Assessment of the minimum radius and time periods of the protection and surveillance zones set in

In Section 3.1 we have considered spontaneous processes proceeding towards equi-librium. After each of these processes has been completed a state of equilibrium is established. The question is how to define equilibrium?

Equilibrium is a thermodynamic state from which no spontaneous (nat-ural) processes can proceed.

In other words, a system is in equilibrium if no changes occur within the system when no interaction with surroundings takes place. An isolated system is in me-chanical equilibriumif no changes occur in pressure, in thermal equilibrium if no changes occur in temperature, in phase equilibrium if no transformation between phases occurs. The system is in chemical equilibrium if no changes occur in the chemical composition of the system. In the forthcoming paragraphs we will derive criteria for systems to be in equilibrium. Firstly, we consider iso-lated systems which do not interact with surroundings (Section 3.4.1) and later on, we consider open and closed systems which interact with surroundings by heat exchange (Section 3.4.2).

3.4.1 Isolated System

Since an isolated system exchanges neither heat nor mass with surroundings, the entropy change of any spontaneous process occurring in the system is:

π = ∆Ssystem = Sf inal state− Sinitial state ≥ 0 (3.46) Thus, any spontaneous process occurring within the system proceeds in the direc-tion of increasing entropy. Consequently, equilibrium is reached when the system entropy is maximum and it cannot be increased any further. Therefore the criteria for an isolated system in equilibrium are

S = S_max (3.47)

or

dS = 0 (3.48)

The just derived criterion of maximum entropy which is applicable to isolated systems (or simply to an adiabatic closed system) is not very useful since most of systems encountered in every-day engineering practise are not isolated and

3.4 General Conditions for Thermodynamic Equilibrium

exchange not only heat but also mass with surroundings. If we consider our Uni-verse⁴ as an isolated system we obtain the well known general statement that the Universe´s entropy increases continuously resulting in a continuously increasing disorder or chaos.

3.4.2 Non-Adiabatic System

When a system involves heat transfer with surroundings, the increase in entropy principle is (see Eq. (3.32)

π = ∆Ssystem− Qt

Tsurr

≥ 0 (3.49)

or

∆S_system≥ Q_t Tsurr

(3.50) where Qt stands for the total amount of heat transferred to the system (see Fig. 3.8). The later consists of the heat entering due to friction (Qf) and the actual heat (Q) supplied from the surroundings to the system, so

Qt= Qf + Q (3.51)

Thus, any considerations on reaching equilibrium require calculations of Qt. This makes the use of inequality (3.50) impractical. Since heat is not a state variable, there exist an infinite number of Q values when the system undergoes a thermo-dynamic process from an initial state to an equilibrium state. This observation is highly disappointing since we cannot derive general equilibrium conditions for a system interacting with the surroundings to reach equilibrium. In-stead we have to restrict ourselves to processes for which we can relate Q to some state variables of the system. In other words, we have to accept some constraints (restrictions) on the way the process is carried out and consequently the derived equilibrium conditions are going to be valid under these constraints only. In this lecture we restrict ourselves to two types of processes; these which proceed under a constant temperature and a constant pressure, and these that proceed under a constant temperature and a constant volume⁵.

4Can we regard our Universe as an isolated system? Perhaps our Universe does interact with other bodies/Universes?

5In thermodynamics one develops criteria for thermodynamic equilibrium under many other constraints; under constant entropy (S) and volume (V), under constant entropy (S) and pressure (p), and many others.

3 Equilibrium Thermodynamics

Before we proceed further, we observe that from Eq. (3.4) one obtains

T · dS = dQf+ dQ (3.52)

and since the friction heat (dQf) is always positive, the following emerges

dQ < T · dS (3.53)

Thus, spontaneous (irreversible) processes proceed towards equilibrium so that the above inequality is always satisfied.

Systems at constant temperature and pressure (T=const., p=const.) Consider a system of a fixed mass. The system is at temperature T and pressure p and interacts with surroundings through exchanging heat and work; however its temperature and pressure remain constant. For any spontaneous processes inequality (3.53) is applicable and since the system remains at a constant pressure, from the first law of thermodynamics, we obtain dQ = dH and therefore:

dH < T · dS (3.54)

or

dH − T · dS < 0 (3.55)

and further

dH − d(T · S) < 0 (3.56)

since temperature remains also constant during the spontaneous process proceed-ing towards equilibrium. Inequity (3.56) can be rearranged into a more elegant form

d(H − T · S) < 0 (3.57)

that defines a new variable G:

G = H − T · S (3.58)

Variable G is called free enthalpy or Gibbs thermodynamic potential. In some textbooks G is simple called Gibbs function.

Thus, in a system that remains at constant pressure and temperature spontaneous processes proceed towards equilibrium in such a way that

dG < 0 (3.59)

and finally, when equilibrium is reached, Gibbs thermodynamic potential G is

3.4 General Conditions for Thermodynamic Equilibrium

minimum:

dG = 0 or G = Gmin (3.60)

Relationships (3.60) formulate equilibrium criteria for a system when its temper-ature and pressure are specified.

We finish this paragraph with a remark. If the system under considerations con-tains only one substance, for example a gas or a solid, specifying the system temperature and pressure determines its volume through the equation of state.

Consequently such a system is always at equilibrium. Relationships (3.60) are useful in determining equilibrium for a system containing a number (minimum two) of species in one or several thermodynamic phases. Such systems rearrange their composition and some or all of the species undergo phase transitions so as to satisfy the minimum Gibbs potential conditions.

Systems at constant temperature and volume (T=const., V=const.) Consider a system of a fixed mass, of volume V and temperature T . The system interacts with surroundings through exchanging heat and work in such a way that its temperature and volume remain constant. Inequity (3.53) is applicable for any spontaneous processes occurring within the system. Since the volume of the system remains constant, from the first law of thermodynamics, we obtain dQ = dU and therefore:

dU < T · dS (3.61)

and further

d(U − T · S) < 0 (3.62)

The above inequity defines a new variable F called free internal energy or Helmholtz free energy. Thus, in a system that remains at constant volume and temperature, spontaneous processes proceed towards equilibrium in such a way that

dF < 0 (3.63)

and finally, when equilibrium is reached, Helmholtz free energy F is minimum:

dF = 0 or F = F_min (3.64)

Relationships (3.64) formulate equilibrium criteria for a system when its temper-ature and volume are specified.

3 Equilibrium Thermodynamics