Factor of proportionality

A solution which obeys Raoult s law over the full range of compositions is called an ideal solution (see Example 7.1). Equation (8.22) describes the relationship between activity and mole fraction for ideal solutions. In the case of nonideal solutions, the nonideality may be taken into account by introducing an activity coefficient as a factor of proportionality into Eq. (8.22). 

The solute molecular weight enters the van t Hoff equation as the factor of proportionality between the number of solute particles that the osmotic pressure counts and the mass of solute which is known from the preparation of the solution. The molecular weight that is obtained from measurements on poly disperse systems is a number average quantity. 

In Sec. 2.2 we saw that the coefficient of viscosity is defined as the factor of proportionality between the shearing force per unit area = F /A and the velocity gradient dv/dy within a liquid [Eq. (2.2)] ... 

Figure 2.1 served as the basis for our initial analysis of viscosity, and we return to this representation now with the stipulation that the volume of fluid sandwiched between the two plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit of separation between the two walls. Next we consider a shearing force acting on this cube of fluid to induce a unit velocity gradient. According to Eq. (2.6), the rate of energy dissipation per unit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with t]q (pure liquid, subscript 0) the factor of proportionality ... 

V is the volume, and F is a factor of proportionality, which is calculable from the elastic properties of the solid. The connection with elasticity was in fact suspected by Sutherland in 1910 (Phil, May., 20, 657), who found that the infra-red frequency of a solid was of the same order as the frequency of an elastic transversal vibration with a wave length equal to the distance between two neighbouring atoms. To every degree of freedom Debye assigns an amount of energy ... 

We can imagine within the fluid two layers separated by dy, over which distance the velocity changes by an amount dv. Therefore dv/dy defines a velocity gradient Newton s law of viscosity states that the shear stress, r = F/A, is proportional to dv/dy. The viscosity rj of the sandwiched fluid is the factor of proportionality ... 

Use these data to evaluate the factor of proportionality in Equation (93). Does this factor seem reasonably constant ... 

Equation (9) shows that the field E and the number of lines per area are directly proportional, with Eq the factor of proportionality. In the presence of a dielectric, er is inserted into Equation (9) to bring it into conformity with Equation (7). Now suppose we apply this idea to a parallel plate capacitor. 

Moraldi gives an estimate of this integral for large times t, on the basis of a dimensional argument. The integral must diverge as t for t — oo it must also be proportional to the cross section for binary interactions which is of the order of a1, the square of the zero of the intermolecular potential functions, V (a) = 0. In other words, the factor of proportionality not specified as yet has units of speed, i.e., the root mean square speed, or... 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

The rate of isotopic exchange is proportional to the difference of contents of labeled atoms in the exchanging molecules. The factor of proportionality was proposed (145) to be named the total rate of exchange since it is equal to the rate of the continuous exchange of atoms that cannot be detected unless the atoms are labeled. The rate of isotopic exchange would be equal to the total rate of exchange if all atoms of an element in one of the species were labeled and those in the other species were normal. 

In Eq. (65) the friction factor was introduced as an empirical factor of proportionality in the calculation of the friction loss head. If Eq. (63) is applied to a length of straight horizontal pipe with no pumps, one finds that... 

Calculation of the critical lateral force is more complicated because traditional LFM does not provide us with easy method to translate current units into force ones. There is no way to define the factor of proportionality until calibration algorithm was developed recently by ourselves [8], The required coefficient depends on design of a microscope, adjustment of the optical system, torsion force constant kL of cantilever and tip height lnp. 

Single-component diffusion under equilibrium conditions can be monitored either by labeling some of the molecules or by following their trajectories. Considering the diffusion flux of the labeled molecules, again a proportionality relation of the type of eq 2 may be established. The factor of proportionality is called the coefficient of self-diffusion (or tracer diffusion). In a completely equivalent way [2], the self-diffusion coefficient may be determined on the basis of Einstein s relation... 

This is Coulomb s law. The units for charge, field strength, and force are made compatible by specifying the units of the factor of proportionality -y. For example, if y = 4-ir/e, where e is the dielectric constant of the medium, the units are in terms of cgs or absolute electrostatic system (esu). Since the dielectric constant for air is essentially 1, for aerosols using the cgs system of units, -y = 4u. 

Besides this electrostatic attraction there is the repulsion which arises when the ions begin to touch each other. This repulsion originates in the general mutual repulsion of the electron clouds of atoms and ions, which always occurs when the clouds penetrate each other and the electrons do not form any atomic bond with a common electron pair (p. 147). The repulsion is difficult to calculate and so Born represented the repulsion energy by B rn, a function which, provided n is large, increases very rapidly with decreasing distance r, corresponding to almost hard spheres. B in this expression is a still undetermined factor of proportionality while the value of the exponent can be deduced from the compressibility n amounts to about 9. 

The constant B from formula (1) consists, like the factor for the Coulomb energy, of a factor of proportionality b for the repulsive energy of an ion pair b rn. In the lattice a kind of Madelung factor K also occurs. Since the repulsive energy decreases so rapidly with increasing distance this interaction is practically restricted to the nearest neighbours in the lattice, so that K can, therefore, be put practically equal to the coordination number. 

To get a sense of the physics behind diffusion, diffusion fluxes and diffusion potentials need to be defined in terms of driving forces. Pick s first law is used to correlate the diffusion flux density with the concentration gradient of the diffiisants, which yields the diffusion coefficient as the factor of proportionality. For a system with two components and one-dimensional diffusion in the z-direction, the particle flux ji is related to the gradient of concentration, 5c,/5z, of these particles as... 

In dimensional variables, equation (100) states that the growth rate of the disturbance is proportional to the product of the transverse wave number with the geometric-mean flame speed v Vq the factor of proportionality, which depends only on the density ratio, may be written as... 

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