Extensive state

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. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

The combination of fundamental variables in equation (l.23) that leads to the variable we call G turns out to be very useful. We will see later that AG for a reversible constant temperature and pressure process is equal to any work other than pressure-volume work that occurs in the process. When only pressure-volume work occurs in a reversible process at constant temperature and pressure, AG = 0. Thus AG provides a criterion for determining if a process is reversible. Again, since G is a combination of extensive state functions... 

The extensive state of knowledge of the electrooxidation of methanol, as presented in this section, offers prospects of tailoring new multimetallic... 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

Consider a material or system that is not at equilibrium. Its extensive state variables (total entropy number of moles of chemical component, i total magnetization volume etc.) will change consistent with the second law of thermodynamics (i.e., with an increase of entropy of all affected systems). At equilibrium, the values of the intensive variables are specified for instance, if a chemical component is free to move from one part of the material to another and there are no barriers to diffusion, the chemical potential, q., for each chemical component, i, must be uniform throughout the entire material.2 So one way that a material can be out of equilibrium is if there are spatial variations in the chemical potential fii(x,y,z). However, a chemical potential of a component is the amount of reversible work needed to add an infinitesimal amount of that component to a system at equilibrium. Can a chemical potential be defined when the system is not at equilibrium This cannot be done rigorously, but based on decades of development of kinetic models for processes, it is useful to extend the concept of the chemical potential to systems close to, but not at, equilibrium. 

The enthalpy is an extensive state function ils value depends only on the state and the amount of the substance and not on its previous history. It has the unils of energy and it is usually expressed in calories (or kilocalories). 

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. 

Disputes relating to access may be referred to the Ministry or a Ministry authorized appointee.23 Due to extensive state ownership in the sector and the organization of these ownership interests with the Ministry, it can be... 

Since F is the symbol for the number of independent intensive variables for a system, it is also useful to have a symbol for the number of natural variables for a system. To describe the extensive state of a system, we have to specify F intensive variables and in addition an extensive variable for each phase. This description of the extensive state therefore requires D variables, where D = F + p. Note that D is the number of natural variables in the fundamental equation for a system. For a one-phase system involving only PV work, D = Ns + 2, as discussed after equation 2.2-12. The number F of independent intensive variables and the number D of natural variables for a system are unique, but there are usually multiple choices of these variables. The choice of independent intensive variables F and natural variables D is arbitrary, but the natural variables must include as many extensive variables as there are phases. For example, for the one-phase system described by equation 2.2-8, the F = Ns + 1 intensive variables can be chosen to be T, P, x, x2,xN. and the D = Ns + 2 natural variables can be chosen to be T, P, ni, n2,..., or T, P, xx, x2,..., xN and n (total amount in the system). 

This form of the fundamental equation, which applies at equilibrium, indicates that the natural variables for this system are T, nAx, and nAfi. Alternatively, P, nAx, and nAp could be chosen. Specification of the natural variables gives a complete description of the extensive state of the system at equilibrium, and so the criterion of spontaneous change and equilibrium is dG < 0 at constant 7( nAz, and nA/l or... 

Duhem s theorem is another rule, similar to the phase rule, but less celebratec It applies to closed systems for which the extensive state as well as the intensiv state of the system is fixed. The state of such a system is said to be completel determined and is characterized not only by the 2 + (iV—l)ir intensive phase rule variables but also by the it extensive variables represented by the masse (or mole numbers) of the phases. Thus the total number of variables is... 

There exists an extensive state function called entropy S that is defined by... 

An alternative measure of the subsystem local information distance relative to the corresponding reference density is defined by the entropy deficiency intensive conjugate of the fragment electron density ( extensive state-function). The Kullback-Lei bier functional of equations (92) and (93) gives ... 

THE SIZE-EXTENSIVE STATE-SPECIEIC MRCC FORMALISM USING AN IMS... 

This will seem like a reasonable conclusion to anyone who recalls our discussion of Euler s Theorem for homogeneous functions in Chapter 2, since V is homogeneous in the first degree in the masses (or mole numbers) of the components NaCl and H2O. It is, in other words, an extensive state variable. 

The energy is an extensive state property of the system under the same conditions of T and p, 10 mol of the substance composing the system has ten times the energy of 1 mol. The energy per mole is an intensive state property of the system. 

Since the energy of the system is an extensive state property, the heat capacity is also. The heat capacity per mole C, an intensive property, is the quantity found in tables of data. If the heat capacity of the system is a constant in the range of temperature of interest, then Eq. (7.19) reduces to the special form... 

H is called the enthalpy of the system, an extensive state property. 

Absolute configurations were first assigned to alkaloids of this section both by chemical transformations to compounds in the lycorine series where Mills rule had been applied and by Klyne s modification of the Hudson lactone rule. This extension states that lactones possessing the absolute configuration of XLVII are more positive in molecular rotation than derivatives in which the lactone ring is opened. If this rule is applicable to the alkaloids of this section, the conversion of homolycorine (XLVIII [M]d -f-268°) to tetrahydrohomolycorine (XLIX [M]d —322°) requires that homolycorine and tetrahydrohomolycorine have the absolute configurations shown in XLVIII and XLIX. These assign-... 

We distinguish between intensive state and extensive state. The intensive state can be identified solely in terms of intensive properties, and therefore it does not involve amounts of material. In contrast, identification of an extensive state must include a value for at least one extensive property, usually either the total amount of material or the total volume. Often only intensive states are needed to perform process analyses, while extensive states are usually needed to perform process designs. 

To change a thermodynamic state, we stand in the surroundings and apply interactions that cross the boundary. So we would like to know the number of orthogonal interactions that are available for changing the extensive state. 

Our initial guess is likely to be that = V, which would mean that the number of properties needed to identify the extensive state is the same as the number of interactions available for manipulating the extensive state. But, in fact, F may differ from V because of constraints. There are competing effects from two kinds of constraints. 

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