Thermodynamics extensive properties

Consider any thermodynamic extensive property, such as volume, free energy, entropy, energy content, etc., the value of which, for a homogeneous system, is completely determined by the state of the system, e.g., the temperature, pressure, and the amounts of the various constituents present thus, ( is a function represented by... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Doubling the number of molecules increases the number of microstates from W to W2, and so the entropy changes from k In W to k In W2, or 2k In W. Therefore, the statistical entropy, like the thermodynamic entropy, is an extensive property. 

We can express the use of all the different units in evolution in the language of thermodynamics. While the genome is defined by a DNA sequence so that each base has a singular intensive property as in a computer code of symbols, by way of contrast, the protein content of a cell is an extensive property being concentration dependent and therefore varies under circumstances such as temperature and pressure although... 

The most important new concept to come from thermodynamics is entropy. Like volume, internal energy and mole number it is an extensive property of a system and together with these, and other variables it defines an elegant self-consistent theory. However, there is one important difference entropy is the only one of the extensive thermodynamic functions that has no obvious physical interpretation. It is only through statistical integration of the mechanical behaviour of microsystems that a property of the average macrosystem, that resembles the entropy function, emerges. 

This relationship is expressed in extensive properties that depend on the extent of the system, as opposed to intensive properties that describe conditions at a point in the system. For example, extensive properties are made intensive by expressing them on a per unit mass basis, e.g. s = S/m density, p 1 /v, v V/m. For a pure system (one species), Equation (1.2) in intensive form allows a definition of thermodynamic temperature and pressure in terms of the intensive properties as... 

The determination of many thermodynamic solution properties of these trihalides, especially enthalpies of solution, requires the use of anhydrous materials. These are by no means always readily available for this group of compounds. Therefore we precede our discussion of thermochemical results with a section on preparative methods, in which we deal summarily with many of the relevant papers published since Taylor s extensively referenced review, published in 1962 (2). 

This simple relationship allows us to express all the thermodynamic variables in terms of our colloid concentration. The Helmholtz free energy per unit volume depends upon concentration of the colloidal particles rather than the size of the system so these are useful thermodynamic properties. If we use a bar to symbolise the extensive properties per unit volume we obtain... 

The units of q are calories per square centimeter per second and those of the thermal conductivity A are calories per centimeter per second per degree Kelvin. It is not the temperature, an intensive thermodynamic property, that is exchanged, but energy content, an extensive property. In this case, the energy density and the exchange reaction, which show similarity, are written as... 

This shows that in the limit of large systems, the thermodynamics of the system is determined by the largest root of the secular equation. In particular, -kgT In as seen in Eq. (7.1.30) is an extensive property, i.e., it is proportional to the number of units. 

It should be emphasized that the criterion for macroscopic character is based on independent properties only. (The importance of properly enumerating the number of independent intensive properties will become apparent in the discussion of the Gibbs phase rule, Section 5.1). For example, from two independent extensive variables such as mass m and volume V, one can obviously form the ratio m/V (density p), which is neither extensive nor intensive, nor independent of m and V. (That density cannot fulfill the uniform value throughout criterion for intensive character will be apparent from consideration of any 2-phase system, where p certainly varies from one phase region to another.) Of course, for many thermodynamic purposes, we are free to choose a different set of independent properties (perhaps including, for example, p or other ratio-type properties), rather than the base set of intensive and extensive properties that are used to assess macroscopic character. But considerable conceptual and formal simplifications result from choosing properties of pure intensive (R() or extensive QQ character as independent arguments of thermodynamic state functions, and it is important to realize that this pure choice is always possible if (and only if) the system is macroscopic. 

The thermodynamic state is therefore considered equivalent to specification of the complete set of independent intensive properties 7 1 R2, Rn. The fact that state can be specified without reference to extensive properties is a direct consequence of the macroscopic character of the thermodynamic system, for once this character is established, we can safely assume that system size does not matter except as a trivial overall scale factor. For example, it is of no thermodynamic consequence whether we choose a cup-full or a bucket-full as sample size for a thermodynamic investigation of the normal boiling-point state of water, because thermodynamic properties of the two systems are trivially related. 

Comparison of the combined first/second law (4.28) with (4.30) leads to the more general and rigorous thermodynamic definitions for the intensive properties T, —P respectively conjugate to the extensive properties S, V ... 

We have previously emphasized (Section 2.10) the importance of considering only intensive properties Rt (rather than size-dependent extensive properties Xt) as the proper state descriptors of a thermodynamic system. In the present discussion of heterogeneous systems, this issue reappears in terms of the size dependence (if any) of individual phases on the overall state description. As stated in the caveat regarding the definition (7.7c), the formal thermodynamic state of the heterogeneous system is wholly / dependent of the quantity or size of each phase (so long as at least some nonvanishing quantity of each phase is present), so that the formal state descriptors of the multiphase system again consist of intensive properties only. We wish to see why this is so. 

As mentioned in the Preface, our goal in Part III is not merely to re-generate the material of Parts I and II (as summarized in Section 8.9) in new mathematical dress. We re-derive (rather trivially) many earlier thermodynamic identities and stability conditions to illustrate the geometrical techniques, but our primary emphasis is on thermodynamic extensions (particularly, to saturation properties, critical phenomena, multicomponent Gibbs-Konowalow-type relationships, higher-derivative properties, and general reversible changes... 

The properties of a substance can be classified as either intensive or extensive. Intensive properties, which include density, pressure, temperature, and concentration, do not depend on the amount of the material. Extensive properties, such as volume and weight, do depend on the amount. Most thermodynamic properties are extensive including energy (E), enthalpy (H), entropy (5), and free energy (G). 

The thermodynamic treatment of an interface generally considers a system composed of the interface (y) and two adjacent homogeneous phases (a and / ). The extensive properties of the systems must be ascribed to these three regions, for example, the Gibbs free energy G and the number of moles of a species in the system fulfill... 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

A thermodynamic property is said to be extensive if the magnitude of the property is doubled when the size of the system is doubled. Examples of extensive properties are volume V and amount of substance n. A thermodynamic property is said to be intensive if the magnitude of the property does not change when the size of the system is changed. Examples of intensive properties are temperature, pressure, and the mole fractions of species. The ratio of two extensive properties is an intensive property. For example, the ratio of the volume of a one-component system to its amount is the molar volume Vm = V/n. 

The internal energy is homogeneous of degree 1 in terms of extensive thermodynamic properties, and so equation 2.2-8 leads to equation 2.2-14. All extensive variables are homogeneous functions of the first degree of other extensive properties. All intensive properties are homogeneous functions of the zeroth degree of the extensive properties. 

This shows that the natural variables of G for a one-phase nonreaction system are T, P, and n . The number of natural variables is not changed by a Legendre transform because conjugate variables are interchanged as natural variables. In contrast with the natural variables for U, the natural variables for G are two intensive properties and Ns extensive properties. These are generally much more convenient natural variables than S, V, and k j. Thus thermodynamic potentials can be defined to have the desired set of natural variables. 

Pressure, volume, temperature, and number of moles are thermodynamic properties or thermodynamic variables of a system—in this case, a gas sample. Their values are measured by experimenters using thermometers, pressure gauges, and other instruments located outside the system. The properties are of two types those that increase proportionally with the size of the system, such as n and K called extensive properties, and those defined for each small region in the system, such as P and T, called intensive properties. Terms that are added together or are on opposite sides of an equal sign must contain the same number of... 

Thermodynamic properties which depend on the amount (mass) of material present (e.g. internal energy U enthalpy H, etc.) are termed EXTENSIVE properties. 

In general thermoeconomic optimization requires the derivation of expressions for entropy production, via nonequilibrium thermodynamics, due to each independent extensive property transport. 

All 29 enumerated problems can be solved on the basis of long-standing studies of many experts. However, we hope that even partial performance of the stated tasks will make the models and methods of the present-day equilibrium thermodynamics the property of a wide circle of researchers and engineers and they would find extensive application in the basic and applied science. 

The thermodynamic properties, U, H, S, A, G are extensive properties because their value change with change in mass (i.e., the number of mole) of the system. In the various thermodynamic equations, the change of state was considered to be due to change of temperature and pressure. This means there is no change in mass of the system and such systems are called closed system. However in the case of an open system containing two or more components, there can be change in the number of moles of various components as well. In that case, an extensive property, say, X, must be a function not only of temperature and the pressure but also of the number of moles of the various components present in the system. 

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