Volume natural variable equations

Starting with the natural variable equation for dU, derive an expression for the isothermal volume dependence of the internal energy, (dUldV)j, in terms of measurable properties (T, V, or p) and a and/or k. Hint You will have to invoke the cyclic rule of partial derivatives (see Chapter 1). 

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 ,. ... 

Gibbs considered the statistical mechanics of a system containing one type of molecule in contact with a large reservoir of the same type of molecules through a permeable membrane. If the system has a specified volume and temperature and is in equilibrium with the resevoir, the chemical potential of the species in the system is determined by the chemical potential of the species in the reservoir. The natural variables of this system are T, V, and //. We saw in equation 2.6-12 that the thermodynamic potential with these natural variables is U[T, //] using Callen s nomenclature. The integration of the fundamental equation for yields... 

Equations of state relate the pressure, temperature, volume, and composition of a system to each other. In this Chapter, we show how to determine other thermodynamic properties of the system from an equation of state. In a typical equation of state, the pressure is given as an explicit function of temperature, volume, and composition. Therefore, the natural variables are the temperature, volume, and composition of the system. That is, once given the volume, temperature, and composition of the system, the pressure is readily calculated from the equation of state. 

The free energy- that has temperature, volume, and mole numbers as its natural variables is the Helmholtz free energy. Before we stated that once the Gibb s free energy of a system is known as a function of temperature, pressure, and mole numbers G(T,p, N, N2,..all the thermodynamics of the system are known. This is equivalent to the statement that once the Helmholtz free energy is known as a function of temperature, volume, and mole numbers of the system A(T, V, Ni,N2, -all the thermodynamics of the system are known. The fundamental equation of thermodynamics can be written in terms of the Helmholtz free energy as... 

The experimental measurements produced concentration-time plots of ethylene oxide and ethylene glycol in the liquid phase, as shown in Figure 8.18. The physical picture of this reaction/reactor system is most closely approximated by the plug-flow gas phase, well-mixed batch liquid phase. The appropriate relationships to model this system are given in equations (8-176) to (8-178), (8-183), and (8-188). The bubble volume is variable, and the nature of the variation changes with the extent of conversion (i.e., concentration of glycol in the liquid phase), however, the pure oxide gas phase allows yg = l. The modified equations specific to this reactor are then... 

Often, the energy in terms of temperature, volume, and mol number U T,V, n) is addressed as the caloric state equation, whereas the volume in terms of temperature, pressure, and mol number V T, p,n) is addressed as the thermal state equation. On the other hand, the energy exclusively expressed in terms of the corresponding natural variables U S, V, n) belongs to the type of a fundamental equation or fundamental form. 

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... 

In variable volume systems the dV/dt term is significant. Although equation 3.0.9 is a valid one arrived at by legitimate mathematical operations, its use in the analysis of rate data is extremely limited because of the awkward nature of the equations to which it leads. Equation 3.0.1 is preferred. 

Naturally, the fixed composition phase transformations treated in this section can be accompanied by local fluctuations in the composition field. Because of the similarity of Fig. 17.3 to a binary eutectic phase diagram, it is apparent that composition plays a similar role to other order parameters, such as molar volume. Before treating the composition order parameter explicitly for a binary alloy, a preliminary distinction between types of order parameters can be obtained. Order parameters such as composition and molar volume are derived from extensive variables any kinetic equations that apply for them must account for any conservation principles that apply to the extensive variable. Order parameters such as the atomic displacement 77 in a piezoelectric transition, or spin in a magnetic transition, are not subject to any conservation principles. Fundamental differences between conserved and nonconserved order parameters are treated in Sections 17.2 and 18.3. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

An adiabatic reactor with a regular flow pattern is much simpler to work with than one with exchange of heat. Except for resistance to exchange of material and heat between fluid and catalyst, the space velocity is the only significant space variable. The natural independent variable to use is space velocity, or the volume traversed divided by the total molar flow. The relation between the concentration and the reciprocal space velocity is given by the equation... 

Equation (12) has the interesting, but often overlooked, consequence that a series of evolving PSDs if expressed in volume may broaden with time yet the same series if expressed in terms of radius may become more monodis-perse. This can be seen from Eq. (12) to arise because each element of nlF) must be multiplied by to yield the distribution n(r) consequently, elements with larger values of V (and r) are weighted more and so become more peaked than elements with smaller F values. It is therefore mandatory to specify the size variable in which a PSD is expressed before commenting on the nature of the evolving PSD. 

As a first approach, it may be natural to employ a collocated grid arrangement and store all the variables at the same set of grid points and to use the same grid volumes for all of them. In this case the number of coefficients that must be computed and stored is minimized, because many of the terms in each of the equations are essentially identical. However, it is not required that... 

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. 

The interrelations of the form of Eqs. 1.6-1 and 1.6-2 are always obeyed in nature, though we may not have been sufficiently accurate in our experiments, or clever enough in other ways to have discovered them. In particular, Eq. 1.6-1 in dicates that if we prepare a fluid such that it has specified values T and V, it wilt always have the same pre.ssure P. What is this alue of the pressure PI To know this we have to either have done the experiment sometime in the past or know the exact functional relationship between T, V, and P for the fluid being considered. What is frequently done for fiuids of scientific or engineering interest is to make a large number of measurement. of P, V, and T and then to develop a volumetric equation of state for the fluid, that is, a mathematical relationship between the variables P, V, and T. Similarly, measurements of U, V, and T are made to develop a thermal equation of state for the fluid. Alternatively, the data that have be eh obtained may be presented directly in graphical or tabular form. (In fact, as will be shown later in this book, it is more convenient to formulate volumetric equations of state in terms of P, V, and T than in terms of P, V, and T, since in this case the same gas constant of Eq. 1.4-3 can be used for all substances. If volume on a per-mass basis V was u.sed, the constant in the ideal gas equation of state would be R divided by the molecular weight of the substance.)... 

A natural starting point is the fundamental equation in a canonical ensemble of c components, for which the independent variables are the temperature , the volume V, and the number of molecules of each species, Nj,i = 1,..., c the potential is the Helmholtz free energy A... 

Sampling Method. The physical properties of natural gas in a sampling cylinder of constant volume are exactly described by the thermal equation of state. The mass in the cylinder can be calculated, if the pressure, the temperature and the composition of gas are known. Since the pressure and the temperature can be measured easily, the only unknown variable is the composition of gas for the calculation of the gas constant R and the compressibihty factor Z. The composition, however, varies only in a small range, so that an averaged value can be used for calculation. This favors the evaluation of the thermal equation of state as a very effective and precise tool in gas sampler technique. 

The mathematical statement of the inverse problem is as follows Given measurements of F x, t), the cumulative volume (or mass) fraction of particles of volume ( x) at various times, determines, b x), the breakage frequency of particles of volume x, and G x x ), the cumulative volume fraction of fragments with volume ( x) from the breakage of a parent particle of volume x. Obviously, the experimental data on F x, t) would be discrete in nature. We assume that G x x ) is of the form (5.2.9) and rely on the development in Section 5.2.1.1 using the similarity variable z = b x)t. Self-similarity is expressed by the equation F x, t) = 0(z), which, when substituted into (6.1.1), yields the equation... 

An important aspect in this definition is the choice of the independent variables. Many analytical equations of state are expressions explicit in pressure that is, temperature, molar volume (or density) and composition x = xi, X2, , x, are the natural independent variables. Therefore, eq 2.32 can be rewritten into ... 

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