Extensive and Intensive Quantities

An important concept is the characterization of thermodynamic quantities as either intensive or extensive. 

Imagine two identical containers filled with the same kind and amount of of gas at the same temperature. What happens if we bring the two containers in contact and allow the gas to fill the combined containers as if it was one. Obviously temperature and pressure do not change. The quantities T and P therefore are called intensive. Volume on the other hand is doubled. Mathematically this means V (x n. Such quantities are said to be extensive, n itself is therefore extensive. Thus far  

The ratio of two extensive quantities, e.g. n/V, again is intensive of course. Another intensive quantity is the chemical potential, ix. Whether one mole of material is added to a large system or to twice as large a system should not matter. This however has implications for the free enthalpy, G. According to Eq. (2.107) we have for a one-component system 

Remark 1 Suppose we consider the potential energy W of a system consisting of N pairwise interacting molecules. Disregarding their spatial arrangement we may write W N /2)V drr . The factor N(N - ) 2 is the number of 

Momentarily we talk about one-component systems and not about mixtures. 

In the study of thermodynamics we can distinguish between variables that are independent of the quantity of matter in a system, the intensive variables, and variables that depend on the quantity of matter. Of the latter group, those variables whose values are directly proportional to the quantity of matter are of particular interest and are simple to deal with mathematically. They are called extensive variables. Volume and heat capacity are typical examples of extensive variables, whereas temperature, pressure, viscosity, concentration, and molar heat capacity are examples of intensive variables. 

It is of the utmost importance to recognize that balances can only be made on extensive variables or quantities. If you double the system, an extensive measure is doubled, whereas an intensive measure remains the same. Thus the mass of two identical bricks is twice the mass of one, but the density, or mass per unit volume, remains the same in the duplicated system because we have doubled both the mass and the volume. The first (mass) behaves like a homogeneous function of degree one, the second (volume) of degree zero. Thus in the simple example used in Example 1, we did not make our balance on the concentration, moles per unit volume = c, but on the amount, moles - Vc. 

We recall that in mechanics the energy is the negative integral of a force by the path. This concept holds ideally for masses that are concentrated in a mathematical point, so-called point masses  

Since a macroscopic system has an extension in space, not all parts of the system can be at the same height. Therefore, the system is not homogeneous. In order to deal with a homogeneous system with gravitational energy, we must consider an infinitely thin shell where all parts of the system can be at the same height. 

The way to create extensive and intensive variables, as exemplified in Eq. (2.4), does not work in general. For example, the volume is an extensive variable, but the mass is captured in the corresponding intensive variable, i.e., the pressure. Therefore, dividing the pressure by mass is not what we would expect. In the sense of thermodynamic homogeneity, scaling in the gravitational field is restricted to a direction in space where no field is actually acting. 

Despite these problems, homogeneity is rather characterized by a certain portion of space where the properties are not changing. Thus, scaling the volume of a region in space where properties such as pressure, charge density, entropy density are constant seems to be a more fundamental process in order to identify extensive and intensive variables. 

The volume is rather a basic concept of a region, as it appears in the derivation of all kinds of equations of continuity. For this reason, we introduce quantities that refer to the unit of volume. We will use the superscript X in order to indicate that this is X per volume. Remember that we are using C for the molar heat capacity (at constant volume), Cv for the specific heat capacity, C for a reduced heat capacity, i.e., divided by a reference value, and now C for the heat capacity density. The quantities X are basically X densities, for example, the mass density or the molar density  

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Extensive and intensive quantities are characterised in that together they can form parameter couples having the dimensions of an energy. For instance ... 

Extensive and Intensive Variables, Partial Molar Quantities... 

The number of unknowns and the number of equations relating these unknowns can become very large in a process-design problem. The number of unknowns and independent equations must be equal in order that a unique solution to a problem exists. Therefore, it is necessary to have a systematic method for enumerating them. The total number of independent extensive and intensive variables associated with each stream in a process is C + 2, where C is the number of independent chemical components in the stream. The quantity and the condition of the stream are completely determined by fixing the flow rate of each component in the stream (or, equivalently, the total flow rate and the mole or mass fractions of C — 1 components) and two additional variables, usually the temperature and pressure, although other choices are possible. This number includes situations where physical and chemical equilibrium exist.t... 

The process relationship for the target quantity 32 (or for a) can be formulated either with extensive or intensive quantities. The difference lies in the choice of the process parameter. The dispersion characteristic formulated with extensive quantities uses as the process parameter the extensive quantity stirrer speed and leads, assuming a given geometry (stirrer type D/d, H/d, h/d = const), to the following dependence (the index d indicating the physical properties of the dispersed phase, no index indicating the physical properties of the continuous phase) ... 

The state of the system is given by a set of values of properly chosen physical variables. To determine unambiguously the state of the simplest system (a pure substance in one phase) one should know two properties (e.g. temperature and pressure) in addition to the quantity (moles). To describe the state of more complex systems one should know more properties (e.g. the concentrations of individual species). The thermodynamic properties of the system depending only on the state and not on the way by which the system has reached the given state, are called state functions. The typical fundamental state functions are temperature, pressure, volume and concentration of the individual components of the system. The thermodynamic properties are usually classified into extensive and intensive ones. The extensive properties are proportional to the quantity of the substance in the system. Therefore, they are additive, i.e. the total extensive property of the system equals the sum of the extensive properties of the individual parts of the system. Typical extensive quantities are weight, energy, volume, number of moles. On the other hand, the intensive properties do not depend on the quantity of the substance in the system (pressure, temperature, concentration, specific quantities, specific resistance, molar heat, etc.). 

Primitive quantities include generalized forces, the concepts of equilibrium and state, and ways to classify properties. The ideas surrounding force, equilibrium, and state are absolutely crucial because they identify those situations which are amenable to thermodynamic analysis. We will have much more to say about these concepts for example, we want to devise quantitative ways for identifying the state of a system and for deciding whether the system is at equilibrium. Although classifications of properties are not crucial, the classifications—extensive and intensive or measurable and conceptual—facilitate our development and study of the subject. 

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. 

In this text, no attempt will be made to use nomenclature to distinguish between extensive and intensive properties or between an extensive property and its mass or mole-normalized specific counterpart. You should be aware, then, that quantities appearing in certain equations may represent an extensive property or its specific (e.g., per-mole) counterpart depending on the situation. For example, AG could have units of kilojoules or kilojoules per mole depending on the context. In general, this should not cause confusion, as the context and units involved will almost always be spelled out explicitly. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

The current 7 is an extensive quantity, in that it depends on the size of the electrode. For this reason, the reaction rate is conveniently referred to the unit surface area (7/S=j, current density). Even so, the current density continues to be an extensive quantity if referred to the geometric (projected) surface area since electrodes are as a rule rough and the real surface does not coincide with the geometric surface [23]. Conversely, b is an intensive quantity, in that it depends only on the reaction mechanism and not on the size of the electtode. The term b is the most important kinetic parameter in electrochemistry also because of the easy and straightforward procedure for its experimental determination. Most electrode mechanisms can be resolved on the basis of Tafel lines only. 

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. 

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. 

Name three extensive and three intensive properties that relate to thermodynamic quantities. What is the basic difference between the two types of properties ... 

Finally, the thermodynamic properties of a system considered as variables may be classified as either intensive or extensive variables. The distinction between these two types of variables is best understood in terms of an operation. We consider a system in some fixed state and divide this system into two or more parts without changing any other properties of the system. Those variables whose value remains the same in this operation are called intensive variables. Such variables are the temperature, pressure, concentration variables, and specific and molar quantities. Those variables whose values are changed because of the operation are known as extensive variables. Such variables are the volume and the amount of substance (number of moles) of the components forming the system. 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

Internal energy is an extensive property of a system. If we double the size of a system, keeping intensive variables such as temperature and pressure constant, we double the system s internal energy. If we divide the internal energy of a system by the number of moles in the system, we obtain the molar internal energy, Um = VIn, which is an intensive quantity. Other molar properties, such as the molar volume, are also indicated by the subscript m. 

Extensive quantities show an exact proportionality to n, only when intensive quantities, like T and P, remain constant. 

The quantities U, H, S, A, G, q and V are extensive and p and T intensive quantities. When an extensive quantity is related to the amount of material in a mole, then it becomes a molar and therefore specific quantity with which the properties of the material under consideration can be described. 

Specific quantities. These, two, are independent of the extension of the system under consideration. They result from extensive quantities when these are related to the unit of mass. So these quantities are also quotients of two extensive quantities and consequently have all the characteristics of intensive quantities. For mixtures the numerical value of these specific quantities is determined by the composition and averaged in accordance with it. Examples ... 

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