Thermodynamics of stationary states of the ideal gas in a heat flow

Abstract

There is a long-standing question as to whether and to what extent it is possible to describe nonequilibrium systems in stationary states in terms of global thermodynamic functions. The positive answers have been obtained only for isothermal systems or systems with small temperature differences. We formulate thermodynamics of the stationary states of the ideal gas subjected to heat flow in the form of the zeroth, first, and second law. Surprisingly, the formal structure of steady state thermodynamics is the same as in equilibrium thermodynamics. We rigorously show that U satisfies the following equation dU= T* dS* -pdV for a constant number of particles, irrespective of the shape of the container, boundary conditions, size of the system, or mode of heat transfer into the system. We calculate S* and T* explicitly. The theory selects stable nonequilibrium steady states in a multistable system of ideal gas subjected to volumetric heating. It reduces to equilibrium thermodynamics when heat flux goes to zero.

Steady thermodynamic fundamental relation for the interacting system in a heat flow

“…Our recent investigations of an ideal gas in a steady state with a heat flow showed a surprising result [22]. We proved that the net heat has an integrating factor and rigorously calculated non-equilibrium 'entropy' and non-equilibrium temperature.…”

“…The integrating factor for the heat differential in the case of the ideal gas discussed previously [22] allowed us to introduce the non-equilibrium entropy and use it to construct the minimum energy principle beyond equilibrium. This principle generalizes thermodynamics' second law beyond equilibrium.…”

“…However, during the transition from one steady state to another, e.g., by a slight change of temperature T 2 or by a motion of the right wall changing L (see Fig. 1), this balance is, in general, disturbed and the net heat may flow to the system changing its internal energy [22]. In the case of a very slow transition between stationary states, the energy changes only by means of mechanical work and heat flow…”

“…Therefore we are in position to ask whether the heat differential dQ has an integrating factor in space T 1 , T 2 , V . For the ideal gas (a = 0) the integrating factor exists [22]. It follows that there exists a function of state, which is constant if the steady state system is "adiabatically insulated" (i.e.…”

“…The parameters T * , a * , b * defined by Eqs. (13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) are not independent. To explain it, we keep in mind that for a given number of particles, three control parameters T 1 , T 2 , V are sufficient to determine the system's energy, work, and net heat differential.…”