# Clarification on thermodynamics concepts of state function, exact differential

1. Jun 16, 2011

I would like someone to explain to me the correlations between these thermodynamics concepts:

1 State function/conservative force/reversibility
2 State function/exact differential

Some functions in the phase space of a system are state function some are not. Is this simply an empirical fact or is there something deeper about it?

AFAIK, heat, for instance, is an inexact differential, but it is also given by the mass times the specific heat times the variation in temperature so it is also expressible as an exact differential. How can it be?

Is the concept of state function only meaningful in a state of equilibrium?

What is the role of time in thermodynamics? Why we always write equation without any reference to time? Is it possible to write an equation of evolution of a thermodynamics system just like the laws of motion in classical mechanics?

2. Jun 16, 2011

### Andy Resnick

This is indeed a deep question. Part of the answer was given by Caratheodry

http://www.jstor.org/stable/2100026

"state space" in thermodynamics is a contact space, not a symplectic space. "conservative" mechanical theories occur in symplectic space (exact differentials, etc), while dissipative systems occur in a contact space. It's possible to embed the contact space into a symplectic structure, but that has a restricted region of validity (AFAIK).

I suspect you have been shown equations of thermostatics- thermodynamics is a fully time-dependent theory, and thermokinetics is a linearized version of thermodynamics. The fully dynamic theory has not been completely formulated in any useful sense- there are still many open questions.

Last edited by a moderator: Apr 26, 2017
3. Jun 16, 2011

### atyy

To heat an object, you not only change its temperature, but also its volume and pressure. The specific heat will generally depend on both volume and pressure (and whatever else). So the amount of energy imparted by heating is path dependent.

While it is not true that heat is both and exact and inexact differential, the above argument is not complete in the sense that we can make an exact differential dS, out of an inexact differential dQ, by dS=dQrev/T.

If a differential satisfies some "Maxwell relation", it is an exact differential. A factor such as 1/T which converts inexact dQrev to exact dS is called an integrating factor. http://pruffle.mit.edu/3.016-2005/Lecture_20_web/node2.html

In defining dS=dQrev/t, we have to specify that the heating occurs during a reversible process. A reversible process is a quasistatic process in which the work done is frictionless. If the process is not quasistatic, then the state variables are not defined during the process, and we don't know how to do the path integral. If work is done against friction, running a force backwards generates as much heat as running it forwards, so dQ/T cannot be zero even though the system is back in its initial state (in a sense, if there is irreversible work, we haven't properly included the heat into our equations for every stage of the process).

Last edited: Jun 16, 2011