User:Milton Beychok/Sandbox

Measurement of density

For a homogeneous solid object, the mass divided by the volume gives the density. The mass is normally measured with an appropriate weighing scale or balance and the volume may be measured directly (from the geometry of the object) or by the displacement of a liquid.

A very simple and commonly used instrument for directly measuring the density of a liquid is the hydrometer. A less common device for measuring fluid density is a pycnometer and a similar device for measuring the absolute density of a solid is a gas pycnometer.

Another instrument used to determine the density of a liquid or a gas is the digital density meter, based on the oscillating U-tube principle.

Density of irregular shaped solid object heavier than water

One of the many statements attributed to the Archimedes principle is that an object immersed in a fluid apparently loses weight by an amount equal to the weight of the fluid displaced. That principle makes it possible to determine of the density of a solid object that is denser than water and is so irregular in shape that its volume cannot be measured directly.

First, the object's dry weight in air, $W_{d}$, is measured. Next, the weight of the object while submerged in water, $W_{s}$, is measured. Then the submerged object is removed from the water, the excess surface water is removed and the wet weight in air, $W_{w}$, is measured. The density of the object, $\rho _{o}$, is then obtained from:

(1)     $\rho _{o}={\frac {m}{V}}={\frac {W_{d}}{(W_{w}-W_{s})/\rho _{water}}}={\frac {W_{d}\cdot \rho _{water}}{W_{w}-W_{s}}}$

and the volumetric porosity fraction, $p$ can be obtained from:

(2)     $p={\frac {W_{w}-W_{d}}{W_{w}-W_{s}}}$

Density of solid objects by x-ray diffraction crytallography

For some crystalline materials, density can be calculated using this equation which is based the molecular mass of the material and on information provided by x-ray crystallography:

(3)     ${\mbox{density}}={\frac {M\cdot N}{L\cdot a\cdot b\cdot c}}$

where:

M is molecular mass

N is the number of ions, atoms or molecules in a unit cell

L is Loschmidt or Avogadro's constant

a, b, c are the lattice parameters

Figure 1: Cubic lattice structure of crystalline sodium chloride

For example, the cubic lattice structure of crystalline sodium chloride (NaCl) is depicted in Figure 1. Each sodium ion is surrounded by six chloride ions and each chloride ion is surrounded by six sodium ions. The sodium chloride molecule, as such, has disappeared. This structural arrangement is in conformity with the x-ray reflection spectraof sodium chloride.

From a study of Figure 1, it can be deduced that each corner chloride ion is common to eight adjacent cubic cells so that only one-eighth of its mass contributes to the mass of the unit cell and thus all eight corner chloride units contribute the mass of one chloride ion. Likewise, the centered chloride ion on each face of the cubic cell is common to two adjacent cubic cells and thus all six of the face-centered chloride ions contribute the mass of three chloride ions. The total mass of the chlorine in the cell is therefore equal to the mass of four chloride ions.

Similarly, it can be deduced that each of the twelve sodium ions on the external faces of the cubic cell is common to four adjacent cubic cells so that only one-fourth of its mass contributes to the mass of the unit cell and thus the twelve external face sodium ions contribute the mass of three sodium ions. The one sodium ion in the center of the cell contributes all of it mass to the mass of the unit cell. The total mass of sodium in the cell is therefore equal to the mass of four sodium ions, and total mass of the cell may be taken as the mass of four molecules of sodium chloride.

The molecular mass of sodium chloride is 58.45 g/mol and the cube's lattice parameters a, b and c (height, width and length of the cell) are each 0.563 nm. Using Avogadro's constant of 6.022×10²³ and equation (3) above:

density of NaCl = ( 58.45 × 4 ) ÷ ( 0.563³ × 6.022×10²³ ) = 217.9 × 10²³ g/nm³ = 2.176 g/cm³

That is within less than 0.5 percent of the 2.165 g/cm³ given as the density of NaCl in the Handbook of Chemistry and Physics.

Density of gases

From the ideal gas law, the density of an ideal gas may be expressed as:

$\rho ={\frac {MP}{RT}}$

and the density of a real gas (as differentiated from an ideal gas) may be expressed as:

(4)     $\rho ={\frac {MP}{ZRT}}$

where $Z$ is the compressibility factor, $R$ is the molar gas constant, $P$ is the pressure, $M$ is the molecular mass, and $T$ is the absolute temperature.

Thus, at an atmospheric pressure of 101.325 kPa, an absolute temperature of 273.15 K ( 0 °C ) and using $R$ as 8.314472 m³·Pa·K^-1·mol^-1, the density of a real gas in kg/m³ is given by:

(5)     $\rho ={\frac {\mathrm {0.0446} \,M}{Z}}$

At atmospheric pressure and 0 °C, most gases may be considered to be ideal and therefore the compressibility factor, $Z$, can be taken to be 1. At other conditions of temperature and pressure, where a gas may deviate from ideal gas behavior, the compressibility factor may be calculated using the van der Waals equation (see Compressibility factor (gases)#The van der Waals equation).