Extensive property entropy

The intensive porperty Temperature is supplemented by a complementary extensive property, entropy. In the case of energy in form of heat it gives the number of degrees of freedom among which the average energy of motion (of the material particles involved), characterized by the temperature, is distributed. Entropy and temperature are complementary state variables. 

Entropy is an extensive property and Sm the molar entropy is often used. 

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. 

Like the extensive properties of a composite system the entropy also is additive over the constituent subsystems,... 

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. 

Any characteristic of a system is called a property. The essential feature of a property is that it has a unique value when a system is in a particular state. Properties are considered to be either intensive or extensive. Intensive properties are those that are independent of the size of a system, such as temperature T and pressure p. Extensive properties are those that are dependent on the size of a system, such as volume V, internal energy U, and entropy S. Extensive properties per unit mass are called specific properties such as specific volume v, specific internal energy u, and specific entropy. s. Properties can be either measurable such as temperature T, volume V, pressure p, specific heat at constant pressure process Cp, and specific heat at constant volume process c, or non-measurable such as internal energy U and entropy S. A relatively small number of independent properties suffice to fix all other properties and thus the state of the system. If the system is composed of a single phase, free from magnetic, electrical, chemical, and surface effects, the state is fixed when any two independent intensive properties are fixed. 

The ith cell has extensive properties of entropy Sh energy Uh volume Vb and mass Nh where... 

Figure 5.1 Schematic plots of (a) constrained entropy Sjj and (b) unconstrained entropy S as functions of a general extensive property X near equilibrium, Xeq. In each case, the negative curvature of the entropy function (constrained or unconstrained) carries it below its equilibrium tangent (dashed line).

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). 

Properties such as internal energy, volume and entropy are called extensive because their values for a given phase are proportional to the mass or volume of the phase. The value of an extensive property of an entire system is the sum of the values of each of the constituent phases. The molar value of an extensive property is that for a properly defined gram-molecular weight or mole of material. The specific value of an extensive property is that per unit weight (eg, one gram of material). A property is called intensive if its value for a given phase is independent of the mass of the phase. Temp and pressure are examples of such intensive properties... 

Since entropy and volume are extensive properties, they can be combined,... 

The entropy and volume are extensive properties, as are the number of moles of each component, but the temperature and pressure are not. Consequently, we may set H as a homogenous function of the first degree in the entropy and mole numbers if the pressure is kept constant. Then, by Euler s theorem,... 

We now move the two surfaces a and b toward each other so that they coincide at some position c to give one two-dimensional surface lying wholly within the real surface. The system is thus divided into two parts, and we assume that the properties of each of the two parts are continuous and identical to the properties of the bulk parts up to the single two-dimensional surface. Certain properties of the system are then discontinuous at the surface. The extensive properties of the two-dimensional surface are defined as the difference between the values of the total system and the sum of the values of the two parts. Thus, we have for the energy, entropy, and mole number of the c components ... 

The system of our choice will usually prevail in a certain macroscopic state, which is not under the influence of external forces. In equilibrium, the state can be characterized by state properties such as pressure (P) and temperature (T), which are called "intensive properties." Equally, the state can be characterized by extensive properties such as volume (V), internal energy (U), enthalpy (H), entropy (S), Gibbs energy (G), and Helmholtz energy (A). 

The state of a system is defined by its properties. Extensive properties are proportional to the size of the system. Examples include volume, mass, internal energy, Gibbs energy, enthalpy, and entropy. Intensive properties, on the other hand, are independent of the size of the system. Examples include density (mass/volume), concentration (mass/volume), specific volume (volume/mass), temperature, and pressure. 

Let P represent some extensive property of the system (e.g., mass, momentum, energy, entropy) and let p represent its intensive counterpart (i.e., per unit mass), such that ... 

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

Equation 9 relates the basic extensive properties, the respective driving potentials, and the entropy production in the diffusion process. The first term, s TT, arises from heat transfer effects while the second term is due to mass transfer. For processes wherein the entropy production due to column heat transfer is small relative to the mass-transfer, the s TE term is negligible and Equation 9 simplifies to... 

The fundamental equations relate all extensive properties of a thermodynamic system, and hence contain all the thermodynamic information on the system. For example, the fundamental equation in terms of entropy is... 

For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V. and the fundamental equations define a fundamental surface of entropy S = S(U,V) in the Gibbs space of S, U, and V. 

Equations of state relate intensive properties to extensive properties, and are obtained from the Euler equation as partial derivatives. In the entropy representation, we have the following equations of state ... 

Since the temperature is not uniform for the whole system, the total entropy is not a function of the other extensive properties of U, V, and N. However, with the local temperature, the entropy of a nonequilibrium system is defined in terms of an entropy density, sk. 

In every nonequilibrium system, an entropy effect exists either within the system or through the boundary of the system. Entropy is an extensive property, and if a system consists of several parts, the total entropy is equal to the sum of the entropies of each part. Entropy balance is... 

Let ikfi, Eu Fi, denote the mass and the specific energy, volume, and entropy, respectively, of the first phase, and M2, JS 2, F2, 5 2 those of the second, M, E, V, and S being the corresponding values for the complex. Then, since these are all extensive properties ( 17.11), with the restrictions mentioned in 3.11 ... 

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