States of a system

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

The starting point for the theory of molecular dynamics, and indeed the basis for most of theoretical chemistry, is the separation of the nuclear and electionic motion. In the standard, adiabatic, picture this leads to the concept of nuclei moving over PES corresponding to the electronic states of a system. 

To define the state yon want to calculate, you must specify the m u Itiplicity. A system with an even ii n m ber of electron s n sn ally has a closed-shell ground state with a multiplicity of I (a singlet). Asystem with an odd niim her of electrons (free radical) nsnally has a multiplicity of 2 (a doublet). The first excited state of a system with an even ii nm ber of electron s usually has a m n Itiplicity of 3 (a triplet). The states of a given m iiltiplicity have a spectrum of states —the lowest state of the given multiplicity, the next lowest state of the given multiplicity, and so on. 

This is Gibbs s phase rule. It specifies the number of independent intensive variables that can and must be fixed in order to estabbsh the intensive equibbrium state of a system and to render an equibbrium problem solvable. 

The concept of equilibrium is central in thermodynamics, for associated with the condition of internal eqmlibrium is the concept of. state. A system has an identifiable, reproducible state when 1 its propei ties, such as temperature T, pressure P, and molar volume are fixed. The concepts oi state a.ndpropeity are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates I and 3 following) is recognized much more indirectly. The number of properties for wdiich values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. 

The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equihbrium states these variables are not aU independent, and fixing a hmited number of them automaticaUy estabhshes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equihbrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

The state of a system may be defined as The set of variables (called the state variables) which at some initial time Iq, together with the input variables completely determine the behaviour of the system for time t > to -... 

Chemistry can be divided (somewhat arbitrarily) into the study of structures, equilibria, and rates. Chemical structure is ultimately described by the methods of quantum mechanics equilibrium phenomena are studied by statistical mechanics and thermodynamics and the study of rates constitutes the subject of kinetics. Kinetics can be subdivided into physical kinetics, dealing with physical phenomena such as diffusion and viscosity, and chemical kinetics, which deals with the rates of chemical reactions (including both covalent and noncovalent bond changes). Students of thermodynamics learn that quantities such as changes in enthalpy and entropy depend only upon the initial and hnal states of a system consequently thermodynamics cannot yield any information about intervening states of the system. It is precisely these intermediate states that constitute the subject matter of chemical kinetics. A thorough study of any chemical reaction must therefore include structural, equilibrium, and kinetic investigations. 

Note that the maximum work depends only upon the initial and final states of a system and not upon the path. 

Equations 2-150 and 2-151 apply to any substance or system and are called equations of stale because they completely determine the state of a system in terms of its thermodynamic properties. 

The state of a system containing a constant amount of material depends upon a few variables, e.g. pressure p, volume V, temperature T. For a given mass of pure substance the volume can be expressed solely as a function of pressure and temperature... 

Another drawback to using Shannon information as a measure of complexity is the fact that it is based on an ensemble of all possible states of a system and therefore cannot describe the information content of a single state. Shannon information thus resembles traditional statistical mechanics - which describes the average or aggregate behavior of, say, a gas, rather than the motion of its constituent molecules - more so than it docs a complexity theory that must address the complexity of individual objects. 

The state of a system is described by giving its composition, temperature, and pressure. The system at the left of Figure 8.1 consists of... 

Entropy, like enthalpy (Chapter 8), is a state property. That is, tine entropy depends only on the state of a system, not on its history. The entropy change is determined by the entropies of the final and initial states, not on the path followed from one state to another. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

Fock Space.—We have already discussed the Hilbert space 3H H in which we postulate the existence of vectors represented by the states of a system of N particles. Let us now build a hyper-Hilbert space by uniting the Hilbert spaces for every possible population the union of the Hilbert space 3f0 for an empty system, the Hilbert space for a system with one particle, etc., without upper limit. This union is called Fock space ... 

The heat absorbed in the change of state of a system at constant volume is equal to the increase of intrinsic energy ... 

Theorem. If A and B are two different states of a system, then the value of the integral... 

Heat, like work, is energy in transit and is not a function of the state of a system. Heat and work are interconvertible. A steam engine is an example of a machine designed to convert heat into work.h The turning of a paddle wheel in a tank of water to produce heat from friction represents the reverse process, the conversion of work into heat. 

The reason is that classical thermodynamics tells us nothing about the atomic or molecular state of a system. We use thermodynamic results to infer molecular properties, but the evidence is circumstantial. For example, we can infer why a (hydrocarbon + alkanol) mixture shows large positive deviations from ideal solution behavior, in terms of the breaking of hydrogen bonds during mixing, but our description cannot be backed up by thermodynamic equations that involve molecular parameters. 

Quantities like V, U, S, H< A, and G are properties of the system. That is, once the state of a system is defined, their values are fixed. Such quantities are called state functions. If we let Z represent any of these functions, then it does not matter how we arrive at a given state of the system, Z has the same value. If we designate Z to be the value of Z at some state l, and Z to be the value of Z at another state 2, the difference AZ = Z2 - Z in going from state l to state 2 is the same, no matter what process we take to get from one state to the other. Thus, if we go from state l through a series of intermediate steps, for which the changes in Z are given by AZ, AZ . AZ,-. and eventually end up in state 2,... 

Similarly, heat is not a state function. The energy transferred as heat during a change in the state of a system depends on how the change is brought about. For example, suppose we want to raise the temperature of 100 g of water from 25°C to 30°C. One way to raise the temperature would be to supply energy as heat by... 

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