ORDINARY MATTER #4

ORDINARY MATTER #4 | Ecency

Greetings to everyone. Sequel to my last post on ORDINARY MATTER #3, this new post will explain how to determine the structures of crystals, structures of other solids, and finally on the ideal gas equations.

DETERMINING THE STRUCTURES OF CRYSTALS

The structure of crystals can be found out by probing them with radiation. The spacing of the layers of atoms is about 0.5 nm, so we need radiation with a wavelength of similar size. X-rays are electromagnetic waves with a wavelength of about 1 nm. When the X-rays hit the crystal, they are diffracted by the planes of atoms, just as light is diffracted by a narrow slit or by a diffraction grating.

Crystal

^{Image by Devanath from Pixabay}

Max von Laue in 1912 was the first to observe this effect. Photographic film is used to detect the diffraction pattern, seen as a series of dots on the film. We can see a similar pattern when we pass a beam of light from a laser through two diffraction gratings crossed at right angles. In a crystal, it is the many parallel planes of atoms that produce the diffraction effect, just like the large number of rulings of the gratings. By measuring the distances between spots on the film, the spacing of the planes can be calculated.

Nowadays, solid state detectors, not film, are used to detect the X-rays. Measured intensities are digitized to store as a computer record, and computer software packages calculate the crystal structures and the positions of the atoms in the planes. Even the molecular structures of complex molecules such as DNA have been worked out from X-ray analysis.

We can use electrons instead of X-rays for studying crystal structures. Electron microscopes give pictures similar to those from an optical microscope. Alternatively, electron microscopes can be set to produce diffraction patterns because electrons can behave as waves.

THE STRUCTURES OF OTHER SOLIDS

Although many materials have a regular crystalline structure, others do not. Such materials are described as amorphous, which means 'without form'. Alternative descriptions used are glassy and solid-liquid. So, not surprisingly, ordinary glass is an amorphous material. It is made from silica (SiO₂) with added oxides.

A diffraction pattern for glass does not show spots. Instead, it shows a couple of very diffuse rings that arise because the separations of the nearest and next-nearest atomic neighbours have different average values. But there is no order at longer range. Glass is hard and brittle and surface cracks weaken it considerably. Glass cutters use this property extensively. They scratch the surface and break the glass along this scratch. As a demonstration, a glass rod is supported at its ends and loaded centrally; the load is increased until the rod breaks. The procedure is repeated, but using a glass rod with a small scratch on the under-surface. The rod now breaks, but with a much smaller load.

Glass softens when heated and then it is possible to shape it, as glass blowers do. It can be drawn out into the very fine fibres that carry light waves in fibre optic communications. Although not a liquid in the true sense, glass flows very slowly. Very old glass windows are thicker at the bottom than at the top as a result of slow flow in the Earth's gravitational field.

Polymers are made up of long chains of atoms forming long molecules. The backbone of the molecule is usually a series of carbon atoms covalently bonded together with other types of atoms attached. There are many examples of polymers in the home - polythene (made from polyethene molecules), nylon, PVC and Perspex are a few examples. The molecular chains can bend and become tangled. Some regions of the polymer may be entirely amorphous, while other regions may be composed of molecules arranged in a regular (crystalline) form. Rubber is such a polymer. The molecules of rubber are more tangled in its original shape than when stretched. Some polymer chains may have actual cross-links.

There are various other types of polymer. Thermosetting polymers can be moulded: they change to the required shape when heated through the so-called glass transition temperature' and then retain this shape when cooled. If reheated, they decompose rather than soften and distort. Such material is melamine which can be used for kitchen utensils. Other polymers, called thermoplastics soften on reheating, become flexible and then melt. Shaped articles can be made from the thermoplastic by moulding or by extruding the material through a tube.

LIQUIDS

We have seen that there is no long-range order between molecules within a liquid. An X-ray picture of a liquid is similar to that for an amorphous solid such as glass. Molecules move around relative to each other and do not keep the same molecules as neighbours. This ability of the molecules to move easily past each other allows liquids to flow. A liquid does not maintain its shape if shear forces are exerted on it, although a liquid can withstand compressive (hydrostatic) forces, because it maintains its volume.

We can see that liquid molecules move easily by observing Brownian motion. The botanist Robert Brown first noticed the effect in 1827 when looking at pollen grains suspended in water. The pollen grains have sufficiently small mass to be buffeted around by the unseen water molecules. A similar arrangement is shown in the diagram below. Smoke particles can be seen moving in air, demonstrating that gas molecules also collide with their neighbours.

SURFACE TENSION

Liquids behave as if there is a very delicate skin over their surface. For example, a pin can be balanced on the 'skin' of a water surface, and some insects can walk on the surface of ponds. This is possible because of surface tension: surface molecules tend to be further apart than underlying molecules, and attractive forces between neighbouring liquid surface molecules are stronger than forces between underlying molecules.

GASES

We have seen that the atoms in a solid vibrate but do not move around. In a liquid, atoms or molecules are able to drift around, but they only alter their mean positions slowly. In contrast, the atoms or molecules in a gas move around rapidly over large distances. There is no pattern to the movement.

As they speed around, the molecules (or atoms) of a gas keep colliding with the walls of their container. As the molecules rebound off the walls, there is a change in their momentum. Each molecule exerts a small force on the walls during a very small interval of time. All these small forces add up so that a large number of such collisions produce a total average force on the walls that is measurable.

Assuming that the molecules are moving equally in all directions, the force per unit area of wall will be the same over all the walls of the container. This force per unit area is the pressure exerted by the gas and it is equal in all directions.

THE IDEAL GAS EQUATION

In the middle of the seventeenth century, Robert Boyle showed experimentally that: The volume of a fixed mass of gas is inversely proportional to the pressure applied to it if the temperature is kept constant. That is, at constant temperature and mass: volume ∝ 1/pressure or pressure ∝ 1/volume.
If p is the pressure and V the volume of the gas, for constant temperature T and constant mass we have:

p ∝ 1/V or pV= constant ..................[Law 1]

The law is known as Boyle's law and applies provided the pressure of the gas is low. In addition to Boyle's law, there are two other gas laws.
The pressure law states that: The pressure of a fixed mass of gas at constant volume is proportional to temperature as measured on the Kelvin scale. That is, for constant volume and mass: pressure ∝ temperature.

p ∝ T or P/T = constant....................[Law 2]

This is a fundamental relationship for an ideal gas (real gases need to be at a low pressure). Note that I have not so far said precisely what I mean by temperature. The next heading below describes how a temperature scale is obtained.

The other law is called Charles’ law: The volume of a fixed mass of gas at constant pressure is proportional to temperature as measured on the Kelvin scale. That is, for constant pressure and mass: volume ∝ temperature.

V ∝ T or V/T = constant.................[Law 3]

Putting the three laws together we obtain the combined gas law. For a constant mass of ideal gas:

pV/T = Constant or P₁V₁/T₁ = P₂V₂/T₂

Subscript 1 refers to the initial conditions for the ideal gas and subscript 2 refers to a later set of conditions.

If a real gas obeys the combined gas law equation, then it is behaving ideally. The equation assumes that there is no attraction, no repulsion and no collisions between the atoms or molecules of the gas. Real gases usually cool if allowed to expand - this property is exploited, for example, to liquefy helium from the gaseous phase at very low temperatures. But an ideal gas will not cool down if it is allowed to expand freely (that is, if it is allowed to expand into a vacuum) and an ideal gas cannot be turned into a liquid.

Finally, we need to know the constant of proportionality in the combined gas law equation. From experiment it is found that the constant depends on the mass of gas involved. If the mass of gas is doubled (this means doubling the number of moles of gas) at constant pressure and temperature, then the volume of gas doubles. So pV/T is proportional to the amount of gas enclosed, that is, proportional to the number of moles n:

pV/T = nR or pV= nRT

where R is the constant of proportionality and is called the molar gas constant. It is a fundamental quantity that can be measured to high accuracy and it has units. The constant can be found from the fact that 1 mole of an ideal gas at standard temperature (273.16 K) and pressure (1.014 × 10⁵ Pa) occupies 0.0224 m³:

R = 1.014 × 10⁵ × 0.0224 Pa m ÷ 1 × 273.16 mol K = 8.31 JK^-1 mol^-1

(Note that Pa m³ ≡ Nm^-2 m³ = Nm ≡ J.
This equation, pV= nRT for n moles of gas, or the alternative form, pV_m = RT. For one mole of gas, where V refers to molar volume, is called the universal gas law equation.

Regarding the Avogadro constant and molar mass; the mole is the unit used to measure the amount of a substance. One mole of any substance contains 6.02 × 10²³ 'elementary entities', which may be atoms or molecules. This number is the Avogadro constant, symbol NA. It was derived by finding experimentally the number of atoms in 0.0120 kg (12 g) of carbon-12. The mass of any other element containing 6.02 × 10²³ atoms or molecules will then be the molar mass, symbol M, of that element. For example, the molar mass of iron is 0.0585 kg and it contains 6.02 × 10²³ atoms. Gaseous oxygen has two atoms per molecule, and here the molar mass, which is 0.0320 kg, contains 6.02 × 10²³ molecules, but twice as many atoms.

I will like to rest a bit here. Till next time when I continue the final episode on this interesting series of Ordinary matter physics, I remain my humble self, @emperorhassy.