Volume displacement, density calculation from

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. 

A student wished to determine the density of an irregular piece of metal and one obtained the following data (a) mass of the metal 10.724 g (b) volume by displacement (1) graduated cylinder with water 31.35 mL, (2) graduated cylinder with water and metal 35.30 mL. Show your calculations for determining the density, and from Table 3.1, identify the metal. 

The porosity of a catalyst or support can be determined simply by measuring the particle density and solid (skeletal) density or the particle and pore volumes. Particle density pp is defined as the mass of catalyst per unit volume of particle, whereas the solid density p, as the mass per unit volume of solid catalyst. The particle volume Vp is determined by the use of a liquid that does not penetrate in the interior pores of the particle. The measurement involves the determination by picnometry of the volume of liquid displaced by the porous sample. Mercury is usually used as the liquid it does not penetrate in pores smaller than 1.2/m at atmospheric pressure. The particle weight and volume give its density pp. The solid density can usually be found from tables in handbooks only in rare cases is an experimental determination required. The same devices as for the determination of the particle density can be used to measure the pore volume V, but instead of mercury a different liquid that more readily penetrates the pores is used, such as benzene. More accurate results are obtained if helium is used as a filling medium [10]. The porosity of the particle can be calculated as ... 

A more accurate procedure is the helium-mercury method. The volume of helium displaced by a sample of catalyst is measured then the helium is removed, and the volume of mercury displaced is measured. Since mercury will not fill the pores of most catalysts at atmospheric pressure, the difference in volumes gives the pore volume of the catalyst sample. The volume of helium displaced is a measure of the volume occupied by the solid material. From this and the weight of the sample, the density of the solid phase, P5, can be obtained. Then the void fraction, or porosity, of the particle, p, may be calculated from the equation... 

The density was measured by Archimedes method. The flexure strength were tested by three point bending way using an INSTRON 5566 test machine (Canton, MA, USA) with a span of 24 mm and a crosshead speed of 0.5 mm/min. The work of fracture was obtained from the characteristic area under the stress-displacement curve divided by the cross section of the specimen. In order to determine the work of fracture effectively, we defined the characteristic area which started from initial point to the 60% drop of the curve. After the test, the silicon volume content was calculated... 

Helium gas pycnometry is commonly used for powders finer than 10 p.m. The small size of the helium molecule enables it to penetrate into very fine pores. The volume occupied by the solid is measured from the volume of gas displaced. The apparent density is then calculated from the mass of the powder used and its measured volume. 

The density of a solid is calculated from its mass and volume. When a solid is completely submerged, it displaces a volume of water that is equal to the volume of the solid. In Figure 2.12, the water level rises from 35.5 mL to 45.0 mL after the zinc object is added. Because the object displaced 9.5 mL (45.0 mL - 35.5 mL) of water, the zinc object has a volume of 9.5 mL. The density of the zinc is calculated as follows ... 

Before turning to many-electron molecules, it is useful to ask Where does the energy of the chemical bond come from In VB theory it appears to be connected with exchange of electrons between different atoms but in MO theory it is associated with delocalization of the MOs. In fact, the Hellmann-Feynman theorem (see, for example, Ch.5 of Ref.[7]) shows that the forces which hold the nuclei together in a molecule (defined in terms of the derivatives of the total electronic energy with respect to nuclear displacement) can be calculated by classical electrostatics, provided the electron distribution is represented as an electron density P(r) (number of electrons per unit volume at point r) derived from the Schrodinger wavefunction k. This density is defined (using x to stand for both space and spin variables r, s, respectively) by... 

Alternatively, the material can be determined by the water-displacement technique. The sample is weighed carefully in air with a sensitive analytical balance. The same sample is then suspended from a wire attached to the balance, then placed in water (or any liquid that will cause it to sink). The weight of the sample in the liquid, less the weight of the wire, is used to determine the mass of liquid displaced by the sample, which provides the volume of the sample. The density can then be calculate. 

An important simplification results if we can consider the bonding between atoms to be a local phenomenon. In this event, we would need to consider only the immediate neighbours of the adsorbate or defect atoms, and we arrive at the cluster models circled in Fig. 1. Of course, some properties of the system will depend on its extended nature. Others, including the variation in total energy with small displacements of atoms, should be described satisfactorily by a cluster calculation. In such cases, the problem has been reduced to one of molecular dimensions, so that the methods of molecular physics or theoretical chemistry could be used. For many systems of interest to the solid-state physicist, where a typical problem might be the chemisorption of a carbon monoxide molecule on the surface of a ferromagnetic metal surface such as nickel, the methods discussed in much of the rest of the present volume are inappropriate. It is necessary to seek alternatives, and this chapter is concerned with one of them, the density functional (DF) formalism. While the motivation of the solid-state physicist is perhaps different from that of the chemist, the above discussion shows that some of the goals are very similar. Indeed, it is my view that the density functional formalism, which owes much of its development and most of its applications to solid-state physicists, can make a useful contribution to theoretical chemistry. 

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