Extensive variables, defined

Before describing these thermodynamic variables, we must talk about their properties. The variables are classified as intensive or extensive. Extensive variables depend upon the amount while intensive variables do not. Density is an example of an intensive variable. The density of an ice crystal in an iceberg is the same as the density of the entire iceberg. Volume, on the other hand, is an extensive variable. The volume of the ocean is very different from the volume of a drop of sea water. When we talk about an extensive thermodynamic variable Z we must be careful to specify the amount. This is usually done in terms of the molar property Zm, defined as... 

As with other extensive variables, we will usually work with the molar function Um defined as... 

The extensive variable Q associated with the electrical potential + in Eqs. (15), (17), and (21) is the thermodynamic surface excess charge density, which is defined by... 

The magnitude of the entropy flux is the entropy transported through unit area per unit time, which is the divergence V Js- It is convenient to define all extensive variables per unit volume (denoted here by lower case symbols)... 

The advantage of the chemical potential over the other thermodynamic quantities, U, H, and G, is that it is an intensive quantity—that is, is independent of the number of moles or quantity of species present. Internal energy, enthalpy, free energy, and entropy are all extensive variables. Their values depend on the extent of the system—that is, how much there is. We will see in the next section that intensive variables such as p., T, and P are useful in defining equilibrium. 

As we proceed, it will become more convenient to work in the intensive variable, mass density p, rather than in numbers of molecules. We will soon discuss intensive and extensive variables, but for now we simply define density as the mass per unit volume, p = M/ V... 

For a system, namely a uniquely identified mass of fluid, it is often appropriate to think of variables or properties that characterize the system as a whole. For example, what is the total mass, momentum, or energy of the system These are called extensive variables or properties. It is reasonable to expect that within a system there may be local spatial variations in variables or properties. The total system property is determined by integrating local distributions over the mass of the system. To accomplish the integration, it is useful to define an intensive variable, which is the extensive variable per unit mass. That is, if the extensive variable is called N, then the associated intensive variable r) is defined as... 

An intensive variable [such as the temperature (T), pressure (P), or individual mole fractions of a single phase (xSi, Xu or y of the hydrate, liquid, or vapor phases, respectively)] is defined as a measured value that is independent of the phase amount. For example, T, P, xSi, xu y or density are intensive variables, while phase masses, volumes, or amounts are extensive variables, and thus not addressed by Gibbs Phase Rule. 

The chemical potential of a species i in a phase with m species is defined as the derivative of the internal energy U of this phase with respect to the number of moles of species i (n,). at constant values of the extensive variables V, S, and the number of moles of the remaining species in the absence of electrical and magnetic fields,... 

When restrictions are placed on a system, values must be assigned to an additional number of extensive variables in order to define the state of a system. If an isolated system is divided into two parts by an adiabatic wall, then the values of the entropy of the two parts are independent of each other. The term T dS in Equation (5.66) would have to be replaced by two terms, T dS and T" dS", where the primes now refer to the separate parts. We see that values must be assigned to the entropy of the two parts or to the entropy of the whole system and one of the parts. Similar arguments pertain to rigid walls and semipermeable walls. The value of one additional extensive variable must be assigned for each restriction that is placed on the system. 

The concept of indifferent states of systems is introduced in Section 5.11. There we define indifferent states as those states that required the assignment of a value to at least one extensive variable in addition to the number of moles of the components in order to define the state. Such systems are considered in more detail in this section. 

When we consider a one-component, two-phase system, of constant mass, we find similar relations. Such two-phase systems are those in which a solid-solid, solid-liquid, solid-vapor, or liquid-vapor equilibrium exists. These systems are all univariant. Thus, the temperature is a function of the pressure, or the pressure is a function of the temperature. As a specific example, consider a vapor-liquid equilibrium at some fixed temperature and in a state in which most of the material is in the liquid state and only an insignificant amount in the vapor state. The pressure is fixed, and thus the volume is fixed from a knowledge of an equation of state. If we now add heat to the system under the condition that the temperature (and hence the pressure) is kept constant, the liquid will evaporate but the volume must increase as the number of moles in the vapor phase increases. Similarly, if the volume is increased, heat must be added to the system in order to keep the temperature constant. The change of state that takes place is simply a transfer of matter from one phase to another under conditions of constant temperature and pressure. We also see that only one extensive variable—the entropy, the energy, or the volume—is necessary to define completely the state of the system. 

The system is thus univariant at this composition, and the temperature is a function of the pressure alone. At a fixed pressure we can transfer matter from one phase to another at constant temperature without changing the composition of the two phases. In doing so, heat must be added to or removed from the system, thus changing the entropy and energy of the system. The volume is determined from equations of state at the fixed temperature and pressure. We find that in such a case we again have to assign values to at least one extensive variable in order to define the state of such a system. 

In this discussion of indifferent states we have always used the entropy, energy, and volume as the possible extensive variables that must be used, in addition to the mole numbers of the components, to define the state of the system. The enthalpy or the Helmholtz energy may also be used to define the state of the system, but the Gibbs energy cannot. Each of the systems that we have considered has been a closed system in which it was possible to transfer matter between the phases at constant temperature and pressure. The differentials of the enthalpy and the Helmholtz and Gibbs energies under these conditions are... 

A one-component system that exists in three phases is indifferent and has no degrees of freedom. In order to define the state of the system then, three extensive variables must be used. We choose for discussion the enthalpy, volume, and number of moles of the components. The enthalpy of the system is additive in the molar enthalpies of the three phases, as is the volume. We can then write three equations ... 

In a system in which there are P pure condensed phases and one chemical reaction at equilibrium, there are (P —1) components. The system is thus univariant and hence indifferent. The state of the system is defined by assigning a value to at least one extensive variable in addition to the mole numbers of the species. The extent of the reaction taking place within the system is dependent upon the value of the additional extensive variable. A simple example is a phase transition of a pure compound when the change of phase is considered as a reaction. We consider the two phases as two species in the one-component system. In order to define the state of the system, we assign values to the volume of the system in addition to the temperature and mole number of the component. For the given temperature and mole number, the number of moles of the component in each phase is determined by the assigned volume. 

Partial molar quantities can be defined as the change of an extensive variable with respect to the mole number of one component at constant temperature, pressure, electric field, and mole numbers of all other components. Then, with Equations (14.73) and (14.74), the change of the partial molar entropy and partial molar volume with the electric field is given by... 

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... 

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. 

TEST 1- Define and give examples of extensive and intensive variables. Define degrees of free-... 

The chemical species set of a state is the set of chemical constituents that are associated with the system description of that state. The values of the attributes chemical-species-set, operating-conditions, and system-volume provide the (n + 2) independent variable quantities that are necessary to define a thermodynamic state. An important feature of this representation is that each state is described by a vector of intensive and extensive variables. The intensive vector defines the operational state of the process, while the extensive vector defines the maximum accumulation of mass and energy that can occur. This is bounded by flowrate, reaction rate and physical size of the process equipment. The values of these variables are accessed through the attributes interval flowrate vector, interval accumu-... 

Non-equilibrium thermodynamics was founded by Onsager. The theory was further elaborated by de Groot and Mazur and Prigogine. The theory is based on the hypothesis of local equilibrium a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this criterion. " Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the flux. - ... 

Later on we shall consider other extensive variables Y which will correspond to intensive variables defined by... 

Derivatives of the extensive variable, E, with respect to three independent, extensive, variables (S,V,N), yield three corresponding dependent, intensive, thermodynamic variables T, p, p, temperature, pressure and chemical potential. It would, therefore, be nice to have a formalism that allows thermodynamic functions to be defined in which the independent variables S or V or N... 

H = H T,p, M, M2, Ms,Mjv), where N is the number of chemical components. An extensive quantity can be divided by the mass of the system constituting a new variable defining a specific quantity ([3], p 103). The specific enthalpy (per unit mass) is then expressed as ... 

The twelve structural parameters defined above are all extensive variables. In order to convert them into intensive variables for use in the correlation for Tg, which is an intensive property, they will all be scaled by the number N of vertices in the hydrogen-suppressed graph of the repeat unit, as described by Equation 2.8 in Section 2.C. In other words, xj/N, x2/N,. .., x12/N, will be used as linear regression variables in the correlation for Tg. 

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