Old quantum theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Old_quantum_theory 

The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics. The main and final accomplishments of the old quantum theory were the determination of the modern form of the periodic table by Edmund Stoner and the Pauli Exclusion Principle which were both premised on the Arnold Sommerfeld enhancements to the Bohr model of the atom.

The main tool of the old quantum theory was the Bohr–Sommerfeld quantization condition, a procedure for selecting out certain states of a classical system as allowed states: the system can then only exist in one of the allowed states and not in any other state.

History

The old quantum theory was instigated by the 1900 work of Max Planck on the emission and absorption of light in a black body with his discovery of Planck’s law introducing his quantum of action, and began in earnest after the work of Albert Einstein on the specific heats of solids in 1907 brought him to the attention of Walther Nernst. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.

In 1910, Arthur Erich Haas develops J. J. Thomson’s atomic model in his 1910 paper that outlined a treatment of the hydrogen atom involving quantization of electronic orbitals, thus anticipating the Bohr model (1913) by three years.

John William Nicholson is noted as the first to create an atomic model that quantized angular momentum as h/2π. Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom.

In 1913, Niels Bohr displayed rudiments of the later defined correspondence principle and used it to formulate a model of the hydrogen atom which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called "space quantization" (German: Richtungsquantelung). This model, which became known as the Bohr–Sommerfeld model, allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. The theory would have correctly explained the Zeeman effect, except for the issue of electron spin. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.

Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. Max Planck introduced the zero point energy and Arnold Sommerfeld semiclassically quantized the relativistic hydrogen atom. Hendrik Kramers explained the Stark effect. Bose and Einstein gave the correct quantum statistics for photons.

The Sommerfeld extensions of the 1913 solar system Bohr model of the hydrogen atom showing the addition of elliptical orbits to explain spectral fine structure.

Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with Werner Heisenberg to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating matrix mechanics.

In 1924, Louis de Broglie introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrödinger's wave mechanics developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory.

In the 1950s Joseph Keller updated Bohr–Sommerfeld quantization using Einstein's interpretation of 1917, now known as Einstein–Brillouin–Keller method. In 1971, Martin Gutzwiller took into account that this method only works for integrable systems and derived a semiclassical way of quantizing chaotic systems from path integrals.

Basic principles

See also: Jun Ishiwara § Quantum physics

The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the quantization condition:

${\displaystyle {\oint }_{H(p,q)=E}{p}_{i}d{q}_i=n_{i}h}$

where the ${\displaystyle p_i}$ are the momenta of the system and the ${\displaystyle q_i}$ are the corresponding coordinates. The quantum numbers ${\displaystyle n_i}$ are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). The integral is an area in phase space, which is a quantity called the action and is quantized in units of the (unreduced) Planck constant. For this reason, the Planck constant was often called the quantum of action.

In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates ${\displaystyle q_i}$ in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.

The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.

This quantization condition is often known as the Wilson–Sommerfeld rule, proposed independently by William Wilson and Arnold Sommerfeld.

Examples

Thermal properties of the harmonic oscillator

The simplest system in the old quantum theory is the harmonic oscillator, whose Hamiltonian is:

${\displaystyle H=\frac{p^2}{2m}+\frac{m{\omega }^2{q}^2}{2}.}$

The old quantum theory yields a recipe for the quantization of the energy levels of the harmonic oscillator, which, when combined with the Boltzmann probability distribution of thermodynamics, yields the correct expression for the stored energy and specific heat of a quantum oscillator both at low and at ordinary temperatures. Applied as a model for the specific heat of solids, this resolved a discrepancy in pre-quantum thermodynamics that had troubled 19th-century scientists. Let us now describe this.

The level sets of H are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule:

${\displaystyle E=n\hslash \omega ,}$

a result which was known well before, and used to formulate the old quantum condition. This result differs by ${\displaystyle \frac{1}{2}\hslash \omega }$ from the results found with the help of quantum mechanics. This constant is neglected in the derivation of the old quantum theory, and its value cannot be determined using it.

The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a Boltzmann weight:

${\displaystyle U=\frac{\sum _n\hslash \omega n{e}^{-\beta n\hslash \omega }}{\sum _n{e}^{-\beta n\hslash \omega }}=\frac{\hslash \omega e^{-\beta \hslash \omega }}{1-e^{-\beta \hslash \omega }},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\beta =\frac{1}{kT},}$

kT is Boltzmann constant times the absolute temperature, which is the temperature as measured in more natural units of energy. The quantity ${\displaystyle \beta }$ is more fundamental in thermodynamics than the temperature, because it is the thermodynamic potential associated to the energy.

From this expression, it is easy to see that for large values of ${\displaystyle \beta }$, for very low temperatures, the average energy U in the Harmonic oscillator approaches zero very quickly, exponentially fast. The reason is that kT is the typical energy of random motion at temperature T, and when this is smaller than ${\displaystyle \hslash \omega }$, there is not enough energy to give the oscillator even one quantum of energy. So the oscillator stays in its ground state, storing next to no energy at all.

This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the specific heat, so the specific heat is exponentially small at low temperatures, going to zero like

${\displaystyle \exp (-\hslash \omega /kT)}$

At small values of ${\displaystyle \beta }$, at high temperatures, the average energy U is equal to ${\displaystyle 1/\beta =kT}$. This reproduces the equipartition theorem of classical thermodynamics: every harmonic oscillator at temperature T has energy kT on average. This means that the specific heat of an oscillator is constant in classical mechanics and equal to k. For a collection of atoms connected by springs, a reasonable model of a solid, the total specific heat is equal to the total number of oscillators times k. There are overall three oscillators for each atom, corresponding to the three possible directions of independent oscillations in three dimensions. So the specific heat of a classical solid is always 3k per atom, or in chemistry units, 3R per mole of atoms.

Monatomic solids at room temperatures have approximately the same specific heat of 3k per atom, but at low temperatures they don't. The specific heat is smaller at colder temperatures, and it goes to zero at absolute zero. This is true for all material systems, and this observation is called the third law of thermodynamics. Classical mechanics cannot explain the third law, because in classical mechanics the specific heat is independent of the temperature.

This contradiction between classical mechanics and the specific heat of cold materials was noted by James Clerk Maxwell in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. Einstein resolved this problem in 1906 by proposing that atomic motion is quantized. This was the first application of quantum theory to mechanical systems. A short while later, Peter Debye gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see Einstein solid and Debye model).

One-dimensional potential: U = 0

One-dimensional problems are easy to solve. At any energy E, the value of the momentum p is found from the conservation equation:

${\displaystyle \sqrt{2m(E-U(q))}=\sqrt{2mE}=p=\text{const.}}$

which is integrated over all values of q between the classical turning points, the places where the momentum vanishes. The integral is easiest for a particle in a box of length L, where the quantum condition is:

${\displaystyle 2{\int }_0^{L}pdq=nh}$

which gives the allowed momenta:

${\displaystyle p=\frac{nh}{2L}}$

and the energy levels

${\displaystyle E_n=\frac{p^2}{2m}=\frac{n^2{h}^2}{8m{L}^2}}$

One-dimensional potential: U = Fx

Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force F binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.

${\displaystyle 2{\int }_0^{\frac{E}{F}}\sqrt{2m(E-Fx)}dx=nh}$

so that the quantum condition is

${\displaystyle \frac{4}{3}\sqrt{2m}\frac{E^{3/2}}{F}=nh}$

which determines the energy levels,

${\displaystyle E_n={\left(\frac{3nhF}{4\sqrt{2m}}\right)}^{2/3}}$

In the specific case F=mg, the particle is confined by the gravitational potential of the earth and the "wall" here is the surface of the earth.

One-dimensional potential: U = 1⁄2kx²

This case is also easy to solve, and the semiclassical answer here agrees with the quantum one to within the ground-state energy. Its quantization-condition integral is

${\displaystyle 2{\int }_{-\sqrt{\frac{2E}{k}}}^{\sqrt{\frac{2E}{k}}}\sqrt{2m\left(E-\frac{1}{2}k{x}^2\right)}dx=nh}$

with solution

${\displaystyle E=n\frac{h}{2\pi }\sqrt{\frac{k}{m}}=n\hslash \omega }$

for oscillation angular frequency ${\displaystyle \omega }$, as before.

Rotator

Another simple system is the rotator. A rotator consists of a mass M at the end of a massless rigid rod of length R and in two dimensions has the Lagrangian:

${\displaystyle L=\frac{M{R}^2}{2}{\dot{\theta }}^2}$

which determines that the angular momentum J conjugate to ${\displaystyle \theta }$, the polar angle, ${\displaystyle J=M{R}^2\dot{\theta }}$. The old quantum condition requires that J multiplied by the period of ${\displaystyle \theta }$ is an integer multiple of the Planck constant:

${\displaystyle 2\pi J=nh}$

the angular momentum to be an integer multiple of ${\displaystyle \hslash }$. In the Bohr model, this restriction imposed on circular orbits was enough to determine the energy levels.

In three dimensions, a rigid rotator can be described by two angles — ${\displaystyle \theta }$ and ${\displaystyle \varphi }$, where ${\displaystyle \theta }$ is the inclination relative to an arbitrarily chosen z-axis while ${\displaystyle \varphi }$ is the rotator angle in the projection to the x–y plane. The kinetic energy is again the only contribution to the Lagrangian:

${\displaystyle L=\frac{M{R}^2}{2}{\dot{\theta }}^2+\frac{M{R}^2}{2}(\sin (\theta )\dot{\varphi }{)}^2}$

And the conjugate momenta are ${\displaystyle p_{\theta }=\dot{\theta }}$ and ${\displaystyle p_{\varphi }=\sin (\theta {)}^2\dot{\varphi }}$. The equation of motion for ${\displaystyle \varphi }$ is trivial: ${\displaystyle p_{\varphi }}$ is a constant:

${\displaystyle p_{\varphi }=l_{\varphi }}$

which is the z-component of the angular momentum. The quantum condition demands that the integral of the constant ${\displaystyle l_{\varphi }}$ as ${\displaystyle \varphi }$ varies from 0 to ${\displaystyle 2\pi }$ is an integer multiple of h:

${\displaystyle l_{\varphi }=m\hslash }$

And m is called the magnetic quantum number, because the z component of the angular momentum is the magnetic moment of the rotator along the z direction in the case where the particle at the end of the rotator is charged.

Since the three-dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the z-component of the angular momentum to be the integers l,m. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox: how can the orientation of the angular momentum relative to the arbitrarily chosen z-axis be quantized? This seems to pick out a direction in space.

This phenomenon, the quantization of angular momentum about an axis, was given the name space quantization, because it seemed incompatible with rotational invariance. In modern quantum mechanics, the angular momentum is quantized the same way, but the discrete states of definite angular momentum in any one orientation are quantum superpositions of the states in other orientations, so that the process of quantization does not pick out a preferred axis. For this reason, the name "space quantization" fell out of favor, and the same phenomenon is now called the quantization of angular momentum.

Hydrogen atom

The angular part of the hydrogen atom is just the rotator, and gives the quantum numbers l and m. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved.

For a fixed value of the total angular momentum L, the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):

${\displaystyle H=\frac{p_r^2}{2}+\frac{l^2}{2r^2}-\frac{1}{r}.}$

Fixing the energy to be (a negative) constant and solving for the radial momentum ${\displaystyle p_r}$, the quantum condition integral is:

${\displaystyle \oint \sqrt{2E-\frac{l^2}{r^2}+\frac{2}{r}}dr=kh}$

which can be solved with the method of residues, and gives a new quantum number ${\displaystyle k}$ which determines the energy in combination with ${\displaystyle l}$. The energy is:

${\displaystyle E=-\frac{1}{2(k+l{)}^2}}$

and it only depends on the sum of k and l, which is the principal quantum number n. Since k is positive, the allowed values of l for any given n are no bigger than n. The energies reproduce those in the Bohr model, except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values.

De Broglie waves

In 1905, Einstein noted that the entropy of the quantized electromagnetic field oscillators in a box is, for short wavelength, equal to the entropy of a gas of point particles in the same box. The number of point particles is equal to the number of quanta. Einstein concluded that the quanta could be treated as if they were localizable objects, particles of light. Today we call them photons (a name coined by Gilbert N. Lewis in a letter to Nature.)

Einstein's theoretical argument was based on thermodynamics, on counting the number of states, and so was not completely convincing. Nevertheless, he concluded that light had attributes of both waves and particles, more precisely that an electromagnetic standing wave with frequency ${\displaystyle \omega }$ with the quantized energy:

${\displaystyle E=n\hslash \omega }$

should be thought of as consisting of n photons each with an energy ${\displaystyle \hslash \omega }$. Einstein could not describe how the photons were related to the wave.

The photons have momentum as well as energy, and the momentum had to be ${\displaystyle \hslash k}$ where ${\displaystyle k}$ is the wavenumber of the electromagnetic wave. This is required by relativity, because the momentum and energy form a four-vector, as do the frequency and wave-number.

In 1924, as a PhD candidate, Louis de Broglie proposed a new interpretation of the quantum condition. He suggested that all matter, electrons as well as photons, are described by waves obeying the relations.

${\displaystyle p=\hslash k}$

or, expressed in terms of wavelength ${\displaystyle \lambda }$ instead,

${\displaystyle p=\frac{h}{\lambda }}$

He then noted that the quantum condition:

${\displaystyle \int pdx=\hslash \int kdx=2\pi \hslash n}$

counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of ${\displaystyle 2\pi }$. Expressed in wavelengths, the number of wavelengths along a classical orbit must be an integer. This is the condition for constructive interference, and it explained the reason for quantized orbits—the matter waves make standing waves only at discrete frequencies, at discrete energies.

For example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls. The condition becomes:

${\displaystyle n\lambda =2L}$

so that the quantized momenta are:

${\displaystyle p=\frac{nh}{2L}}$

reproducing the old quantum energy levels.

This development was given a more mathematical form by Einstein, who noted that the phase function for the waves, ${\displaystyle \theta (J,x)}$, in a mechanical system should be identified with the solution to the Hamilton–Jacobi equation, an equation which even William Rowan Hamilton in the 19th century believed to be a short-wavelength limit of a sort of wave mechanics. Schrödinger then found the proper wave equation which matched the Hamilton–Jacobi equation for the phase, this is the famous equation that bears his name.

Kramers transition matrix

The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, Hendrik Kramers was able to find heuristics for describing how emission and absorption should be calculated.

Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency:

${\displaystyle X_n(t)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{ik\omega t}{X}_{n;k}}$

The index n describes the quantum numbers of the orbit, it would be n–l–m in the Sommerfeld model. The frequency ${\displaystyle \omega }$ is the angular frequency of the orbit ${\displaystyle 2\pi /T_n}$ while k is an index for the Fourier mode. Bohr had suggested that the k-th harmonic of the classical motion correspond to the transition from level n to level n−k.

Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to ${\displaystyle |X_k{|}^2}$, as it would be in classical mechanics. The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels.

This idea led to the development of matrix mechanics.

Limitations

The old quantum theory had some limitations:

• The old quantum theory provides no means to calculate the intensities of the spectral lines.
• It fails to explain the anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected).
• It cannot quantize "chaotic" systems, i.e. dynamical systems in which trajectories are neither closed nor periodic and whose analytical form does not exist. This presents a problem for systems as simple as a 2-electron atom which is classically chaotic analogously to the famous gravitational three-body problem.

However it can be used to describe atoms with more than one electron (e.g. Helium) and the Zeeman effect. It was later proposed that the old quantum theory is in fact the semi-classical approximation to the canonical quantum mechanics but its limitations are still under investigation.

Period 2 element

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Period_2_element

A period 2 element is one of the chemical elements in the second row (or period) of the periodic table of the chemical elements. The periodic table is laid out in rows to illustrate recurring (periodic) trends in the chemical behavior of the elements as their atomic number increases; a new row is started when chemical behavior begins to repeat, creating columns of elements with similar properties.

The second period contains the elements lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon. In a quantum mechanical description of atomic structure, this period corresponds to the filling of the second (n = 2) shell, more specifically its 2s and 2p subshells. Period 2 elements (carbon, nitrogen, oxygen, fluorine and neon) obey the octet rule in that they need eight electrons to complete their valence shell (lithium and beryllium obey duet rule, boron is electron deficient.), where at most eight electrons can be accommodated: two in the 2s orbital and six in the 2p subshell.

Periodic trends

Calculated atomic radii of period 2 elements in picometers.

Period 2 is the first period in the periodic table from which periodic trends can be drawn. Period 1, which only contains two elements (hydrogen and helium), is too small to draw any conclusive trends from it, especially because the two elements behave nothing like other s-block elements. Period 2 has much more conclusive trends. For all elements in period 2, as the atomic number increases, the atomic radius of the elements decreases, the electronegativity increases, and the ionization energy increases.

Period 2 only has two metals (lithium and beryllium) of eight elements, less than for any subsequent period both by number and by proportion. It also has the most number of nonmetals, namely five, among all periods. The elements in period 2 often have the most extreme properties in their respective groups; for example, fluorine is the most reactive halogen, neon is the most inert noble gas, and lithium is the least reactive alkali metal.

All period 2 elements completely obey the Madelung rule; in period 2, lithium and beryllium fill the 2s subshell, and boron, carbon, nitrogen, oxygen, fluorine, and neon fill the 2p subshell. The period shares this trait with periods 1 and 3, none of which contain transition elements or inner transition elements, which often vary from the rule.

Chemical element			Block	Electron configuration
3	Li	Lithium	s-block	[He] 2s1
4	Be	Beryllium	s-block	[He] 2s2
5	B	Boron	p-block	[He] 2s2 2p1
6	C	Carbon	p-block	[He] 2s2 2p2
7	N	Nitrogen	p-block	[He] 2s2 2p3
8	O	Oxygen	p-block	[He] 2s2 2p4
9	F	Fluorine	p-block	[He] 2s2 2p5
10	Ne	Neon	p-block	[He] 2s2 2p6

Lithium

Main article: Lithium

 

Lithium metal floating on paraffin oil

Lithium (Li) is an alkali metal with atomic number 3, occurring naturally in two isotopes: ⁶Li and ⁷Li. The two make up all natural occurrence of lithium on Earth, although further isotopes have been synthesized. In ionic compounds, lithium loses an electron to become positively charged, forming the cation Li⁺. Lithium is the first alkali metal in the periodic table, and the first metal of any kind in the periodic table. At standard temperature and pressure, lithium is a soft, silver-white, highly reactive metal. With a density of 0.564 g⋅cm⁻³, lithium is the lightest metal and the least dense solid element.

Lithium is one of the few elements synthesized in the Big Bang. Lithium is the 33rd most abundant element on earth, occurring in concentrations of between 20 and 70 ppm by weight, but due to its high reactivity it is only found naturally in compounds.

Lithium salts are used in the pharmacology industry as mood stabilising drugs. They are used in the treatment of bipolar disorder, where they have a role in treating depression and mania and may reduce the chances of suicide. The most common compounds used are lithium carbonate, Li₂CO₃, lithium citrate, Li₃C₆H₅O₇, lithium sulphate, Li₂SO₄, and lithium orotate, LiC₅H₃N₂O₄·H₂O. Lithium is also used in batteries as an anode and its alloys with aluminium, cadmium, copper and manganese are used to make high performance parts for aircraft, most notably the external tank of the Space Shuttle.

Beryllium

Main article: Beryllium

 

Large piece of beryllium

Beryllium (Be) is the chemical element with atomic number 4, occurring in the form of ⁹Be. At standard temperature and pressure, beryllium is a strong, steel-grey, light-weight, brittle, bivalent alkali earth metal, with a density of 1.85 g⋅cm⁻³. It also has one of the highest melting points of all the light metals. Beryllium's most common isotope is ⁹Be, which contains 4 protons and 5 neutrons. It makes up almost 100% of all naturally occurring beryllium and is its only stable isotope; however other isotopes have been synthesised. In ionic compounds, beryllium loses its two valence electrons to form the cation, Be²⁺.

Small amounts of beryllium were synthesised during the Big Bang, although most of it decayed or reacted further to create larger nuclei, like carbon, nitrogen or oxygen. Beryllium is a component of 100 out of 4000 known minerals, such as bertrandite, Be₄Si₂O₇(OH)₂, beryl, Al₂Be₃Si₆O₁₈, chrysoberyl, Al₂BeO₄, and phenakite, Be₂SiO₄. Precious forms of beryl are aquamarine, red beryl and emerald. The most common sources of beryllium used commercially are beryl and bertrandite and production of it involves the reduction of beryllium fluoride with magnesium metal or the electrolysis of molten beryllium chloride, containing some sodium chloride as beryllium chloride is a poor conductor of electricity.

Due to its stiffness, light weight, and dimensional stability over a wide temperature range, beryllium metal is used in as a structural material in aircraft, missiles and communication satellites. It is used as an alloying agent in beryllium copper, which is used to make electrical components due to its high electrical and heat conductivity. Sheets of beryllium are used in X-ray detectors to filter out visible light and let only X-rays through. It is used as a neutron moderator in nuclear reactors because light nuclei are more effective at slowing down neutrons than heavy nuclei. Beryllium's low weight and high rigidity also make it useful in the construction of tweeters in loudspeakers.

Beryllium and beryllium compounds are classified by the International Agency for Research on Cancer as Group 1 carcinogens; they are carcinogenic to both animals and humans. Chronic berylliosis is a pulmonary and systemic granulomatous disease caused by exposure to beryllium. Between 1% – 15% of people are sensitive to beryllium and may develop an inflammatory reaction in their respiratory system and skin, called chronic beryllium disease or berylliosis. The body's immune system recognises the beryllium as foreign particles and mounts an attack against them, usually in the lungs where they are breathed in. This can cause fever, fatigue, weakness, night sweats and difficulty in breathing.

Boron

Main article: Boron

 

Boron chunks

Boron (B) is the chemical element with atomic number 5, occurring as ¹⁰B and ¹¹B. At standard temperature and pressure, boron is a trivalent metalloid that has several different allotropes. Amorphous boron is a brown powder formed as a product of many chemical reactions. Crystalline boron is a very hard, black material with a high melting point and exists in many polymorphs: Two rhombohedral forms, α-boron and β-boron containing 12 and 106.7 atoms in the rhombohedral unit cell respectively, and 50-atom tetragonal boron are the most common. Boron has a density of 2.34⁻³. Boron's most common isotope is ¹¹B at 80.22%, which contains 5 protons and 6 neutrons. The other common isotope is ¹⁰B at 19.78%, which contains 5 protons and 5 neutrons. These are the only stable isotopes of boron; however other isotopes have been synthesised. Boron forms covalent bonds with other nonmetals and has oxidation states of 1, 2, 3 and 4. Boron does not occur naturally as a free element, but in compounds such as borates. The most common sources of boron are tourmaline, borax, Na₂B₄O₅(OH)₄·8H₂O, and kernite, Na₂B₄O₅(OH)₄·2H₂O. it is difficult to obtain pure boron. It can be made through the magnesium reduction of boron trioxide, B₂O₃. This oxide is made by melting boric acid, B(OH)₃, which in turn is obtained from borax. Small amounts of pure boron can be made by the thermal decomposition of boron bromide, BBr₃, in hydrogen gas over hot tantalum wire, which acts as a catalyst. The most commercially important sources of boron are: sodium tetraborate pentahydrate, Na₂B₄O₇ · 5H₂O, which is used in large amounts in making insulating fiberglass and sodium perborate bleach; boron carbide, a ceramic material, is used to make armour materials, especially in bulletproof vests for soldiers and police officers; orthoboric acid, H₃BO₃ or boric acid, used in the production of textile fiberglass and flat panel displays; sodium tetraborate decahydrate, Na₂B₄O₇ · 10H₂O or borax, used in the production of adhesives; and the isotope boron-10 is used as a control for nuclear reactors, as a shield for nuclear radiation, and in instruments used for detecting neutrons.

Boron is an essential plant micronutrient, required for cell wall strength and development, cell division, seed and fruit development, sugar transport and hormone development. However, high soil concentrations of over 1.0 ppm can cause necrosis in leaves and poor growth. Levels as low as 0.8 ppm can cause these symptoms to appear in plants particularly boron-sensitive. Most plants, even those tolerant of boron in the soil, will show symptoms of boron toxicity when boron levels are higher than 1.8 ppm. In animals, boron is an ultratrace element; in human diets, daily intake ranges from 2.1 to 4.3 mg boron/kg body weight (bw)/day. It is also used as a supplement for the prevention and treatment of osteoporosis and arthritis.

Carbon

Main article: Carbon

 

Diamond and graphite, two different allotropes of carbon

Carbon is the chemical element with atomic number 6, occurring as ¹²C, ¹³C and ¹⁴C. At standard temperature and pressure, carbon is a solid, occurring in many different allotropes, the most common of which are graphite, diamond, the fullerenes and amorphous carbon. Graphite is a soft, hexagonal crystalline, opaque black semimetal with very good conductive and thermodynamically stable properties. Diamond however is a highly transparent colourless cubic crystal with poor conductive properties, is the hardest known naturally occurring mineral and has the highest refractive index of all gemstones. In contrast to the crystal lattice structure of diamond and graphite, the fullerenes are molecules, named after Richard Buckminster Fuller whose architecture the molecules resemble. There are several different fullerenes, the most widely known being the "buckeyball" C₆₀. Little is known about the fullerenes and they are a current subject of research. There is also amorphous carbon, which is carbon without any crystalline structure. In mineralogy, the term is used to refer to soot and coal, although these are not truly amorphous as they contain small amounts of graphite or diamond. Carbon's most common isotope at 98.9% is ¹²C, with six protons and six neutrons. ¹³C is also stable, with six protons and seven neutrons, at 1.1%. Trace amounts of ¹⁴C also occur naturally but this isotope is radioactive and decays with a half life of 5730 years; it is used for radiocarbon dating. Other isotopes of carbon have also been synthesised. Carbon forms covalent bonds with other non-metals with an oxidation state of −4, −2, +2 or +4.

Carbon is the fourth most abundant element in the universe by mass after hydrogen, helium and oxygen and is the second most abundant element in the human body by mass after oxygen, the third most abundant by number of atoms. There are an almost infinite number of compounds that contain carbon due to carbon's ability to form long stable chains of C — C bonds. The simplest carbon-containing molecules are the hydrocarbons, which contain carbon and hydrogen, although they sometimes contain other elements in functional groups. Hydrocarbons are used as fossil fuels and to manufacture plastics and petrochemicals. All organic compounds, those essential for life, contain at least one atom of carbon. When combined with oxygen and hydrogen, carbon can form many groups of important biological compounds including sugars, lignans, chitins, alcohols, fats, and aromatic esters, carotenoids and terpenes. With nitrogen it forms alkaloids, and with the addition of sulfur also it forms antibiotics, amino acids, and rubber products. With the addition of phosphorus to these other elements, it forms DNA and RNA, the chemical-code carriers of life, and adenosine triphosphate (ATP), the most important energy-transfer molecule in all living cells.

Nitrogen

Main article: Nitrogen

 

Liquid nitrogen being poured

Nitrogen is the chemical element with atomic number 7, the symbol N and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere. The element nitrogen was discovered as a separable component of air, by Scottish physician Daniel Rutherford, in 1772. It occurs naturally in form of two isotopes: nitrogen-14 and nitrogen-15.

Many industrially important compounds, such as ammonia, nitric acid, organic nitrates (propellants and explosives), and cyanides, contain nitrogen. The extremely strong bond in elemental nitrogen dominates nitrogen chemistry, causing difficulty for both organisms and industry in breaking the bond to convert the N
₂ molecule into useful compounds, but at the same time causing release of large amounts of often useful energy when the compounds burn, explode, or decay back into nitrogen gas.

Nitrogen occurs in all living organisms, and the nitrogen cycle describes movement of the element from air into the biosphere and organic compounds, then back into the atmosphere. Synthetically produced nitrates are key ingredients of industrial fertilizers, and also key pollutants in causing the eutrophication of water systems. Nitrogen is a constituent element of amino acids and thus of proteins, and of nucleic acids (DNA and RNA). It resides in the chemical structure of almost all neurotransmitters, and is a defining component of alkaloids, biological molecules produced by many organisms.

Oxygen

Main article: Oxygen

Oxygen is the chemical element with atomic number 8, occurring mostly as ¹⁶O, but also ¹⁷O and ¹⁸O.

Oxygen is the third-most common element by mass in the universe (although there are more carbon atoms, each carbon atom is lighter). It is highly electronegative and non-metallic, usually diatomic, gas down to very low temperatures. Only fluorine is more reactive among non-metallic elements. It is two electrons short of a full octet and readily takes electrons from other elements. It reacts violently with alkali metals and white phosphorus at room temperature and less violently with alkali earth metals heavier than magnesium. At higher temperatures it burns most other metals and many non-metals (including hydrogen, carbon, and sulfur). Many oxides are extremely stable substances difficult to decompose—like water, carbon dioxide, alumina, silica, and iron oxides (the latter often appearing as rust). Oxygen is part of substances best described as some salts of metals and oxygen-containing acids (thus nitrates, sulfates, phosphates, silicates, and carbonates.

Oxygen is essential to all life. Plants and phytoplankton photosynthesize water and carbon dioxide and water, both oxides, in the presence of sunlight to form sugars with the release of oxygen. The sugars are then turned into such substances as cellulose and (with nitrogen and often sulfur) proteins and other essential substances of life. Animals especially but also fungi and bacteria ultimately depend upon photosynthesizing plants and phytoplankton for food and oxygen.

Fire uses oxygen to oxidize compounds typically of carbon and hydrogen to water and carbon dioxide (although other elements may be involved) whether in uncontrolled conflagrations that destroy buildings and forests or the controlled fire within engines or that supply electrical energy from turbines, heat for keeping buildings warm, or the motive force that drives vehicles.

Oxygen forms roughly 21% of the Earth's atmosphere; all of this oxygen is the result of photosynthesis. Pure oxygen has use in medical treatment of people who have respiratory difficulties. Excess oxygen is toxic.

Oxygen was originally associated with the formation of acids—until some acids were shown to not have oxygen in them. Oxygen is named for its formation of acids, especially with non-metals. Some oxides of some non-metals are extremely acidic, like sulfur trioxide, which forms sulfuric acid on contact with water. Most oxides with metals are alkaline, some extremely so, like potassium oxide. Some metallic oxides are amphoteric, like aluminum oxide, which means that they can react with both acids and bases.

Although oxygen is normally a diatomic gas, oxygen can form an allotrope known as ozone. Ozone is a triatomic gas even more reactive than oxygen. Unlike regular diatomic oxygen, ozone is a toxic material generally considered a pollutant. In the upper atmosphere, some oxygen forms ozone which has the property of absorbing dangerous ultraviolet rays within the ozone layer. Land life was impossible before the formation of an ozone layer.

Fluorine

Main article: Fluorine

 

Liquid fluorine in ampoule

Fluorine is the chemical element with atomic number 9. It occurs naturally in its only stable form ¹⁹F.

Fluorine is a pale-yellow, diatomic gas under normal conditions and down to very low temperatures. Short one electron of the highly stable octet in each atom, fluorine molecules are unstable enough that they easily snap, with loose fluorine atoms tending to grab single electrons from just about any other element. Fluorine is the most reactive of all elements, and it even attacks many oxides to replace oxygen with fluorine. Fluorine even attacks silica, one of the favored materials for transporting strong acids, and burns asbestos. It attacks common salt, one of the most stable compounds, with the release of chlorine. It never appears uncombined in nature and almost never stays uncombined for long. It burns hydrogen simultaneously if either is liquid or gaseous—even at temperatures close to absolute zero. It is extremely difficult to isolate from any compounds, let alone keep uncombined.

Fluorine gas is extremely dangerous because it attacks almost all organic material, including live flesh. Many of the binary compounds that it forms (called fluorides) are themselves highly toxic, including soluble fluorides and especially hydrogen fluoride. Fluorine forms very strong bonds with many elements. With sulfur it can form the extremely stable and chemically inert sulfur hexafluoride; with carbon it can form the remarkable material Teflon that is a stable and non-combustible solid with a high melting point and a very low coefficient of friction that makes it an excellent liner for cooking pans and raincoats. Fluorine-carbon compounds include some unique plastics. it is also used as a reactant in the making of toothpaste.

Neon

Main article: Neon

 

Neon discharge tube

Neon is the chemical element with atomic number 10, occurring as ²⁰Ne, ²¹Ne and ²²Ne.

Neon is a monatomic gas. With a complete octet of outer electrons it is highly resistant to removal of any electron, and it cannot accept an electron from anything. Neon has no tendency to form any normal compounds under normal temperatures and pressures; it is effectively inert. It is one of the so-called "noble gases".

Neon is a trace component of the atmosphere without any biological role.