State of the system defined

The component can be represented over the initial I) and intermediate J) states of the system defined by Wn, which is a normal-ordered Hamiltonian with respect to the I) state ... 

Let us consider a state of the system defined by particular values of the variables T, p, w, and calculate the variation in pressure... 

Consider a system that consists of two standard playing dice, with the state of the system defined by the sum of the values shown on the top faces, (a) The two arrangements of top faces shown here can be viewed as two possible microstates of the system. Explain, (b) To which state does each microstate correspond (c) How many possible states are there for the system (d) Which state or states have the highest entropy Explain, (e) Which state or states have the lowest entropy Explain. 

Consider a system that consists of two standard playing dice, with the state of the system defined by the sum of the values shown on the top faces, (a) The two arrangements of top... 

The microscopic state of the system defines coordinates, momenta, spins for every particle in the system. Each point in phase space corresponds to a microscopic state. There are, however, many microscopic states, in which the states of particular molecules or bonds are different, but values of the macroscopic observables are the same. For example, a very large number of molecular configurations and associated momenta in a fluid can correspond to the same number of molecules, volume, and energy. All points of the harmonic oscillator phase space that are on the same ellipse in Fig. 5 have the same total energy. 

One then speaks of Fas a state fiinction because it is a fiinction only of those variables that define the state of the system, and not of the path by which the state was reached. An especially important feature of such fiinctions is that if one writes DF as a fiinction of several variables, say v, y, z,... 

If the adiabatic work is independent of the path, it is the integral of an exact differential and suffices to define a change in a function of the state of the system, the energy U. (Some themiodynamicists call this the internal energy , so as to exclude any kinetic energy of the motion of the system as a whole.)... 

Flere the subscripts and/refer to the initial and final states of the system and the work is defined as the work perfomied on the system (the opposite sign convention—with as work done by the system on the surroundings—is also in connnon use). Note that a cyclic process (one in which the system is returned to its initial state) is not introduced as will be seen later, a cyclic adiabatic process is possible only if every step is reversible. Equation (A2.1.9), i.e. the mtroduction of t/ as a state fiinction, is an expression of the law of conservation of energy. 

Here =MkT. In a real system the thennal coupling with surroundings would happen at the surface in simulations we avoid surface effects by allowing this to occur homogeneously. The state of the surroundings defines the temperature T of the ensemble. 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

It is known that polymers may exist in various stationary states, which are defined by the amount and distribution of intermolecular bonds in the sample at definite network structure. The latter is defined by the conditions of storage, exploitation, and production of the network. That is why T values may be different. The highest value is observed in the equilibrium state of the system. In this case it is necessary to point out, that the ph value becomes close to the ph one at n,. 

Mixing of fluids is necessary in many chemical processes. It may include mixing of liquid vith liquid, gas with liquid, or solids with liquid. Agitation of these fluid masses does not necessarily imply any significant amount of actual intimate and homogeneous distribution of the fluids or particles, and for this reason mixing requires a definition of degree and/or purpose to properly define the desired state of the system. 

Other thermodynamic functions described above in that the change in free energy AG is determined solely by the initial and final states of the system. The maximum work, or maximum available energy, defined in terms of the Gibbs free energy G, which is now called the free enthalpy, is... 

It is to be remarked that these operators can act only on states of the system expressed in occupation number representation, as explicitly appearing in the definitions, Eqs. (8-105), (8-106), (8-112), and (8-114). We can multiply any one of these operators by a scalar factor, so that we can also define the following operators ... 

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

In the investigation of the properties of the free energy no assumption has been made as to the nature of the external work At. Let us now assume that there is some function 12 of the variables defining the physical and chemical state of the system, such that ... 

We can represent states of the system (with constant values specified for all the variables except 9 and at) by a set of isotherms as shown in Figure 2.1 la. Two isotherms, 9 and 92 are shown, with 92 < 9t. State I, which is defined by 9 and A], can be connected to states T and 1" by a series of reversible isothermal processes (horizontal lines in the figure). We remember that heat is absorbed or evolved along a reversible isothermal path, and we will assume that this flow of heat is a continuous function of at along the isotherms, with the absorption or liberation depending upon the direction in which at is varied. That is, suppose... 

Generally, for a pure substance in which the composition is constant, only two of the thermodynamic quantities listed above need be specified as independent variables to uniquely define the system. In the presence of significant gravitational, electric, or magnetic fields, or where the surface area or chemical composition of the system is variable, additional quantities may be needed to fix the state of the system. We will limit our discussion to situations where these additional variables are held constant, and hence, do not need to be considered. 

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,... 

PES), which is different for each electronic state of the system (i.e. each eigenfunction of the BO Schrodinger equation). Based on these PESs, the nuclear Schrodinger equation is solved to define, for example, the possible nuclear vibrational levels. This approach will be used below in the description of the nuclear inelastic scattering (NIS) method. 

The state of a physical system is defined by a normalized fimction P of the spatial coordinates and the time. This function contains all the information that exists on the state of the system. 

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