Entropy natural variable equations

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

This fundamental equation for the entropy shows that S has the natural variables U, V, and n . The corresponding criterion of equilibrium is (dS) 0 at constant U, V, and n . Thus the entropy increases when a spontaneous change occurs at constant U, V, and ,. At equilibrium the entropy is at a maximum. When U, V, and , are constant, we can refer to the system as isolated. Equation 2.2-13 shows that partial derivatives of S yield 1/T, P/T, and pJT, which is the same information that is provided by partial derivatives of U, and so nothing is gained by using equation 2.2-13 rather than 2.2-8. Since equation 2.2-13 does not provide any new information, we will not discuss it further. 

Step 2. Use the total differential of specific enthalpy in terms of its natural variables, via Legendre transformation of the internal energy from classical thermodynamics, to re-express the pressure gradient in the momentum balance in terms of enthalpy, entropy, and mass fractions. Then, write the equation of change for kinetic energy in terms of specific enthalpy and entropy. 

Again, this is not a useful spontaneity condition unless we can keep the system isen-tropic. Because p and S must be constant in order for the enthalpy change to act as a spontaneity condition, p and S are the natural variables for enthalpy. Equation 4.4 does suggest why many spontaneous changes are exothermic, however. Many processes occur against a constant pressure that of the atmosphere. Constant pressure is half of the requirement for enthalpy changes to dictate spontaneity. However, it is not sufficient, because for many processes the entropy change is not zero. 

Note that the fundamental equations (f) express a change of internal energy dll when the entropy dS and the volume dV are changed. It is said that S, V) are the natural variables of the internal energy U = U S,V), because dU has a particularly simple relation to dS and dV. Correspondingly, S,p) axe the natural variables for enthalpy H = and (T,p) are the natural variables for the... 

Relative activation enthalpies (Aif) in Table 2 were converted to o% kx k ) at 298 K, and were plotted against Hammett a constants. Here, we used enthalpies, because the size of the entropy and hence the free energy depend much on low frequencies, which are less reliable than higher frequencies, especially for compounds with weak interactions such as TS (8). The use of free energy (AG ) gave similar correlations with more scattered points. As for the Hammett o constant, we used dual-parameter o constants in the form of the Yukawa-Tsuno equation (LArSR equation) (9) as defined in eq 3. Here, the apparent a constant (aapp) has a variable resonance contribution parameter (r), which varies depending on the nature of the reaction examined for t-cumyl... 

The thermodynamic functions have been defined in terms of the energy and the entropy. These, in turn, have been defined in terms of differential quantities. The absolute values of these functions for systems in given states are not known.1 However, differences in the values of the thermodynamic functions between two states of a system can be determined. We therefore may choose a certain state of a system as a standard state and consider the differences of the thermodynamic functions between any state of a system and the chosen standard state of the system. The choice of the standard state is arbitrary, and any state, physically realizable or not, may be chosen. The nature of the thermodynamic problem, experience, and convention dictate the choice. For gases the choice of standard state, defined in Chapter 7, is simple because equations of state are available and because, for mixtures, gases are generally miscible with each other. The question is more difficult for liquids and solids because, in addition to the lack of a common equation of state, limited ranges of solubility exist in many systems. The independent variables to which values must be assigned to fix the values of all of the... 

The partial derivative in Equation 9.5 is taken at constant temperature, pressure, and number of moles per unit volume n, n, and n, of continuous phase, dispersed phase, and surfactant in the microemulsion. However, the number of droplets N may vary. This equation applies even when no excess phase is present and would be used to find the equilibrium droplet radius in that case. In Equation 9.6, the partial derivative is taken at constant N but with variable n. It is applicable only when excess disperse phase is present. Using both these equations, the above authors found that with a given amount of surfactant present, more droplets of smaller size are predicted than would be expected based on the pertinent natural radius. That is, solubilization with an excess of the dispersed phase is somewhat less than if droplets assumed their natural radius. In this case the energy required to bend the film beyond its natural radius is offset by the decrease in free energy associated with the increased entropy of the more numerous droplets. 

Equations 35 and 36 are called thermokinematic functions of state. (Note that the variable s was introduced along with Eq. 23 in order to facilitate elimination of and jr from Eqs. 19 and 20 respectively. A more natural way to eliminate these variables would be to simply multiply Eq. 19 by a, then subtract from Eq. 20. In the latter procedure entropy s would never be defined, rather a function for "internal availability" b(e,v) would arise. The choice of introducing s was made in order that the traditional results would be obtained.)... 

To complete our thermodynamic description of pure component systems, it is therefore necessary that we (1) develop an additional balance equation for a state variable and (2) incorporate into our description the unidirectional character of natural processes. In Sec. 4.1 we show that both these objectives can be accomplished by introducing a single new thermodynamic function, the entropy. The remaining sections of this chapter are concerned with illustrating the properties and utility of this new variable and its balance equation. 

Equation 4 is the equation of state including the reaction coordinate, T", as an independent variable the material is thus assumed to be in mechanical and thermal equilibrium, but not necessarily in phase equilibrium. Equation 5 expresses the stress as the sum of the equilibrium pressure and the viscous stress, which is assumed proportional to the velocity gradient [Eq. (6)]. The rate equation [Eq. (7)] assumes that the rate of change of the reaction coordinate is proportional to the chemical affinity. A, defined as the difference in the electrochemical potentials of the two phases at the same pressure and temperature. The exact forms of the rate equations, Eqs. (6) and (7), are not essential to the argument concerning the nature of the critical points provided only that they are valid in the limit as the shock end states are approached. They may thus be considered as first order expansions about equilibrium of more general nonlinear rate equations. Whatever rate equations are assumed they must reduce to the forms shown in order to conform to the thermodynamic requirement that entropy production be second order as equilibrium is approached. 

In fluid mechanics it might be natural to employ mass based thermodynamic properties whereas the classical thermodynamics convention is to use mole based variables. It follows that the extensive thermodynamic functions (e.g., internal energy, Gibbs free energy, Helmholtz energy, enthalpy, entropy, and specific volume) can be expressed in both ways, either in terms of mass or mole. The two forms of the Gibbs-Duhem equation are ... 

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