Intensive and Extensive Variables

Another distinction that we make among the thermodynamic functions is to describe p, V, T, U, and 5 as the fundamental properties of thermodynamics. The other quantities, H, A, and G are derived properties, in that they are defined in terms of the fundamental properties, with 

4 State Functions and Exact Differentials Inexact Differentials and Line Integrals 

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. 

Most often, we will choose the independent variables to be those quantities we control in the laboratory. The usual thermodynamic choices are (p and T) or (Vand T), Then, we measure changes in the thermodynamic properties of the system as these variables are altered. Thus, for a pure substance, writing 

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, 

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Various properties into Intensive and Extensive variables, listed ... 

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... 

Recall from the general relationships between intensive and extensive variables, an extensive variable is found by integrating the intensive variable over the mass of a system or, (using the mass density p of the system) integrating over the volume of the system, Eq. 2.18 ... 

As previously remarked (Sidebar 10.1), intensive vectors have privileged status in Ms The asymmetry between intensive and extensive variables can already be recognized in the U-based (or S-based) fundamental equation of Gibbs... 

Chemical thermodynamics deals with the physicochemical state of substances. All physical quantities corresponding to the macroscopic property of a physicochemical system of substances, such as temperature, volume, and pressure, are thermodynamic variables of the state and are classified into intensive and extensive variables. Once a certain number of the thermodynamic variables have been specified, then all the properties of the system are fixed. This chapter introduces and discusses the characteristics of intensive and extensive variables to describe the physicochemical state of the system. 

In this Section the internal energy function has been introduced in the form E - E(T,V), whereas in Section 1.18 it has been formulated as E - E(S,V). Considering that thermodynamic functions of state should be useful in deriving various intensive and extensive variables, are the two formulations equivalent If not, which one is more fundamental In a similar vein discuss the relation between H... 

Distinguish between intensive and extensive variables, giving examples of each. Use the Gibbs phase rule to determine the number of degrees of freedom for a multicomponent multiphase system at equilibrium, and state the meaning of the value you calculate in terms of the system s intensive variables. Specify a feasible set of intensive variables that will enable the remaining intensive variables to be calculated. 

In Section 7.4a we outline the calculation of the work (or more precisely, the rate of energy transferred as work) required to move fluid through a continuous process system, and in Section 7.4b we review the concepts of intensive and extensive variables introduced in Chapter 6 and introduce the concept of specific properties of a substance. Section 7.4c uses the results of the two preceding sections to derive the energy balance for an open system at steady state. 

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-... 

In what follows we shall extend our considerations to cover a complete specification of the state of a system in terms of both intensive and extensive variables. To do this we must consider closed systems and ask ourselves how many variables must be fixed to determine completely the equilibrium state of the system, that is to determine the intensive and extensive variables characterizing each phase. 

Understand the general concept of equilibrium, which is very important in the application of thermodynamics in chemical engineering Understand the difference between intensive and extensive variables Understand that total mass and total energy are conserved in any process... 

With the distinction now made between intensive and extensive variables, it is possible to rephrase the requirement for the complete specification of a thermodynamic state in a more coherent manner. The experimental observation is that the specification of two state variables uniquely determines the values of aifother state variables of an equilibrium, single-component, single-phase system. [Remember, however, that to determine the size of the system, that is, its mass or total volume, one must also specify the mass of the system, or"the value of one other extensive parameter (total volume, - total energy, etc.).] The implication of this statement is that for each substance there exist, in principle, equations relating each state variable to two others. For example. 

Furthermore, since the same fluid in the same state of aggregation is present in regions I and II, and since we have already established that the temperature, pressure, and molar Gibbs energy each have the same value in the two regions, the value of any state property must be the same in the two subsystems. It follows that any thermodynamic derivative that can be reduced to combinations of intensive variables must have the same value in the two regions of the fluid. The second derivatives (Eq. 7.2-2b), as we will see shortly, are combinations of intensive and extensive variables. However, the quantities NSxy, wherex and v denote U, V, or N, are intensive variables. Therefore, it follows that... 

Specification of the Equilibrium State Intensive and Extensive Variables Equations of State IS... 

It is possible to subdivide the properties used to describe a thermodynamic system (e.g., T, P, V,U,...) into two main classes termed intensive and extensive variables. This distinction is quite important since the two classes of variables are often treated in significantly different fashion. For present purposes, extensive properties are defined as those that depend on the mass of the system considered, such as volume and total energy content, indeed all the total system properties (Z) mentioned above. On the other hand, intensive properties do not depend on the mass of the system, an obvious example being density. For example, the density of two grams of water is the same as that of one gram at the same P, T, though the volume is double. Other common intensive variables include temperature, pressure, concentration, viscosity and all molar (Z) and partial molar (Z, defined below) quantities. ... 

Summarizing, we recall a common and simple definition of intensive and extensive variables. If two identical systems are gathered, the extensive term will double, whereas the intensive term will remain the same. Examples are the pairs (T, S), (-p, V), iix, n), (cp, q), i.e., (temperature, entropy), (pressure, volume), (chemical potential, mol number), (electrical potential, electrical charge). Often the extensive quantity may flow into or out of the system. 

We discuss again the topic of intensive and extensive variables in a formal way, namely in terms of exchange of energy. A change in energy form can be written in differential form, such as... 

In equilibriim thermodynamics the energy of a system may be considered to be a homogeneous bilinear function of pairs of intensive and extensive variables, either of which can be considered as the independent variable. For example, either pressure or volume may be considered as an independent variable depending upon the environment. The difference between the heat capacity at constant volume and at constant pressure is well known in equilibrium thermodynamics. Thus, in a single component equilibrium system where temperature. 

Typical notation distinguishes between intensive and extensive variables. Intensive quantities such as temperature and pressure do not scale with the system size extensive quantities such as internal energy and entropy do scale with system size. For example, if the size of a box of gas molecules is doubled and the number of molecules in the box doubles, then the internal energy and entropy double while the temperature and pressure are constant. Frequently, certain conventions are used to denote intensive versus extensive properties, such as using lowercase for intensive variables (e.g., p for pressure) and uppercase for extensive variables (e.g., S for entropy). Unfortunately, such nomenclature is not standardized and is frequently inconsistent. For example, temperature is almost always represented with an uppercase T, although it is an intensive quantity. 

Duhem s law, which relates to all the intensive and extensive variables, and is applicable only to closed systems. 

The involvement of several components in a phase consequently introduces new couples of intensive and extensive variables (one couple per component) chemical potentials and amounts of matter (see section 2.2.4), and then the involvement of new variables compositions, partial molar variables (see section 2.3), chemical potentials (see section 2.6) and mixing variables. 

Let us consider a partial molar variable. In a uniform solution, this variable is a function of the variables of the problem, i.e. some intensive and extensive variables belong to other conjugate couples of J. If we have one mole of a solution overall, by having the molar fractions jc, as the composition... 

Explain why F values are not necessarily strictly additive. (Hint. Consider the properties of intensive and extensive variables.)... 

Divide the following physical variables into intensive and extensive variables, respectively. Then show by examples how to convert the extensive variables into intensive variables. 1) The mass m (kg) of a system of substances. 2) The modulus of elasticity E (MPa) of a steel specimen. 3) The viscosity 77 (Pa s) of a given saline solution. 4) The electric charge Q (C) on a charged condenser plate. 5) The molar mass M (g/mol) of a chemical compound. 6) The cement content C (kg) in a given concrete specimen. 7) The elongation (m) of a loaded test steel rod. 8) The coefficient of thermal expansion a (K ) of pyrex glass. 

Consider a nonreacting system containing specified amounts of its k components that reaches equilibrium at some temperature 7 and pressure P and let be the number of phases present. What information is needed for the complete specification of this system in terms of both intensive and extensive variables In this case, thus, the complete determination of its equilibrium state, requires - in addition to 7, P, and the composition of each phase - the amounts of each phase. Hence ... 

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