Fundamental equations energy, differential form

The differential of the integrated form (equation 2.2-14) of the fundamental equation for the internal energy is... 

The fundamental differential form for Gibbs energy can be integrated to express Gibbs energy as a sum of contributions by the components. From Equation (4.99),... 

This is the fundamental equation for the Gibbs free energy of mixture in terms of temperature, pressure, and number of moles of all components. An alternative form of this equation, which will be useful in subsequent developments, is obtained by developing an expression for the differential of the ratio G /RT, which is dimensionless. We start by calculating the differential of this ratio ... 

We now can develop a similar thermodynamic treatment for the surface phase a, which is the interface between the phases a and (, and is in equilibrium with these two phases. When the adsorbed phase is treated as a two dimensional surface, fundamental equations in classical thermodynamics can still be applied. Applying the same procedure to surface free energy, we will obtain the Gibbs adsorption equation. This is done as follows. The total differentiation of the surface free energy takes the form similar to eq. (2.3-1) with P dV being replaced by 7idA ... 

Substituting the definitions given in Equation (7.3) into Equation (7.2) gives the differential form of the fundamental energy equation ... 

It is obvious that the Hartree-Fock equations are rather complicated integro-differential equations of a non-linear nature with bifurcations etc., and it was hence of fundamental importance when Ueb and Simon [21] in 1977 could show the mathematical existence of solutions to these equations. There are still some mathematical problems associated with the Hartree-Fock scheme, particularly the connection between the starting point of the c culatlons and the final result, which is usually associated with a "local minimum" of the energy . We note further that the concept of "self-consistency is related to some form of "numerical convergence" in a specified number... 

This differential equation is the fundamental population balance. This equation together with mass and energy balances for a system form a dynstmic multidimensional accounting of a process where there is a change in the particle size distribution. This equation is completely general and is used when the particles are distributed along both external and internal coordinate space. External coordinate space is simply the position x, y, and z in Cartesian coordinates. Internal coordinates Xj are, for example, the shape, chemical composition, and the size of the particles. More convenient and more restrictive forms of the population balance will be subsequently developed. 

This variety of reactor types reflects the complex interaction between chemical reaction and mass and heat transfer. In spite of this complexity, in principle every reactor can be described by the fundamental balance equations for the preservation of mass, energy, and impetus. These form a system of five partial differential equations (PDFs), coupled through temperature, concentation, and three rate vectors [Damkohler 1936, Platzer 1996, Adler 2000]. 

When the ideal gas law is written in terms of energy, a partial differential equation is obtained. The situation becomes more simple, if we want to get the free energy as fundamental form. Once we have an expression for the free energy, we can use the Legendre transformation to get the ordinary energy. We illustrate now this type of... 

---