Mass density, definition

The 1,000 in the denominator is the mass density of water in kg/m at the temperature of 4°C used as the reference temperature in the definition of specific gravity. All units in this equation must be in the mks (meter-kilogram-second) system. 

In many applications, the particles will be composed of multiple chemical species. In such cases, it is necessary to introduce a vector of internal coordinates p whose components are the mass of each chemical species. Obviously, the sum of these internal coordinates is equal to the particle mass. By definition, if pa, is the mass of component a, then integration over phase space leads to a component disperse-phase mass density ... 

A word of caution involves the definition of mole. As indicated above, SI moles are gram moles, the mass in grams of 6.02 x 10 molecules. There is an inconsistency in the SI system of units that may cause problems when converting molar densities and molar flow rates to mass densities and mass flow rates. A point in the system with molar concentrations (i.e., molar densities) of a and b has a mass density of p = qMa + bMs, where Ma and Mb are the molecular weights of the two components. 

The easiest way to convert between concentrations is to take a careful look at the units of both what you want and what you have, and ask what physical properties (i.e. molar mass, density) you could use to interchange them. In this example, we want to convert a molarity into a mass fraction. We have from the definitions that ... 

Fractals can be defined in two different ways. The first definition considers a fractal to be a structure that is self-similar at any scale. The second definition considers a fractal to be a stmcture with a noninteger Hausdorff dimension. Let M be the fractal mass, i.e., the number of points of the fractal. If the mass density is constant, the mass is proportional to the fractal volume. The latter is proportional to, where L is the fractal length and the Hausdorff, or fractal, dimension. Both definitions... 

Another common type of problem that uses ratios occurs when two quantities are related to each other either by some chemical or physical property. One such property is mass density. Mass density is defined as the mass of a substance per unit volume. This definition is itself a ratio that allows us to convert between mass and volume, as shown in Example Problem 1.6. 

The specific gravity, denoted d, S.G., or simply G, is a dimensionless physical quantity equal to the ratio of the mass density of the material at a given temperature (tj to the mass density of a reference fluid selected as a standard at a given temperature (t ). Since the mass density of materials varies with temperature, for a precise definition the temperature of both materials must be stated ... 

There are several particle density definitions available. Depending on the appKcation, one definition may be more suitable than the others. For nonporous particles, the definition of particle density is straightforward, i.e., the mass of the particle, M, divided by the volume of the particle, Vp, as shown in Eq. (15). 

The characteristic time defined in (9.21) establishes a time scale for surface evolution of the kind discussed in the preceding section. Its definition depends on a number of parameter values that are not measurable and, therefore, are not known with any certainty. To get some idea of its magnitude, estimate the value of the characteristic time for the particular case of a Si surface with a mismatch strain of Cin = 0.008 at a temperature of T = 600 °C. Base the estimate on the unit cell dimension of a = 0.5431 nm for the diamond cubic crystal structure, and on the following values of macroscopic material parameters an elastic modulus of E = 130 GPa, a Poisson ratio of = 0.25, a mass density oi p = 2328kg/m, and the surface energy of 70 = 2J/m. Assume that 10% of the surface atoms are involved in the mass transport process at any instant so that = 0.1. 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

Molar masses and molecular dimensions of polymers are accessible not only through scattering techniques but also from viscosity measurements. Indeed, the response of macromolecules to the application of hydrodynamic forces can give information about their volumes and their dimensions and thus indirectly about their molar masses. By definition, the viscosity of a liquid is proportional to the product of the flow time of a characteristic volume times its density ... 

This definition is in terms of a pool of liquid of depth h, where z is distance normal to the surface and ti and k are the liquid viscosity and thermal diffusivity, respectively [58]. (Thermal diffusivity is defined as the coefficient of thermal conductivity divided by density and by heat capacity per unit mass.) The critical Ma value for a system to show Marangoni instability is around 50-100. 

To determine the number of moles of solute from the definition of molality, m = (moles solute)/(kg solvent), first find the mass of solvent using its density ... 

As was pointed out earlier. Equation (1.6) allows us to find the attraction field everywhere, but it requires a volume integration, that in general is a rather cumbersome procedure. Fortunately, in many cases the calculation of the field g(p) can be greatly simplified. First, consider an elementary mass with density 6 q), located in the volume AV. Now let us start to increase the density and decrease the volume in such a way that the mass remains the same. By definition, these changes do not make a noticeable influence on the field because the observation point p is far away. In the limit, when... 

By definition, the Laplacian of U represents the divergence of the attraction field, and, correspondingly, its value characterizes the density of masses at same point. Now the following question arises. What does the Laplacian tells us about the behavior of the potential To answer this question we first consider the simplest case, when U depends on one argument, x, Fig. 1.7a. Then, we can represent the derivatives as ... 

The behavior of g as a function of R is shown in Fig. 1.12c, and, of course, it is a continuous function. Now let us mentally decrease the thickness h and increase the volume density so that the mass remains the same. In such a way we arrive at a distribution of masses with a surface density, and this replacement does not change the field outside the shell, but it leads to a discontinuity of the field at the surface masses. It is instructive to demonstrate why the field inside the shell, Relementary surfaces dS and dS2- By definition we have ... 

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