delta Q + \delta W$? Would this operation be correct?

$\Delta U=\int dU= \int (\delta Q + \delta W)=\int \delta Q + \int \delta W= \Delta Q + \Delta W$

 

[Andrew Mason]

Q and W are path dependent whereas U depends only on the thermodynamic state. So the heat flow and work done in going between two states that are even slightly (infinitesimally) different does not define a precise value for δQ or δW. However if you define the path and express Q and W in terms of state functions of the system, then you can denote them as exact differentials eg:

• $dQ_{rev} = TdS$ or
• dQ = dU + PdV where dW = PdV or
• dQ = dU where volume is constant or
• dW = dU where the process is adiabatic.

AM