Properties independent

In this section, we will develop a web of property relationships whereby we can relate the thermodynamic properties we need to solve problems to properties we can measure in the lab. We want to relate fundamental and derived thermodynamic properties, such as u, s, h, a, and g, to things we can measure, such as measured properties P, v, T or to quantities for which measured data are typically reported, for example, c , Cp, p, and k. We will exploit the rigor of mathematics to allow us to develop an intricate web of these relationships. As with searching for sites on the Internet, there is usually more than one way to obtain our final answer some are quicker, while others are slower. 

We limit our present discussion to constant composition systems we will learn about mixtures that can change in composition in Chapter 6. Recall that the state postulate says that for systems of constant composition, values of two independent, intensive properties completely constrain the state of the system. In mathematical terms, the change in any intensive thermodynamic property of interest, z, can be written in terms of partial derivatives of the two independent intensive properties, x and y, as follows  

The form of Equation (5.4) is general and we can use it to express any of the properties we examined in Section 5.1 in terms of two independent properties. For example, say we want to calculate the change in internal energy for a first-law analysis of a closed system. We may choose to relate the differential change in internal energy, dit, to the measured properties temperature, T, and molar volume, v. In the form of Equation (5.4), we write  

In Equation (5.5), the intensive properties T and v constrain the state of the system we call these two the independent properties. For brevity, we will use the notation  

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Computerized psychometric chart. User provides two independent properties of moist air and program calculates the remaining properties. 

Computes thermodynamic properties of air, argon, carbon monoxide, carbon dioxide, hydrogen, nitrogen, oxygen, water vapor, and products of combustion for hydrocarbons. Computes all properties from any two independent properties. 

Computes properties of air-water vapor mixtures for HVAC, combustion, aerodynamic, and meteorological applications. Any two independent properties may be inpuicd by... 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

Up to now we have considered non-dynamical equilibrium properties, namely time-independent properties but the richness of a liquid is related to its flow, gradients, and dynamics. We will briefly consider a few dynamical properties of liquid water here and refer the interested readers to Refs. 31, 46, and 47 for details and others. 

Many solvents do not possess the simple structure that allows their effects to be modeled by the Langevin equation or generalized Langevin equation used earlier to calculate the TS trajectory [58, 111, 112]. Instead, they must be described in atomistic detail if their effects on the effective free energies (i.e., the time-independent properties) and the solvent response (i.e., the nonequilibrium or time-dependent properties) associated with the... 

State-of-the-art polymeric materials possess property distributions in more than one parameter of molecular heterogeneity. Copolymers, for example, are distributed in molar mass and chemical composition, while telechelics and macromonomers are distributed frequently in molar mass and functionality. It is obvious that n independent properties require n-dimensional analytical methods for accurate (independent) characterization of the different structural parameters. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

Evidence for Equilibrium. The lines in Figure 1, as well as those from data obtained on the other two elastomers, have a zero slope over three decades of time, an indication that the data represent elastic (time-independent) properties. Furthermore, the moduli of the elastomers at 5°C, evaluated as illustrated by Figures 1 and 2 and based on the dimensions of the specimens at 5°C, are less than those at 30°C. The ratio Ejq/Ej is 1.06 for the LHT-240 and Tri-NCO elastomers and 1.08 for the TIPA elastomer. These values are only slightly less than 303/278 ( = 1.09) which is essentially the theoretical value for the ratio of elastic moduli at 30° and 5°C. Actually, the data at 5°C showed a slight amount of relaxation during three decades of times. But relaxation was complete at 30°C, as indicated by the above remarks. 

This parameter is then a potential-independent property of the electrolyte at a given temperature over some unspecified potential difference range. 

Chemical reactions have two independent properties, their energy and their rate. Table 1-8-2 compares these two properties. AG represents the amount of energy released or required per mole of reactant. The amount or sign of AG indicates nothing about the rate of the reaction. 

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. 

One of the great benefits of polyurethane is versatility. With only slight changes in chemistry, one can make products ranging from soft furniture cushions to automobile bumpers and infinite numbers of other products. Depending on the application, a polyurethane chemist can vary density and stiffness to achieve acceptable product performance. The chemistry is in fact much more versatile than is required. Figure 2.19 covers soft foams, rigid foams, and other polyurethanes. We will provide more details later in this chapter, particularly as to how the independent properties of density and stiffness relate to end uses. 

Table 8). Evidently, stoichiometry and transport energetics are substrate-independent properties of the pump. The kinetics of the transport, however, is very much affected, by the substrate. This substrate specificity is not very well understood. It has been... 

And now the precise effect that we are formulating of the randomness of the forces acting over the x-interval is that they cause the independence property so that, according to information theory,... 

Indeed, this interdependence of all but a small number of independent properties ( degrees of freedom ) is observed to be one of the most striking and universal features of all known gaseous substances. If we consider a fixed quantity of a pure substance ( simple gas ) in the... 

Let us first attempt to establish an operational mechanical temperature scale for that is based solely on mechanical concepts, such as pressure or volume, that are assumed to be well established. (Such a scale may be of little practical utility, but it satisfies the thermodynamicist s penchant for orderly logic.) To this end, we recall from IL-1 (Table 2.1) that only two properties suffice to uniquely fix the value of (as well as all other properties) of a simple gas. We may therefore choose P and V as these independent properties, and express by the functional relationship... 

If all independent properties of S belong to one class or the otherd, then S (and all its composites) may be called macroscopic. 

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. 

Ty [see (11.27)—(11.30)]. As described previously, scalar products for S) and V) (i.e., involving properties CP, fiT, aP) are obtained by matrix inversion from those for T) and — P) (i.e., involving Cv, /3S, 1 v). The vector-algebraic procedure to be described will automatically express any desired derivative in terms of the six properties in Table 12.2, and these expressions may subsequently be reduced (if desired) to involve only three independent properties by identities previously introduced [cf. (11.39)-(11.42)], consistent with the /(/ + l)/2 rule. ... 

It is shown that the melting rates for copper, evaluated upon the assumptions of temperature-dependent and temperature-independent properties, differ by approximately 10%. This is in agreement with the computation of Hamill and Bankoff (H4) in the solidification of a semi-infinite copper melt. The temperature distribution is found by integration of Eq. (117), and bounds are established for the amount of material melted in a manner similar to that employed by Landau. As expected, the largest melting rate possible is the steady-state one. 

It is an experimental fact that every thermodynamic system possesses a delinite number n of independent properties that determine its state. Consequently, an equation of state is a relation between n properties (mutually independent chosen totherwi.se arbitrarily) as the independeni properties I c, vi. . v of ihe system and one more properly, the dependent property y Hence the equation of stale is a function of the form... 

In addition to the three quantum numbers discussed above, experimental evidence reqnires an additional quantum number ms, which by analogy to classical mechanics is attributed to an intrinsic (i.e.. position-independent) property of the electron called spin. Unlike the other quantum numbers, however, it can assume only two values ( ). As we shall see, this fact determines the orbital population of the many-electron atom. 

TEMPERATURE SCALES AND STANDARDS. That property of systems which determines whether they are in thermodynamic equilibrium. Two systems are in equilibrium when their temperatures (measured on die same temperature scale) are equal, The existence of the property defined as temperature is a consequence of the zeroth law of thermodynamics. The zerodi law of thermodynamics leads to the conclusion that in the case of all systems there exist functions of their independent properties j , such dial at equilibrium... 

During the production phase, the positions, velocities, and accelerations created at each step in time were put on magnetic tapes. These tapes were later analyzed for the time-dependent and independent properties of the system. From a statistical mechanical standpoint, the data on these tapes may be viewed in one of two ways ... 

The 600 blocks of positions and velocities represent an ensemble of 600 points in the phase space rN, of the entire system being studied or an ensemble of 6007V points in the reduced phase or p space of a single molecule. This approach was taken in calculating time-independent properties. For example, the mean square force on a molecule, , was given by... 

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