Intensive state

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). 

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. 

For a system containing N chemical species distributed at equihbrium among 7C phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N — mole fraciions in each phase. The number of these variables is 2 -t- (V — 1)7T. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. 

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 general XT E problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N — 1 liquid-phase mole fractions, and N — 1 vapor-phase mole fractions. (Note that Xi = 1 and y = 1, where x, and y, represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to estabhsh the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by siiTUiltaneous solution of the N equihbrium relations ... 

The first BioCD took its inspiration from the compact disc. The compact disc was invented in 1970 by Claus Campaan of Phillips Laboratory. The concept is purely digital and uses null interferometers that are far from quadrature, as appropriate for the readout of two binary intensity states. The interferometers were common-path and stable, as required for the mechanical environment of portable compact disc readers. The original BioCD used the same physics as the compact disc, but modified the on-disc microstructures to change from the digital readout to an analog readout that operated in quadrature for sensitive detection of surface-bound proteins7,8. Because the quadrature condition is established by diffraction off of microstructures on the disc, this is called the microdiffraction-class (MD-Class) of BioCD. 

It is important to distinguish between the intensive (state) properties (functions) and the extensive properties (functions). 

The internal energy per unit mass e is an intensive (state) function. Enthalpy h, a compound thermodynamic function defined by Equation 1.8, is also an intensive function. 

Each such null vector may be considered an invariant or symmetry of the thermodynamic system, because it corresponds to an operation (change of extensive variables Xt) that produces no response in any intensive state variable and thus leaves the thermodynamic state unaltered (Sidebar 7.2). As described in Sidebar 10.3, these invariants also play a role somewhat analogous to overall rotations and translations ( null eigenmodes of the Hessian matrix) in the theory of molecular vibrations. 

Let us refer to Figure 5-7 and start with a homogeneous sample of a transition-metal oxide, the state of which is defined by T,P, and the oxygen partial pressure p0. At time t = 0, one (or more) of these intensive state variables is changed instantaneously. We assume that the subsequent equilibration process is controlled by the transport of point defects (cation vacancies and compensating electron holes) and not by chemical reactions at the surface. Thus, the new equilibrium state corresponding to the changed variables is immediately established at the surface, where it remains constant in time. We therefore have to solve a fixed boundary diffusion problem. 

Since these considerations are independent of the nature of the sample, the results are valid for all crystals which are exposed to a sudden change of intensive state variables. The meaning of the chemical diffusion coefficient D must, however, be carefully investigated in each case (see Section 5.4.4). At 1000°C, Dv for simple transition-metal oxides is on the order of 10 7 cm2/s. This gives for cubic samples of 10-3 cm3 a defect relaxation time of approximately 1 h according to Eqn. (5.86). 

From Eqn. (14.3), one can easily derive the cross relations between extensive and intensive state variables as, for example,... 

A lot of thermodynamics makes use of the important concept of state function, which is a property with a value that depends only on the current state of the system and is independent of the manner in which the state was prepared. For example, a beaker containing 100 g of water at 25°C has the same temperature as 100 g of water that has been heated to 100°C and then allowed to cool to 25°C. Internal energy is also a state function so the internal energy of the beaker of water at 25°C is the same no matter what its history of preparation. State functions may be either intensive or extensive temperature is an intensive state function internal energy is an extensive state function. 

Experience shows that for a system that is a homogeneous mixture of Ns substances, Ns + 2 properties have to be specified and at least one property must be extensive. For example, we can specify T, P, and amounts of each of the Ns substances or we can specify T, P, and mole fractions x, of all but one substance, plus the total amount in the system. Sometimes we are only interested in the intensive state of a system, and that can be described by specifying Ns + 1 intensive properties for a one-phase system. For example, the intensive state of a solution involving two substances can be described by specifying T, P, and the mole fraction of one of substances. 

The intensive variables T, P, and nt can be considered to be functions of S, V, and dj because U is a function of S, V, and ,. If U for a system can be determined experimentally as a function of S, V, and ,, then T, P, and /q can be calculated by taking the first partial derivatives of U. Equations 2.2-10 to 2.2-12 are referred to as equations of state because they give relations between state properties at equilibrium. In Section 2.4 we will see that these Ns + 2 equations of state are not independent of each other, but any Ns+ 1 of them provide a complete thermodynamic description of the system. In other words, if Ns + 1 equations of state are determined for a system, the remaining equation of state can be calculated from the Ns + 1 known equations of state. In the preceding section we concluded that the intensive state of a one-phase system can be described by specifying Ns + 1 intensive variables. Now we see that the determination of Ns + 1 equations of state can be used to calculate these Ns + 1 intensive properties. 

In addition to being a function of T, the partition function is also a function of V, on which the quantum description of matter tells us that the molecular energy levels, , depend. Because, for single-component systems, all intensive state variables can be written as functions of two state variables, we can think of q(T, V) as a state function of the system. The partition function can be used as one of the independent variables to describe a single-component system, and with one other state function, such as T, it will completely define the system. All other properties of the system (in particular, the thermodynamic functions U, H, S, A, and G) can then be obtained from q and one other state function. 

The Gibbs phase rule allows /, the number of degrees of freedom of a system, to be determined. / is the number of intensive variables that can and must be specified to define the intensive state of a system at equilibrium. By intensive state is meant the properties of all phases in the system, but not the amounts of these phases. Phase equilibria are determined by chemical potentials, and chemical potentials are intensive properties, which are independent of the amount of the phase that is present. The overall concentration of a system consisting of several phases, however, is not a degree of freedom, because it depends on the amounts of the phases, as well as their concentration. In addition to the intensive variables, we are, in general, allowed to specify one extensive variable for each phase in the system, corresponding to the amount of that phase present. 

The phase rule [Eq. (2.7)] yields F = 2 —ic + N = 2 — 2 + 2 = 2. Specification of T and P fixes the intensive state, and thus the phase compositions, of the system. Since the liquid phase is pure species 1, addition of species 2 to the system increases its amount in the vapor phase. If the composition of the vapor phase is to be unchanged, some of species 1 must evaporate from the liquid phase, thus decreasing the moles of liquid present. 

In accord with the phase rule, the system has 2 degrees of freedom. Once T and P are specified, the intensive state of the system is fixed. Provided the two phases are still present, their compositions cannot change. 

Internal energy (through the enthalpy, defined in Sec. 2.5) is useful for the calculation of heat and work quantities for such equipment as heat exchangers, evaporators, distillation columns, pumps, compressors, turbines, engines, etc., because it is a state function. The tabulation of all possible Q s and W s for all possible processes is impossible. But the intensive state functions, such as specific volume and specific internal energy, are properties of matter, and they can be measured and their values tabulated as functions of temperature and pressure for a particular substance for future use in the calculation of Q or W for any process involving that substance. The measurement, correlation, and use of these state functions is treated in detail in later chapters. 

The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed. These are therefore phase-rule variables, but they are not all independent. The phase rule gives the number of variables from this set which must be arbitrarily specified to fix all remaining phase-rule variables. 

The characteristic feature of all these processes is the coexistence of two phases. According to the phase rule, a two-phase system consisting of a single species is univariant, and its intensive state is determined by the specification of just one intensive property. Thus the latent heat accompanying a phase change is a function of temperature only, and is related to other system properties by an exact thermodynamic equation ... 

The phase rule for nonreacting systems, presented without proof in Sec. 2.8 results from application of a rule of algebra. The number of phase-rule variable which must be arbitrarily specified in order to fix the intensive state of a syste at equilibrium, called the degrees of freedom F, is the difference between t total number of phase-rule variables and the number of independent equatio that can be written connecting these variables. 

Solution According to the phase rule (see Sec. 15.8), the system has two degrees of freedom. Specification of both the temperature and the pressure leaves no other degrees of freedom, and fixes the intensive state of the system, independent of the initial amounts of reactants. Therefore, material-balance equations do not enter into the solution of this problem, and we can make no use of equations that relate compositions to the reaction coordinate. Instead, phase equilibrium relations must... 

When JV = 2, the phase rule becomes F = 4 - it. Since there must be at least one phase (it = 1), the maximum number of phase-rule variables which must be specified to fix the intensive state of the system is three namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be... 

At equilibrium the system consists of liquid and vapor in equilibrium, and the intensive state of the system is fixed by the specification of T and P. Therefore, one must first determine the phase compositions, independent of the ratio of reactants. These results may then be applied in the material-balance equations to find the equilibrium conversion. 

As the pure substance, hydrogen H2 may exist in various physical phases as vapor, liquid and solid. As with any pure substance, the thermodynamic state of hydrogen is completely defined by specifying two independent intensive state variables. All other variables of state can then be determined by using one of the three relationships of state. 

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