Dimensionless units

The use of dimensionless units simplifies the construction of programs for simulation. All the variables we have discussed can be made dimensionless by normalizing them to a standard value. For instance (denoting the dimensionless unit with a double dagger), we can write  

The heterogeneous rate constant is converted to dimensionless form by multiplying through by the appropriate simulation units. 

The equation for diffusion from the first spatial cell  

If all calculations are made in dimensionless units, how would one get back to real units With dimensional analysis, it can be seen that  

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The difference between using this approximation and the true value is only minor and is often ignored. Note that both a and 5 are dimensionless units. 

Dimensionless Quantities. Certain quantities, eg, refractive index and relative density (formerly specific gravity), are expressed by pure numbers. In these cases, the corresponding SI unit is the ratio of the same two SI units, which cancel each other, leaving a dimensionless unit. The SI unit of dimensionless quantities may be expressed as 1. Units for dimensionless quantities such as percent and parts per million (ppm) may also be used with SI in the latter case, it is important to indicate whether the parts per million are by volume or by mass. 

In this section we shall expand upon the problem of one-dimensional motion in a potential V x). Although it is a textbook example, we use here the less traditional Feynman path-integral formalism, the advantage of which is a possibility of straightforward extension to many dimensions. In the following portion of the theory we shall use dimensionless units, in which h = i,k = 1 and the particle has unit mass. 

Hh and Hg are the height of heat and mass transfer units, respectively, in multiples of the cell height, 1. H and M are the same in dimensionless units. 

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

On the basis of the values of AS° derived in this way it appears that the chelate effect is usually due to more favourable entropy changes associated with ring formation. However, the objection can be made that and /3l-l as just defined have different dimensions and so are not directly comparable. It has been suggested that to surmount this objection concentrations should be expressed in the dimensionless unit mole fraction instead of the more usual mol dm. Since the concentration of pure water at 25°C is approximately 55.5 moldm , the value of concentration expressed in mole fractions = cone in moldm /55.5 Thus, while is thereby increased by the factor (55.5), /3l-l is increased by the factor (55.5) so that the derived values of AG° and AS° will be quite different. The effect of this change in units is shown in Table 19.1 for the Cd complexes of L = methylamine and L-L = ethylenediamine. It appears that the entropy advantage of the chelate, and with it the chelate effect itself, virtually disappears when mole fractions replace moldm . ... 

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

Figure 3. First control trial. The temperature and reactant flow rate profile are shown in dimensionless units for the first pilot plant control trial. The PID algorithm and batch start-up control strategy were modified as a result of this trial.

Figure 4. Results of the final control trial. Three pilot plant runs were made to fine tune the PID algorithm and control strategy. This illustrates the excellent temperature control achieved in the final trial, using the same dimensionless units used in Figure 3.

The measured growth rates are illustrated by the circles in Fig. 7. The interface velocity is plotted versus the interface temperature T. The value of T is always greater than Tq because of the release of the latent heat at the interface. Dimensionless units for T and the velocity are used here. The maximum velocity corresponds to 80m /s for argon. The most surprising aspect is the rapid crystallization at low temperatures. Most materials exhibit sharply reduced rates at low temperatures, as expected for an activated growth process. That is, the kinetics can be represented as the product of an Arrhenius factor F(T) and a term that accounts for the net production of crystalline material as a result of the atoms ordering and disordering at the interface,... 

For a fixed molar ratio (ns/riAh equal to 0.05887, the temperature as applied in experiment E4, and a batch time of 347.8 dimensionless units, the feed rate of B (and thus the feed time) was optimized by computation to find tj = 323.19 dimensionless units. A run was carried out at these conditions. The data collected from this experiment were then used for re-estimation of the kinetic parameters. The new kinetic model was used to evaluate the new optimum feed rate for the same total amount of B. The optimum batch time reduced to 275.36 and the feed time to 242.75 units. Table 5.4-19 summarizes the results for three successive optimizations and re-estimations. Evidently, even a very simplified kinetic model can be successfully used in search for an optimum provided that kinetic parameters are updated based on every subsequent run carried out at the optimum conditions evaluated from the preceding set of kinetic parameters. 

Under these conditions, corresponding to so-called cr-control [37], the elastic capacitor is described by the Hamiltonian, Eq. (7). In dimensionless units this becomes... 

Dimensionless units are also used that are valid only for systems that operate at standard pressure (/ llm = 1). The actual units are as follows ... 

Typical values of H for gasoline components range between 20 and 500 atm (0.03 to 0.30 in dimensionless units at the standard condition [Patm = 1]). 

This arbitrary value is the measured height on the trial curve of the isocyanate absorbance corresponding to the time when the experimental data reaches its maximum. In this way a curve is obtained which has a maximum value of 1.0 in dimensionless units. Since the units of the measurement cancel, the measurement can be done using any convenient units. 

Table I. Input parameters of the GL model for D = 3 (All parameters are given in dimensionless units. See Eq.(2).

Figure 19. The root-mean-square (RMS) position of each segment of both acyl chains of SOPC lipids is plotted as a function of the average position of the segment. The sn tail is given by the closed symbols, and the sn2 tail is given by the open symbols. Various numbers of the tail segments and of the backbone segments are indicated. The lines are drawn to guide the eye. The arrow points to the position of the unsaturated bond, (a) SCF results (conversion from dimensionless units to real units is approximately a factor of 0.2 nm), (b) MD results (the average over the sides of the bilayer is taken)...

The lateral compressibility, i.e. the relative area change upon an imposed membrane tension, decreases slightly more than linearly with the chain length. This means that it is more difficult to expand the membrane surface area of a long-chained lipid than a shorter one. In Figure 20 dimensionless units are used, which means that the surface tension is given in units kT/as. 

Within the applicability of the semiclassical approximation, the propagator (108) is rather insensitive to the particular value of the width parameters jj, but this parameter can of course affect the numerical efficiency of the calculation. In the numerical studies presented below, we have chosen the width y as the width of the harmonic ground state of the jth vibrational mode. In the dimensionless units used here, this choice corresponds to y = 1 for all degrees of freedom. 

A iH NMR spectrum is a graph of resonance frequency (chemical shift) vs. the intensity of Rf absorption by the sample. The spectrum is usually calibrated in dimensionless units called "parts per million" (abbreviated to ppm) although the horizontal scale is a frequency scale, the units are converted to ppm so that the scale has the same numbers irrespective of the strength of the magnetic field in which the measurement was made. The scale in ppm, termed the 6 scale, is usually referenced to the resonance of some standard substance whose frequency is chosen as... 

You need the Henry s law constants for benzene and DDT. Determine these values in the units of atmm /g, atmm /mole, and dimensionless units. 

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