Thermodynamic equilibrium isolated system

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

Equilibrium is a very important concept in discussions of thermodynamics. An isolated system is at equilibrium when it has no tendency to change—a condition that is called internal equilibrium. This implies that the system is at mechanical equilibrium (i.e., it has no tendency for bulk movement of material), thermal equilibrium [i.e., it has no tendency for transport of energy (without bulk movement of material)], and material equilibrium [i.e., it has no tendency for material to change form (such as by a phase transformation or a chemical reaction)]. 

The second law of thermodynamics An isolated system, if not already in its state of thermodynamic equilibrium, spontaneously evolves toward it. Thermodynamic equilibrium has the greatest entropy among the states accessible to the system. Perpetual motion machines of the second kind are thus impossible. 

It is an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system that when the entropy of the system has reached a maximum, the system will be in a state of equilibrium. Although this principle has by no means escaped the attention of physicists, its importance does not seem to have been duly appreciated. Little has been done to develop the principle as a foundation for the general theory of thermodynamic equilibrium (my italics). ... 

To derive the condition for thermodynamic equilibrium, we start with an isolated system consisting of two subsystems as shown in Figure 5.6. Subsystem A is the one of primary interest in that it is the one in which the chemical process is occurring. Subsystem B is a reservoir in contact with subsystem A in such a way that energy in the form of heat or work can flow between the two subsystems. If left alone, the system will come to equilibrium. Energy will be transferred between the subsystems so that the temperature and pressure will be... 

Generally, in a system that is energetically and materially isolated from the environment without a change in volume (a closed system), the entropy of the system tends to take on a maximum value, so that any macroscopic structures, except for the arrangement of atoms, cannot survive. On the other hand, in a system exchanging energy and mass with the environment (an open system), it is possible to decrease the entropy more than in a closed system. That is, a macroscopic structure can be maintained. Usually such a system is far from thermodynamic equilibrium, so that it also has nonlinearity. 

Now the previously isolated subsystem is allowed to exchange x with an external reservoir that applies a thermodynamic force Xr. Following the linear analysis (Section II F), denote the subsystem thermodynamic force by Xs(x) (this was denoted simply X(x) for the earlier isolated system). At equilibrium... 

Mass transfer is a kinetic process, occurring in systems that are not at equilibrium. To understand mass transfer from a thermodynamic perspective, consider the isolated system shown in Figure 1. The system is bounded by an impermeable insulating wall which prevents the transfer of matter, heat, or mechanical energy between the system and the external environment. The system is subdivided into... 

Work can be completely converted to heat, but—and this is important—a complete conversion of heat to work is not possible in an isothermal system. This problem is dealt with by the second law of thermodynamics, with its statement on entropy The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium. ... 

Remember that G, H and S are all thermodynamic functions of state, i.e. they depend only on the initial and final states of the system, not on the ways the last is reached. As we have seen, for AG = 0 the reaction has reached equilibrium (and in isolated systems AS has reached a maximum). If AG < 0 the reaction was spontaneous, but if AG > 0 the reaction could not have taken place unless energy was provided from other coupled source. If the source is external then the system is not isolated it is closed if there is no exchange of material or open if there is such exchange. In both cases the environmental changes must be taken into account. 

An example of such a system is one in which the internal energy, U, and the volume, V, are constant. If these are the only two constraints on the system then, at thermodynamic equilibrium, the entropy, S, is at a maximum. On the other hand, if entropy and volume are constant for the isolated system then, at thermodynamic equilibrium, the internal energy is at a minimum. See also Closed System Open System... 

The second law of thermodynamics states that an isolated system in equilibrium has maximum entropy. This is the basis for a variational principle often used in determining the equilibrium state of a system. When the system contains several elements which are allowed to exchange mass with each other, the variational principle yields the condition that all elements must have equal chemical potential once equilibrium is established. 

It was the principal genius of J. W. Gibbs (Sidebar 5.1) to recognize how the Clausius statement could be recast in a form that made reference only to the analytical properties of individual equilibrium states. The essence of the Clausius statement is that an isolated system, in evolving toward a state of thermodynamic equilibrium, undergoes a steady increase in the value of the entropy function. Gibbs recognized that, as a consequence of this increase, the entropy function in the eventual equilibrium state must have the character of a mathematical maximum. As a consequence, this extremal character of the entropy function makes possible an analytical characterization of the second law, expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, processes, perpetual motion machines, and the like. 

This equation has the same form as (3.7) for an isolated system the stationary solution of the master equation ps is identical with the thermodynamic equilibrium pe. 

In equilibrium thermodynamics, entropy maximization for a system with fixed internal energy determines equilibrium. Entropy increase plays a large role in irreversible thermodynamics. If each of the reference cells were an isolated system, the right-hand side of Eq. 2.4 could only increase in a kinetic process. However, because energy, heat, and mass may flow between cells during kinetic processes, they cannot be treated as isolated systems, and application of the second law must be generalized to the system of interacting cells. 

Thermodynamic systems are parts of the real world isolated for thermodynamic study. The parts of the real world which are to be isolated here are either natural water systems or certain regions within these systems, depending upon the physical and chemical complexity of the actual situation. The primary objects of classical thermodynamics are two particular kinds of isolated systems adiabatic systems, which cannot exchange either matter or thermal energy with their environment, and closed systems, which cannot exchange matter with their environment. (The closed system may, of course, consist of internal phases which are each open with respect to the transport of matter inside the closed system.) Of these, the closed system, under isothermal and iso-baric conditions, is the one particularly applicable for constructing equilibrium models of actual natural water systems. 

Isolated Systems Isolated systems exchange neither energy nor matter with the environment. The simplest example from chemical or biological engineering is the adiabatic batch reactor. Isolated systems naturally tend towards their thermodynamic equilibrium with time. This state is characterized by maximal entropy, or the highest possible degree of disorder. 

The state function which measures disorder is the entropy, S, and the second law of thermodynamics may be stated as follows The entropy of the universe or of an isolated system always increases when a spontaneous irreversible process occurs entropy remains constant in a reversible process, i.e., a process which remains at equilibrium for every step along the way,... 

Homogeneous systems, such as cooking oil, exist in thermodynamic equilibrium and the properties of these systems are determined by their chemical composition.1 Heterogeneous systems are not in thermodynamic equilibrium. The properties of these systems are governed by both the chemical composition and the internal framework formed by the spatial arrangement of the individual chemical components present. The formation of steric, electrostatic and covalent forces between these individual components can have a dramatic effect on the properties of the product. Ice cream is a classic example, which is primarily composed of ice cream mix and air. The pure components of ice cream have different structural properties in the mixture than they do in their isolated form. Frozen ice cream has a certain consistency and texture, which is quite different from any of the individual frozen ingredients. 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

One spectacular use of a rotaxane was to illustrate James Clerk Maxwell s 19th century thought experiment known as Maxwell s demon as shown in Fig. 1.23. Maxwell proposed several experiments that would violate the Second Law of Thermodynamics to show how the entropy of an isolated system could be reduced without expending energy. To do so he invoked the idea of a demon who effortlessly operates a frictionless door between two compartments which contain particles at different temperatures. Whenever a particle approaches the door the demon decides whether to open it and allow the particle through. In this way the particles can be sorted so that one compartment contains only hot particles and the other only cold. A similar example can be envisaged in which particles at an equilibrium pressure are moved from one compartment to the other to increase its pressure without any work apparently being done. 

First of all relying directly on the second law we will try to give the interpretation of the Prigogine theorem. Taking into account that the traditional variables of equilibrium thermodynamics are the parameters of state and, wishing to reveal the formalized relations between both thermodynamics, let us consider two situations sequentially (1) when some parameters of interaction that hinder the attainment of final equilibrium between the open subsystem and other parts of the isolated system that contains this subsystem are set (2) when flows are taken constant for the flow exchange between the open subsystem and the environment. It is obvious that both situations can be reduced to the case of fixing individual forces which is normally considered in the nonequilibrium thermodynamics. 

Let us consider an isolated system described by the hamiltonian H. The system is assumed in thermodynamical equilibrium at temperature T. Thus, its density operator has the form... 

Entropy in an isolated system increases dS/dt> 0 until it reaches equilibrium dS/dt = 0, and displays a direction of change leading to the thermodynamic arrow of time. The phenomenological approach favoring the retarded potential over the solution to the Maxwell field equation is called the time arrow of radiation. These two arrows of time lead to the Einstein-Ritz controversy Einstein believed that irreversibility is based on probability considerations, while Ritz believed that an initial condition and thus causality is the basis of irreversibility. Causality and probability may be two aspects of the same principle since the arrow of time has a global nature. 

The second law of thermodynamics then states that the entropy of an isolated system (a system with constant U and constant V) increases (d5 > 0) for a spontaneous processes or stays constant (d5 = 0) at equilibrium. In Eq. 11.9, no internal processes ate considered S is normally regarded as an external variable even though it is not possible to control its value externally without knowledge of the properties of the system). If internal processes are considered, an internal variable must be included, namely, /a, the chemical potential (to be discussed shortly). 

The deduction of a criterion for the evolution of an open system to its stationary state resembles the classical thermodynamic problem of predict ing the direction of spontaneous irreversible evolution in an isolated system According to the Second Law of thermodynamics, in the latter case the changes go only toward the increase in entropy, the entropy being maximal at the final equilibrium state. 

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