Proportionality and Density

16 Write a mathematical expression indicating that one quantity is directly proportional to another quantity. 

18 Given the values of two quantities that are directly proportional to each other, calculate the proportionality constant, including its units. 

19 Write the defining equation for a proportionality constant and identify units in which it might be expressed. 

20 Given two of the following for a sample of a pure substance, calculate the third mass, volume, and density. 

Consider the following experiment. The mass of a clean, dry container for measuring liquid volume was measured and found to be 26.42 g. Then 5.2 mL of cooking oil was placed into the container. The mass of the oil and the container was 31.06 g. The 

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Chapter 4 provides an opportunity for you to apply the chemical calculation skills you learned as a result of studying Chapter 3. Scientific notation, dimensional analysis, metric units, significant figures, temperature, proportionality, and density are needed to understand the concepts and work the problems in this introduction to gases. You may find that you occasionally need to review Chapter 3 as you study this chapter. If so, don t be concerned. All successful science students review and refine their understanding of prior material-even content from prior coursework—as they learn new ideas. In fact, we selected the topics of Chapter 4, in part to give you a chance to apply your calculating skills immediately after you learned them. 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

For compressible fluids at isothermal conditions, more accurate ccxrputations may be obtained by assuming proportionality between pressure and density, whereby eqn (3) is slightly modified ... 

D3 and D4 receptors are distributed in proportionately higher densities in limbic areas. 

There is an essential difference between the decomposition rates expressed by the quantities J and k. Unlike J, which does not depend on the particle size, k is inversely proportional to the initial dimensions of the particle. For pro = 1 (e.g., for p = 2,000 kg m and rg = 0.5 mm = 5 x 10 m), the rates J and k are numerically equal. The difference between these rates increases proportionately with increasing size and density of the particles. Equation 3.32 permits conversion from relative values of the rate constants k expressed in per second to the absolute rates J in units of kg m s. This opens up an attractive possibility for the interpretation of data obtained by traditional measurement of the a—t kinetic curves in terms of the Langmuir vaporization equations. 

The main difference between the kinetics in this regime and in the one considered above is the noticeable competition between reactions (a) and (J), as was illustrated in the case of the t] = 0-33 exponential growth data in Fig. 2.8. By equations (2.11) and (2.12), this results in being keenly dependent upon the product of [O2] and the difference Ika — A /[M]). As a function of gas pressure at a given temperature, t( departs radically from inverse proportionality to density and in fact exhibits a minimmn, approximately at k/[M] = ka- Correspondingly the temperature dependence of is different for each density of reactants and diluents, and its correspondence with equation (2.15) and the activation energies of elementary chain reaction steps vanishes. Data demonstrating this behaviour are exhibited in Fig. 2.10, as individual curves of log t vs 1 jT for each postshock pressure. 

At even quite moderate pressures of a few atmospheres the static permittivites of gases show significant deviations from simple linear proportionality to density expected for ideal gases without significant effects of molecular interactions. As with the equation of state for pressure as a function of density these deviations can be described by a virial series in powers of density with second and higher order dielectric virial coefficients. To introduce these in convenient form for theoretical analysis a macroscopic spherical sample in vacuum is assumed with a uniform field (before insertion of the sample) from external chargee. As the macroscopic in the sample is then given by =... 

In this equation, P/RT has the role of a proportionality constant. Hence, molar mass and density are directly proportional to each other at a given temperature and pressure. 56. Butane, C4Hjo, has a higher molar mass, 58.12 g/mol, than propane, CjHj, 44.09 g/mol. At a given temperature and pressure, density... 

Dense fluid transport property data are successfully correlated by a scheme which is based on a consideration of smooth hard-sphere transport theory. For monatomic fluids, only one adjustable parameter, the close-packed volume, is required for a simultaneous fit of isothermal self-diffusion, viscosity and thermal conductivity data. This parameter decreases in value smoothly as the temperature is raised, as expected for real fluids. Diffusion and viscosity data for methane, a typical pseudo-spherical molecular fluid, are satisfactorily reproduced with one additional temperamre-independent parameter, the translational-rotational coupling factor, for each property. On the assumption that transport properties for dense nonspherical molecular fluids are also directly proportional to smooth hard-sphere values, self-diffusion, viscosity and thermal conductivity data for unbranched alkanes, aromatic hydrocarbons, alkan-l-ols, certain refrigerants and other simple fluids are very satisfactorily fitted. From the temperature and carbon number dependency of the characteristic volume and the carbon number dependency of the proportionality (roughness) factors, transport properties can be accurately predicted for other members of these homologous series, and for other conditions of temperature and density. Furthermore, by incorporating the modified Tait equation for density into... 

The analysis leading to Equation 6.12 shows a dependence of the flux on the mean speed. For Equations 6.13 and 6.14, this dependence is rolled into the constant of proportionality, and so thermal conductivity, viscosity, and diffusion coefficients have a dependence on temperature. These coefficients and their temperature dependence may be important in reaction probabilities for systems in which gradients exist for the densities of the reactants. 

As shown in Table 13.4, the pH value and density of bio-oil in this work are comparable with other literatures. In addition, the bio-oil produced has moderate value of viscosity which favored the range of bio-oil application. Based on the ultimate analysis, the oxygen content measured in this work is almost the same in other findings which resulted in proportionate HHV. Hence, the slow pyrolysis bio-oil produced from this work has a high potential to substitute conventional fossil fuel produced from fast pyrolysis process. 

The estimation of the molar volumes and densities of RTILs can be considerably simplified, requiring only two parameters valid for all the RTILs considered in this book [288]. A proportionality was found, shown for 162 RTILs with a large variety of anions in Fig. 6.5, between N (v+ + v ) and the experimental V(298 K), with a squared correlation coefficient of fcorr = 0.9895. Therefore, a good estimation of V (298 K) is ... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

A general observation, verified for nearly all soHds, is a proportionality between current density and field strength, known as Ohm s law. The electrical conductivity is this proportionality constant and is defined as... 

The precise numerical values of the calculated electron densities are unimportant, as the most important feature is the relative electron density thus, the electron density at the pyrazine carbon atom is similar to that at an a-position in pyridine and this is manifest in the comparable reactivities of these positions in the two rings. In the case of quinoxaline, electron densities at N-1 and C-2 are proportionately lower, with the highest electron density appearing at position 5(8), which is in line with the observation that electrophilic substitution occurs at this position. 

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