The Fundamental Equation

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 

The extensive properties of U and X are the canonical variables. The fundamental equation in terms of internal energy U 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. 

Differential forms of the fundamental equations contain the intensive thermodynamic properties. For example, dS and dU are 

the first-order partial derivatives are the intensive properties T, /, and Y. In terms of the intensive properties, Eqs. (1.51) and (1.52) become 

The thermodynamic properties of a system in a gravitational field are emphasized in the following sections. However, because of the similarity of the two fields, the concepts and equations for the centrifugal field are the same. 

The work done on a system of mass m in raising the system from r, to r2 in the gravitational field is 

The differential quantity of work done in moving dn moles of a component of the system from one homogenous region to another is given by 

This equation is the fundamental equation for the energy function, and one such equation is applicable to each homogenous region. 

The fundamental equations for the enthalpy and the Gibbs and Helmholtz energies are obtained from Equation (14.6) by the same methods used previously. Then we have, for each homogenous region, 

---

Schrodinger wave equation The fundamental equation of wave mechanics which relates energy to field. The equation which gives the most probable positions of any particle, when it is behaving in a wave form, in terms of the field. 

Equation II-7 is the fundamental equation of capillarity and will recur many times in this chapter. 

One of the fundamental equations of thermo dynamics concerns systems at equilibrium and relates the equilibrium constant K to the dif ference in standard free energy (A6°) between the products and the reactants... 

The method thus outlined allows the development of a conceptual understanding of the limits of operation of a humidification column. For actual design, the simplifications used herein may be avoided by handling the fundamental equations numerically by computer. 

Simulation of aerosol processes within an air quaUty model begins with the fundamental equation of aerosol dynamics which describes aerosol transport (term 2), growth (term 3), coagulation (terms 4 and 5), and sedimentation (term 6) ... 

Thermodynamic Analyses of Cycles The thermodynamic quahty measure of either a piece of equipment or an entire process is its reversibility. The second law, or more precisely the entropy increase, is an effective guide to this degree of irreversibility. However, to obtain a clearer picture of what these entropy increases mean, it has become convenient to relate such an analysis to the additional work that is required to overcome these irreversibihties. The fundamental equation for such an analysis is... 

Fundamental Equations A complete development of the fundamental equations is presented elsewhere (Growl aud Louvar, 1990, pp. 129-144). The model begins by writing an equation for the conservation of mass of the dispersing material ... 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

The fundamental equations of quantum chemistry are usually expressed in units designed to simplify their form by eliminating fundamental constants. The atomic unit of length is the Bohr radius ... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Richards, et. al. comment that while the exact relationship between the rule found by their genetic algorithm and the fundamental equations of motion for the solidification remains unknown, it may still be possible to connect certain features of the learned rule to phenomenological models. 

This is the fundamental equation of colorimetry and spectrophotometry, and is often spoken of as the Beer-Lambert Law. The value of a will clearly depend upon the method of expression of the concentration. If c is expressed in mole h 1 and / in centimetres then a is given the symbol and is called the molar absorption coefficient or molar absorptivity (formerly the molar extinction coefficient). 

The new quantum mechanics contradicts this independent electron model as it is often called. In Heisenberg s formulation of quantum mechanics the fundamental equation is,... 

Equations (7), (9), and (10) are different forms of the Gihbs-Helmholtz equation, which is the fundamental equation of electrochemistry. 

We attempt here to describe the fundamental equations of fluid mechanics and heat transfer. The main emphasis, however, is on understanding the physical principles and on application of the theory to realistic problems. The state of the art in high-heat flux management schemes, pressure and temperature measurement, pressure drop and heat transfer in single-phase and two-phase micro-channels, design and fabrication of micro-channel heat sinks are discussed. 

The fundamental equation of sedimentation equihbrium can be manipulated to define a new function with dimensions of molar mass (g/mol) called M r). M (r) at a radial position r is defined by... 

The subscript 3 is used to designate the pendant double bonds. Therefore, the fundamental equation is given by ... 

The EMA method is similar to the volume-averaging technique in the sense that an effective transport coefficient is determined. However, it is less empirical and more general, an assessment that will become clear in a moment. Taking mass diffusion as an example, the fundamental equation to solve is... 

The A2 parameter ean be ealeulated from chromatographic data by transfomfing the fundamental equation (Equation 4.22) and substituting values obtained from ehromatograms for eoneentration 0.3, 0.5, and 0.7 of one of the components, or by transformation of Equation 4.22 to the linear form assuming that... 

These are the fundamental equations for the design of thick cylinders and are often referred to as Lame s equations, as they were first derived by Lame and Clapeyron (1833). The constants A and B are determined from the boundary conditions for the particular loading condition. 

A further substitution from these equations into Eq. (5.2.13) yields the fundamental equation of electrochemical kinetics ... 

The essential information on the structure of data can be extracted by means of the fundamental equation of data analysis... 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

---