Dimensionless Relative Variables

We inspect now the thermodynamic variables and functions of the various energy forms by making the respective extensive variable dimensionless. 

We illustrate the principle first with the kinetic energy. The kinetic energy U of a mass point with mass m expressed in terms of the momentum P is U P) = P l(2m), which is well known in classical mechanics. We form the differential, thinking that the mass is constant and obtain 

Here dP/P is a dimensionless increase of the momentum, the relative increase of the momentum. Consequently, the first term P /m on the right-hand side of Eq. (2.67) is an energy. Formally P /m is the increase of the kinetic energy when the relative increase of the momentum dP/P equals 1. 

We could further substitute In E = z and further P = expz into Eq. (2.67) but we will leave it as is. However, we can state that the differential is of the order P.  

Inspecting Table 2.4 we find that the kinetic energy runs with the order p /m, the gravitational energy runs with the order mgh, the volume energy runs with the order C S)/V , and finally the thermal energy runs with the order eap S/Cv)S. 

---

P relative dimensionless potential variable = / ° yVn components of approximate of dimensionless overpotential... 

To normalize the governing equations, we introduce a dimensionless position, z = x/a, and two dimensionless dependent variables,/ =/// and u = ua/DD. Note that the normalized velocity m is equivalent to a local Peclet number, indicating the relative magnitudes of the advective and diffusive fluxes of the reactive species. Applying these definitions to the transport equations yields the dimensionless governing equations... 

As dimensionless concentration variable (f> is used throughout. In case of hard colloidal particles the quantity volume fraction. For polymers and penetrable hard spheres (j> refers to the relative concentration with respect to overlap (see (1.24))... 

For a better comparison between the behavior of different gases, it is necessary to choose a proper reference state. Gases depart from ideality when the temperature approaches the critical temperature. It is therefore more logical to compare gases at a temperature which is relative to their T. A set of three dimensionless reduced variables was introduced for this purpose this is the principle of corresponding states ... 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

Figure 7 shows the results of measurements of adsorption density by Parsonage, etal. [77] on a series of eighteen block copolymers, with poly(2-vinylpyridine) [PVP] anchors and polystyrene [PS] buoys, adsorbed from toluene (selective for PS) of variable molecular weight in each block. The results are presented as the reciprocal square of Eq. 28, that is, as a dimensionless number density of chains ct (d/Rg A)-2. For all but the copolymers of highest asymmetry, Eq. 28 is in good agreement with the data of Fig. 7. The high asymmetry copolymers are in the regime of the data of curves (a) and (c) of Fig. 3 where the large relative size...

Again, it must be noted that evaluating the rotational components of U and. S requires relatively little in the way of molecular information. All that is required is the principal moments of inertia, which derive only from the molecular structure. Thus, any methodology capable of predicting accurate geometries should be useful in the construction of rotational partition functions and the thermodynamic variables computed therefrom. Also again, the units chosen for quantities appearing in the partition function must be consistent so as to render q dimensionless. 

Stability relates to the behaviour of a system when it is subjected to a small perturbation away from a given stationary state (or if fluctuations occur naturally). If the perturbation decays to zero, the system has some in-built tendency to return back to the same state. In this case it is described as locally stable. (The qualification local means that very large perturbations may have different consequences.) We will introduce the relatively simple mathematical techniques required to determine this local stability of a given state in Chapter 3. It will also be useful before then to reduce the reaction rate equations (2.1)—(2.3) to their simplest possible form by introducing dimensionless variables and quantities. 

For solving Eq. (1.4), the first step is to change the conventional variables (concentrations, distance, times), into dimensionless ones in order to get a more general approach and to avoid problems relative to particular units. An adequate set of dimensionless variables used in this problem is... 

---