Mass, calculation from density

The approximate molar mass, calculated from the gas density data, is 89 g/mol. The empirical formula, calculated from the percentage composition data, is C2H3O with the empirical formula unit mass of 43.0. The exact molar mass must be (2)(43) = 86.0 g/mol since this is the only multiple of 43.0 (whole-number multiple) reasonably close to the approximate molecular formula of 89 g/mol. The molecule must be the equivalent of 2 empirical formulas CqHgO. 

Figure 9.2. Size dependence of the bulk density of icy satellites. Solid and broken lines show the compressed density curves for pure ice and an ice-rock mixture (60 % ice + 40 % rock by mass) calculated from Lupo and Lewis f6]. Bulk densities of icy satellites belonging to each planet are shown for comparison.

Organizing data Using your measurements for the volumes from item 5 and the recorded mass, calculate the density of each ball. 

The approximate crystallinity of a polymer sample can be calculated from density measurements. Based on the two-phase model of polymer behavior, the mass or volume fraction of crystallinity can be calculated by measuring the density of a polymer sample, if the densities of the amorphous material and of pure crystals are known, as indicated by the following equations ... 

From equations (1.34), (1.35) the total mass adsorbed (m ) can be calculated from density measurements and calculations of the sorptive gas (p, p ) and given data of the corresponding helium experiments (mHj,p[,J and after choosing or calculating the reference density (pj ) of the sorbate phase. 

To elucidate consequences of all four different model assumptions (P1...P4) for the volume of a porous sorbent/sorbate system, adsorption data (Q) of nitrogen N2 (5.0) on activated carbon (AC) Norit R1 Extra which have been measured gravimetrically at 298 K, and the adsorbed masses calculated from these data are shown in Figure 1.22, [1.50]. The lowest curve presents the Q-data. Note that these decrease at high gas densities as then seemingly the increase in buoyancy of the sorbent/sorbate sample is larger than the uptake of nitrogen gas, cp. Eq. (1.31). The other curves present data as follows ... 

These data present absolute amounts of adsorbed masses calculated from 2-databy using proposition (P3), i. e. optimization procedure (1.38) with the Langmuirian isotherm (1.49). The data fit is reasonably well, cp. Table 1.7. Also data increase monotonously with increasing gas density, i. e. the stability condition (1.47)... 

Molar masses, glass transition temperatures, and fractional free volume (calculated from densities and Paul and Park s series of increments) of the polymers are summarized in Tables I and II (6-8). Fractional free volume (Vf) calculated by the method of Park and Paul (P) differs for different gases, depending on the diameters of the gas molecules. Here, the values for O2 and N2 are given as examples. As expected, the values for Vf increase with decreasing kinetic diameter of the gas molecules. 

Density, p, is defined as the mass per unit volume at constant temperature and pressure, so the unit for density is kg/ rri. Density for ILs decreases with temperature in their liquid range and increases with pressure, both linearly (Gardas et al., 2007 Rilo et al., 2010c). The reciprocal value of density is the specific volume, which is usually refered as the molar volume, Vm, calculated from density and the molar mass value, M,... 

The reactor volume is calculated from Mj and the bulk density of the catalyst material, (-r ) depends not only on composition and temperature, but also on the nature and size of the catalyst pellets and the flow velocity of the mixture. In a heterogeneous reaction where a solid catalyst is used, the reactor load is often determined by the term space velocity, SV. This is defined as the volumetric flow at the inlet of the reactor divided by the reaction volume (or the total mass of catalyst), that is... 

Tlic density (p) of a substance is tlic ratio of its mass to its volume and may be expressed in units of pounds per cubic foot (Ib/ft ) or kilograms per cubic meter (kg/nv ). For solids, density can be determined easily by placing a known mass of the substance in a liquid and measuring tlie displaced volume. The density of a liquid can be measured by weighing a known volume of the liquid in a gradmitcd cylinder. For gases, tlie ideal gas law (to be discussed in Section 4.6) can be used to calculate tlie density from tlie temperature, pressure, and molecular weight of tlie gas. 

Air at sea level exerts a static pressure, due to the weight of the atmosphere, of 1013.25 mhar. The density, or specific mass, at 20°C is 1.2 kg/m . Densities at other conditions of pressure and temperature can he calculated from the Gas Laws ... 

The label on a bottle of concentrated hydrochloric acid. The label gives the mass percent of HCI in the solution (known as the assay] and the density (or specific gravity) of the solution. The molality, molarity, and mole fraction of HCI in the solution can be calculated from this information. 

STRATEGY We calculate the density of the metal by assuming first that its structure is ccp (fee) and then that it is bcc. The structure with the density closer to the experimental value is more likely to be the actual structure. The mass of a unit cell is the sum of the masses of the atoms that it contains. The mass of each atom is equal to the molar mass of the element divided by Avogadro s constant. The volume of a cubic unit cell is the cube of the length of one of its sides. That length is obtained from the radius of the metal atom, the Pythagorean theorem, and the geometry of the cell. 

Our task is to estimate the volume occupied by one atom of lithium. As usual, the mole is a convenient place to begin the calculations. Visualize a piece of lithium containing one mole of atoms. The molar mass, taken from the periodic table, tells us the number of grams of Li in one mole. The density equation can be used to convert from mass to volume. Once we have the volume of one mole of lithium, we divide by the number of atoms per mole to find the volume of a single atom. 

Because the beam mass per unit length m is calculated from m = pbt p = density), inserting Eq. 27 into Eq. 26 results in... 

An alternative approach is to use the fact that an MD calculation samples the vibrational modes of the polymer for a period of time, f, from 0 to fmax and to calculate from the trajectory, the mass weighted velocity autocorrelation function. Transforming this function from the time domain into the frequency domain by a Fourier transform provides the vibrational density of states g(v). In practice this may be carried out in the following way ... 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

Find the density p from an equation of state at Pi,TSi for the phase densities pc and pL, and the reciprocal of the homogeneous specific volume vH. Calculate the mass flux from... 

To determine the density of a liquid, the mass in grams of a measured volume of liquid in milliliters is determined. The density is calculated from these measurements. The volume may be measured by a variety of devices, such as a graduated cylinder, pipet, or buret. Very precise determinations of volume are measured with pycnometers. These devices hold a specified volume. A cap or stopper with a capillary overflow tube ensures repeatability in the measurement. See Figure 15.15. 

The key to any reaction experiment is moles. The numbers of moles may be calculated from various measurements. A sample may be weighed on a balance to give the mass, and the moles calculated with the formula weight. Or the mass of a substance may be determined using a volume measurement combined with the density. The volume of a solution may be measured with a pipet, or calculated from the final and initial readings from a buret. This volume, along with the molarity, can be used to calculate the moles present. The volume, temperature, and pressure of a gas can be measured and used to calculate the moles of a gas. You must be extremely careful on the AP exam to distinguish between those values that you measure and those that you calculate. 

The mass of a solvent or solution may be calculated from the volume and density. 

If kinetic data are to be used, it is necessary to transform the variables to conform with those of the partial equilibrium model. The units used in the model equations for and nj are moles formed/kg of solution. Thus the mass of solution in the reacting system from which the kinetic data comes must be known. Frequently, one will know the volume and have to approximate the density. A relation between and t is also needed. For this, the mass of solid originally present must be known. The amount of solid reacing, -ANg, for a time interval At can be obtained from rate curves or calculated from an integrated rate equation. The fraction of the original mass reacting in the time interval gives an approximate value of 5, e.g.,... 

Volume of the unit cell (VcM). This may be calculated from the unit cell constants by using the following formulae. From this the density (/>) of the compound may be obtained p = Mcc /FccM, where MceH is the mass of the matter in the unit cell, that is the sum of the masses of all the atoms contained. The mass of an individual atom is calculated in grams by the ratio of the atomic molar mass and the Avogadro number. 

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