Intensive variable Internal energy

Postiilate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. Tnese molar or unit-mass properties, represented by the plain symbols U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V U S ) do depend on system size, and are extensive. For a system containing n moles of fluid, M = nM, where M is a molar property. 

In thermodynamics the state of a system is specified in terms of macroscopic state variables such as volume, V, temperature, T, pressure,/ , and the number of moles of the chemical constituents i, tij. The laws of thermodynamics are founded on the concepts of internal energy (U), and entropy (S), which are functions of the state variables. Thermodynamic variables are categorized as intensive or extensive. Variables that are proportional to the size of the system (e.g. volume and internal energy) are called extensive variables, whereas variables that specify a property that is independent of the size of the system (e.g. temperature and pressure) are called intensive variables. 

In general, dw is written in the form (intensive variable)-d(extensive variable) or as a product of a force times a displacement of some kind. Several types of work terms may be involved in a single thermodynamic system, and electrical, mechanical, magnetic and gravitational fields are of special importance in certain applications of materials. A number of types of work that may be involved in a thermodynamic system are summed up in Table 1.1. The last column gives the form of work in the equation for the internal energy. 

In a more or less complex fashion, equations 1.97, 1.103, and 1.107 describe the response of the crystal, in terms of energy, to modification of the intensive variable P. We note that, in all cases, the differential forms combine internal energy and volume and are thus analytical representations of the potential diagram seen above for Ni2Si04 in figure 1.12. 

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. 

In the Eulerian view, a fluid is characterized by fields of intensive variables or properties r). For example, the internal energy (or temperature, for a constant specific heat) is assumed to be a continuous function of time and space, r)(t,x). Because rj is a continuous differentiable function, the following expansion is generally valid ... 

Refer to Fig. 2.3 and assume that r) is an intensive variable, like specific internal energy. At any spatial location, namely (r, 9), the height of the surface represents the magnitude of r), for example, internal energy. The gradient represents the local slope of the surface in the r and 9 directions,... 

Consider next the energy equation, neglecting kinetic and gravitational-potential energy. Here the extensive variable is the internal energy of the gas E and the intensive variable is the specific internal energy e. The first law of thermodynamics provides the system energy balance... 

The fundamental equation for U is in agreement with the statement of the preceding section that for a homogeneous mixture of Ns substances, the state of the system can be specified by Ns + 2 properties, at least one of which is extensive. The total number of variables involved in equation 2.2-8 is 2NS + 5. Ns + 3 of these variables are extensive (U, S, V, and (nj), and Ns + 2 of the variables are intensive (T, P, /.q ). Note that except for the internal energy, these variables appear in pairs, in which one property is extensive and the other is intensive these are referred to as conjugate pairs. These pairs are given later in Table 2.1 in Section 2.7. When other kinds of work are involved, there are more than 2Ns + 5 variables in the fundamental equation for U (see Section 2.7). 

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. 

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. 

Thermodynamic principles arise from a statistical treatment of matter by studying different idealized ensembles of particles that represent different thermodynamic systems. The first ensemble that we study is that of an isolated system a collection of N particles confined to a volume V, with total internal energy E. A system of this sort is referred to as an NVE system or ensemble, as N, V, and E are the three thermodynamic variables that are held constant. N, V, and E are extensive variables. That is, their values are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size of the system. 

Equation (1.107) relates the total change in internal energy to the sum of the products of intensive variables T. P, F, lib if/, and the changes in extensive properties (capacities) of dS, dV, dl, dNb and de. The Bronsted work principle states that the overall work A W performed by a system is the sum of the contributions due to the difference of extensive properties AK across a difference of conjugated potentials A). 1-. 2... 

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... 

This expression shows that volume and entropy serve as thermodynamic variables, or as control variables, for the internal energy function of the system E = E(S, V). Nevertheless, the intensive variables are those of the surroundings, and are therefore well defined, even when the processes in the system proper are far removed from equilibrium. For the present we exclude other types of work that are treated in Chapter 5. We defer the generalization of the present treatment to the case of open systems to Section 1.20. 

The successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

The sign of the derivatives can be determined simply by writing down the variables in two rows, the intensive variables above, the extensive variables below so that conjugate pairs are in the same column. The signs attached to the intensive variables are those of the corresponding terms in the fundamental equation (4.23) for the internal energy. 

Variables of the kind with which the phase rule is concerned are called phase-rule variables, and they are intensive properties of the system. By this we mean properties that do not depend on the quantity of material present. If you think about the properties we have employed so fer in this book, you have the feeling that pressure and temperature are independent of the amount of material present. So is concentration, but what about volume The total volume of a system is called an extensive property because it does depend on how much material you have the specific volume, on the other hand, the cubic meter per kilogram, for example, is an intensive property because it is independent of the amount of material present. In Chap. 4 we take up additional intensive properties, such as internal energy and enthalpy. You should remember that the specific (per unit mass) values of these quantities are intensive properties the total quantities are extensive properties. 

Define or explain the following terms energy, system, closed system, nonflow system, open system, flow system, surroundings, property, extensive property, intensive property, state, heat, work, kinetic energy, potential energy, internal energy, enthalpy, initial state, final state, point (state) function, state variable, cyclical process, and path function. 

To calculate the internal energy per unit mass 0) firom the variables that can be measured, we make use of a special property of internal energy, namely, that it is an exact differential (because it is a point or state property, a matter to be described shortly) and, for a pure component, can. be expressed in terms of just two intensive variables according to the phase rule for one phase ... 

Intensive—independent of quantity of material extensive—dependent of quantity of material. Measurable—temperature, pressure unmeasurable—internal energy, enthalpy. State variable—difference in value between two states depends only on the states path variable—difference in two states depends on trajectory reaching in the final state. 

The thermodynamic variables, 7, S,H, A, and G introduced above are extensive quantities like the volume V. Thus, the amount of internal energy in a sulfuric acid solution depends on whether one has 250mL beaker, a 4L bottle, or a full railway tank car. Just as for volume, it is necessary to define intensive variables giving the internal energy per gram, Ug, or the internal energy per mole, C/m. Since the present discussion is concerned with chemistry, we will use only the molar quantities t/m, 5j , A,, and which are defined from the corresponding... 

This simply shows that there is a physical relationship between different quantities that one can measure in a gas system, so that gas pressure can be expressed as a function of gas volume, temperature and number of moles, n. In general, some relationships come from the specific properties of a material and some follow from physical laws that are independent of the material (such as the laws of thermodynamics). There are two different kinds of thermodynamic variables intensive variables (those that do not depend on the size and amount of the system, like temperature, pressure, density, electrostatic potential, electric field, magnetic field and molar properties) and extensive variables (those that scale linearly with the size and amount of the system, like mass, volume, number of molecules, internal energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables are not. 

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