Metals

Most of the elements of the Periodic Table are metals. Metallic character increases to the left of the Periodic
Table and non-metals are found on the right.

  • Metals react primarily by donating one or more electrons (their valence electrons).
  • We can thus define metallicity as the tendency of a chemical to donate electrons in chemical reactions.
  • Metallic behaviour increases down a group and so Francium (Fr) is the most metallic element and the
    most reactive metal, exploding violently on contact with water.
  • Metalloids or semi-metals occur at the boundary between metals and non-metals. The exact boundary
    between metal and non-metal depends on the property measured and which elements are classed as
    metalloids is partly a matter of opinion, but boron (B), silicon (Si), germanium (Ge), arsenic (As), antimony
    (Sb), Tellurium (Te) and Pollonium (Po) are usually classed as metalloids. However, metals like aluminium
    (Al) and tin (Sn) exhibit some very non-metallic behaviours and the non-metals carbon (C) and iodine (I)
    have some metallic properties, for example.
Metallic Bonding

  • Occurs in elemental metals and alloys and some non-metals like graphite.
  • Sodium has only one valence electron and a melting point of 97.7 degrees C and is a soft metal.
  • Magnesium has 2 valence electrons and a m.p. = 650 C and is quite hard.
  • Iron has 3 valence electrons and a m.p. = 1538 C and is a hard metal.
  • Tungsten has 6 valence electrons and a m.p. = 3422 C.

Did you know? Tungsten carbide is the hardest metal alloy available (it is an alloy of tungsten and carbon).
Steel is iron carbide (an alloy of iron and carbon). Adding small amounts of carbon hardens many metals.

A crude but elegant and useful model for metallic bonding is to imagine the atoms losing their valence
electrons (those electrons from their incomplete outer electron shells) and forming a lattice of positively
charged ions, more-or-less fixed in place, surrounded by a sea of free or delocalised electrons, the shed
valency electrons, which are free to move about through the metal. You may picture this as ball-bearings in
thick treacle - the treacle representing the electron sea and the ball-bearings the ion cores of the atoms.
Metals
Metals come in a variety of colours and vary considerably in metallic lustre. Can you name a metal which matches
the colour of each of the balls?
Above: most elements of the periodic table are metals. We have the s-block metals (dark grey), the d-block and
transition metals (blue) and all the elements in green left of the staggered red line are metals, except for the
semi-metals already mentioned. The  f-block (lanthanides and actinides) and all the trans-uranic elements
(elements with an atomic number greater than 92) are also metals. Only those elements to the right of the red
line are non-metals.
Opposite charges attract: the sea of electrons are negatively charged, teh ions positively charged, and it is teh
attraction between the ions and the electrons which holds the atoms together.
Physical Properties of Metals

  • Conductivity: the delocalised electrons in metals are free to move within the metal and when electrodes are
    connected across a block of metal, these electrons move away from the negative electrode toward the
    positive electrode, carrying electric current with them. The electrons also conduct heat. Metals are good
    conductors of electricity and heat. Silver is the best electrically conducting metal.
  • Ductility and malleability: the delocalised electrons act like a ‘liquid glue’ and the metal atoms are free to
    move past one another without breaking the metal – metals are ductile (can be drawn into wires) and
    malleable (can be beaten into shape).
  • Hardness and density: metallic bonds are strong so metals are hard and most have high melting / boiling
    points. The atoms are closely packed (like oranges in a box) and so metals have high density. Osmium (Os)
    is the densest, lithium (Li) the least dense
  • Metallic lustre: the delocalised electrons are good at reflecting (or re-emitting) light that strikes the metal
    surface and so metals are shiny (lustrous).

Metal Crystals

Solid metals are crystalline, meaning that their atoms are held in well-defined 3D arrays. The basic atomic
structure of many metals can be understood by assuming the atoms to be spherical and that they will pack
together as closely as possible (pulled together by the metallic bonds). The diagram below illustrates the two close-
packed structures that result:
hexagonal close-packed (hcp) and cubic close-packed (ccp), ccp is also
called
face-centred cubic (fcc).
Metals can be pictured as: a lattice of fixed positive ions in a 'sea' of electrons.
One should not take these models too literally - we are still dealing with atoms and not true ions as such. A 6+ ion
is very unlikely to exist under normal conditions! However, the fact that tungsten (W) has 6 valence electrons
which can participate in metallic bonding accounts for the great hardness and very high melting temperature of
this metal, which has made it useful in light-bulbs (though less so these days). the table below illustrates the
trends:
Note that the addition of a second layer of close-packed spheres results in
the formation of 4-sided
tetrahedral spaces and 8-sided octahedral
spaces
. The third layer may then be added above the tetrahedral sites, that
is directly above the first/bottom layer, forming an hcp arrangement, or
directly above the octahedral sites, forming a fcc / ccp arrangement. In hcp,
the structure repeats every two layers (the ABABAB... structure) whilst in ccp
it repeats every three layers (ABCABCABC... structure). Click the image on
teh right to see the result without transparency of the layers applied. Note
that all the atoms are the same and the colour-coding merely indicates
different layers within the crystal.
Left: the ABAB... arrangement of
hexagonal close-packing, as seen in
metals like beryllium (Be),
magnesium (Mg), titanium (Ti),
cobalt (Co) and zinc (Zn).
Left: the ABCABC... arrangement of
cubic close-packing, as seen in
metals like calcium (Ca), copper
(Cu), silver (Ag), gold (Au), nickel
(Ni), platinum (Pt), lead (Pb) and
lanthanum (La).
Note that in the close-packed structures, the height between the centres of spheres stacked in adjacent layers is
about 0.816 of an atomic diameter, and not 1 diameter, since the layers sit in the depressions of the layer beneath
them, allowing the layers to pack closer together. The value of 0.816 can be derived from trigonometry (see
diagram below). Observe carefully that each atom (except those on the edges and faces) is in contact with 12
others - the
coordination number is 12 for both ccp and hcp. In these close-packed structures, the spheres
occupy 74% of the volume, the maximum for packed spheres.
The metalloid Pollonium (Po) is the only element known to crsytalise in a simple cubic (sc) lattice:
In real metals the structures given above are often not exact, since metal atoms are usually not perfect spheres and
varying degrees of deviation from these ideal structures occurs. For example in zinc and cadmium, the hcp lattice is
distorted: rather than each atom having 12 equidistant neighbours, each atom has 6 nearest neighbours in the same
plane, which is close-packed, and six neighbours, three from the layer above and three from the layer below, which are
10% more distant.

Allotropy in Metals

Many elements, including metals, crsytallise in one or more forms. Which form is stable depends on conditions
(temperature and pressure). For example, iron (Fe) will adopt the bcc form below 906 degrees C (alpha-iron),
switching to ccp/fcc between 906 and 1401 degrees (gamma-iron) and switching again to a bcc above 1401 degrees
(delta-iron) and then it melts at 1528 C.

Other metallic crystal structures

Some metals adopt a structure other than the four discussed above, especially when exhibiting non-metallic behaviour.
For example, the normal allotrope of tin, white tin or beta-tin, which is metallic and ccp, switches to a less-dense
non-metallic covalent form below 18 C, called grey or alpha-tin which has the tetrahedral structure of diamond.

Alloys

Alloys are solid solution mixtures of metals or of a metal and one or more non-metals. If the metals alloy then they will
mix when molten and crystallise together. Steel, for example, is an alloy of iron with small amounts of steel (and
frequently other additives like vanadium). Carbon atoms are much smaller than the iron atoms and in steel the carbon
atoms fill some of the octahedral sites between the iron atoms (causing a slight distortion to the iron lattice). Such an
alloy, in which one component occupies either the tetrahedral or octahedral spaces/interstices is called an
interstitial
alloy
. If the atoms of the alloy components are similar in size then some atoms of one metal may be replaced by atoms
of the other component without uses the interstices. Such an alloy is called a
substitution alloy. Brass is an example
of a substitution alloy, in which about one-third of the copper atoms are replaced by zinc. Alloying alters and often
'improves' the properties of the alloy. Steel is much harder and has a much higher tensile strength than iron. Brass is
much harder than copper and rings when struck (useful in making bells). Duraluminum (duralumin) is an alloy
containing about  4.4% copper, 1.5% magnesium, 0.6% manganese and 93.5% aluminium by mass. The lightness and
strength of duralumin has enabled its use in aircraft frames. The bonding in alloys is presumed to be metallic, and so
at least some alloys are more than simple physical solutions. Alloys may be stoichiometric (i.e. with the components
present in simple ratios) or non-stoichiometric (for example in an interstitial alloy all the octahedral sites could be filled,
giving a definite stoichiometry, otherwise a variable fraction may be filled). What type of bonding exists between iron
and carbon in steel? I am not sure anyone has the answer, but it is probably ionic.

Hydrides, nitrides and Carbides

Many metals react directly with hydrogen, forming hydrides, with nitrogen forming nitrides, and with carbon to form
carbides. Groups 1 and 2 of the Periodic Table (s-block metals) such Ca form ionic nitrides, carbides and hydrides,
whilst those of p-block metals tend to be covalent and those of d-block metals are metallic or interstitial alloys. A similar
trend is also seen in the metal borides.
Metal oxides are primarily ionic and contain oxide ions (with some degree of bond polarisation or covalent character).
This is to be expected, since oxygen is a highly electronegative element and metals are electropositive and so easily
donate electrons to oxygen. However, the oxides of less reactive or more noble metals, such as platinum (Pt) and gold
(Au), Pauling's electronegativity values predict covalent bonds. (For example, the electronegativity of Au is 2.4, that of
O 3.5, and 3.5 - 2.4 = 1.1, which gives a predicted 26% ionic character, or 74% covalent character, for a single Au-O
bond).
Metals react directly with a number of elements, including oxygen, nitrogen and carbon. First we shall look at the
reactions of metals with oxygen, to form oxides.

Metal Oxides and Hydroxides

Metal oxides are typically prepared by heating the metal in air. Most metal oxides and hydroxides are basic, and so
react with acids, but those of the metalloids and some other elements with semi-metallic character are
amphoteric
(meaning they can behave as a base or an acid depending on the reaction). Some examples of metal oxides and their
chemical reactions are given below:
We need to standardise things, so we have used 1 molar (1 M or 1 mol/L or 1 mol/dm3) solutions of metal ions and
we need a fixed temperature, say 298 K (25C). Notice how the equation for the cell is written, with zinc metal / zinc ion
half-cell on one side and the copper metal / copper ion half-cell on the other, divided by a line (|| in this case). By
convention the cell equation is written so that electrons flow from the left half-cell to the right half-cell. In this case the
electrode potential (E, the emf in Volts) will be positive, and in this case E = +1.10 V. If we had written the equation in
reverse, with the copper half-cell on the right, then we would have E = -1.10V, indicating that electrons flow from right
to left.

The voltage generated by this cell can be used to do useful work and several cells could be connected together in
series to form a power battery. The voltage is generated by the following chemical reactions:
Since, in this cell, zinc is doing the metal thing and losing its electrons (forcing copper to behave as a non-metal so
to speak) we can say that zinc is more metallic than copper, or less crudely, that
zinc is a more reactive metal
than copper
. Note also that zinc is oxidised and copper ions educed: oxidation is loss of electrons, reduction
is gain of electrons
(OIL RIG). Zinc has a higher oxidation potential (ability to become oxidised) than copper
(and zinc has a lower reduction potential).

Rather than considering every possible pairwise comparison between the metals, it is easier to compare each metal
in turn to a standard electrode or half-cell, and the one most commonly used is the
standard hydrogen electrode
(using hydrogen gas as the electrode, and acid as the electrolyte). We can then compare the
standard electrode
potentials
(or standard redox potentials) of the metals, on a scale in which the hydrogen half-cell is given a value of
zero volts (voltage is only relative, which is why a squirrel can walk alone an aerial power-line without frying!). This
then gives us a measure of the tendency of a metal to react like a metal, that is to lose electrons, and this gives us a
metal reactivity series. this series is, with the more reactive metals at the top, as follows:
Metal Reactivity series

React with water and acids

francium (Fr)       
caesium (Cs)                      
rubidium (Rb)
potassium (K)
sodium (Na)
lithium (Li)
barium (Ba)
strontium (Sr)
calcium (Ca)

React with dilute acids

magnesium (Mg)
aluminium (Al)
manganese (Mn)
zinc (Zn)
chromium (Cr)
iron (Fe)
cadmium (Cd)
cobolt (Co)
nickel (Ni)
tin (Sn)
lead (Pb)

Hydrogen (H2) goes here for comparison

Highly unreactive metals (may be oxidised by strong acids)

antimony (Sb)
bismuth (Bi)
copper (Cu)
mercury (Hg)
silver (Ag)
gold (Au)
platinum (Pt)
This reactivity series helps predict the reactions of metals with dilute acids. When a metal reacts with an acid, it is
essentially losing electrons to the hydrogen ions, e.g.
That is, those reactive metals (metals with oxidation potentials above hydrogen, that is with positive oxidation
potentials with hydrogen's potential set equal to zero) displace hydrogen from acids, producing hydrogen gas and a
metal slat of the acid (the salts formed above being iron(II) sulphate from sulphuric acid and iron(II) chloride from
hydrochloric acid). Unreactive metals may still react with certain acids, especially when those acids are concentrated,
for example copper will dissolve in concentrated nitric acid. However, in these reactions the acid is not really behaving
like an acid, but is behaving as an oxidising agent, oxidising the metal in a different type of reaction (see the coinage
metals below).

Q. Would you place an acidic drink in an aluminium can?

A. You might think not, after all aluminium reacts with acids, including say the phosphoric acid in certain fizzy drinks,
however, aluminium is much less reactive than expected because it reacts with air to form oxide which rapidly coats
aluminium in an unreactive layer of aluminium oxide. Many metals tarnish in air, by forming the oxide (or sometimes
peroxide or nitride) and these oxide layers can create barriers to reactions with the metal, though not all oxide layers
are as stubborn as that on aluminium.
Note that some expected displacement reactions do not happen! Sometimes a more reactive metal will not displace a
less reactive metal, perhaps because there is a barrier to the reaction, such as the presence of an oxide coat on the
metal (or perhaps some other condition creating unfavourable entropy). Displacement reactions can make beautiful
demonstrations.

s-block Metals

The s-block metals, those in Groups 1 and 2 of the Periodic Table are especially reactive. Consider, for example the
reactions of the Group 1 metals with water: a small piece of sodium fizzes, zipping around the surface of the water
evolving hydrogen; potassium will catch fire and burn as the hydrogen gas produced ignites (this makes a good lab
demonstration), whilst caesium explodes on contact with water!

Of the p-block metals, aluminium is also particularly reactive and its reaction with bromine makes an excellent lab
demonstration!

Links to Youtube videos: (to be added).
Below: a smaller fcc lattice demonstrating the meaning of face-centred cubic: the spheres form a cube with an extra
sphere in the middle of each face. In the bottom row the close-packed layers have been colour-coded as before:
The Reactivity of Metals

Metals typically react with non-metals by forming ionic bonds. This happens because metals are electropositive
(have low electronegativity) and non-metals are highly electrongeative.
Electronegativity is the ability of an atom to
draw electrons from a bond towards itself and when a highly electronegative element (with a large pull on the bonding
electrons) is bonded to an element with low electronegativity (and a weak hold on the bonding electrons) the more
electronegative element easily gains the electrons and the bond becomes ionic, that is electrons are transferred from
one atom to the other, the atoms become electrically charged ions held together by electrostatic attraction (Coulomb
force) - an ionic bond.

In reality, however, most bonds are partly ionic and partly covalent. Recall that a covalent bond results between two
atoms when they share a pair of bonding electrons. In a perfect covalent bond, such as when an element bonds to
itself covalently, e.g. C-C, the bond is completely covalent and not at all ionic (except even here external forces may
polarise the bond). Usually, however, one atom in the bond is more electronegative than the other and steals more
than its fair share of the electrons and the bond has a certain percentage of ionic character and is said to be
polarised. There then exists a whole spectrum of bonds from pure covalent to ionic (though a pure ionic bond can not
exist).

Metals can also form covalent bonds, sometimes with each other or with a non-metal of similar electronegativity. For
example, in NaCl, sodium has a very low electronegativity of 0.9, chlorine has a high electronegativity of 3.0, the
difference, 3.0 - 0.9 = 2.1, is large and the bond is ionic (2.1 gives a bond expected to be 67% ionic). Of course metals
also bond with themselves by forming metallic bonds (which can be thought of as special 'delocalised' covalent bonds
involving all the atoms in the crystal). Metallic bonds can also exist between different metals in alloys.

We can use electropositivity as one measure of metallicity (there are other measures) in that we can say the more
electropositive an element is, the more metallic it is. It is also generally true that more electropositive elements will be
more reactive, as they more easily lose electrons to form bonds. Another measure is the redox potential (reduction
potential or oxidation potential) which is the ability of an element to gain or lose electrons to another element when
both elements are present in solution. It can be measured by connecting two 'half-cells' together, in which each
half-cell is an electrode of the metal to be tested immersed in a solution of a salt of that metal, called the
electrolyte.
Two such half-cells make a complete
electrochemical cell. If the two metals and hence the two electrolytes are
different then they can be connected electrically by a salt bridge (a connecting tube of electrolyte, a glass tube usually
containing concentrated potassium chloride in agar jelly). A voltmeter connected across the two metal electrodes then
measures the
electromotive force (emf) of the cell, that is the driving force that causes electrons to flow around the
circuit from one electrode to the other. An example of an electrochemical cell is shown below:
Hydroxides are also either basic (containing hydroxide ions) or amphoteric:
Displacement Reactions

The reactivity series also helps predict certain other kinds of chemical reactions involving metals. In displacement
reactions a more reactive metals displaces a less reactive metal from one of its salts (akin to how a metal displaces
hydrogen from an acid), e.g.
Energy Band Theory (and why mercury is a liquid)

Quantum physics explains many of the properties of materials, including metals. It explains why copper and silver
are such good conductors of electricity (silver has the highest electrical conductivity of any element under normal
conditions) and why gold melts at a much lower temperature than tungsten.

If you are familiar with benzene in organic chemistry, or with conjugated organic molecules in which single and
double bonds alternate, then you may know that  some of the covalent bonding electrons in these molecules
delocalise, generating in benzene a delocalised ring of pi-electrons, a giant
molecular orbital that spans all 6 c
atoms and which can be pictured as a little electric circuit. In a metal a similar phenomenon happens: the atomic
orbitals of neighbouring atoms overlap and delocalise, forming a huge 'sea' of delocalised electrons over the
whole crystal. Some of the properties of the metal can then be explained by waves of electric charge in this
electron sea. In essence, all the atoms in a metal crystal become a single giant delocalised 'covalent' molecular
orbital, which we call metallic bonding. (If you need to then review atomic orbitals now, molecular orbitals will be
discussed in a future article).

In a metal, this atomic orbital overlap produces an
energy band of many closely-spaced overlapping atomic
orbitals. The energy band of greatest interest is that formed by overlapping atomic orbitals in the incomplete
outer shell of electrons, the
valence band. (Review atomic orbitals and shells). In nickel, for example, each Ni
atom has two 4s electrons in its valence shell and these are the electrons that contribute to metallic bonding. The
valence band of a nickel crystal therefore contains 2N states, where N is the number of atoms in the crystal, each
contributing two valence electrons. (Recall that an actual piece of metal consists of many microscopic metal
crystals packed together, but how the orbitals overlap between neighbouring crystals I do not know).

In a typical solid there are a number of energy bands separated in energy from one-another by large gaps,
called forbidden gaps, where electrons tend not to be found with that energy. Typically, each subshell forms a
separate energy band (though they may overlap to give hybrid bands). So, in nickel, the six 3p electrons overlap
to form a band with 6N quantum states, which is narrower than the 4s band and lies in energy below the 4s band.
The closeness of the energy levels within each band makes them essentially continuous, in the sense that
electrons can easily move up and down within an energy band, by gaining or losing small amounts of energy to
jump up or down an energy level.
Let us consider three special cases: caesium (Cs) and mercury (Hg) have the lowest melting points of all the
common metals. Cs melts at 28.7 C, so it would melt in your hand, except i would not recommend trying this as it
will react violently with your moist skin and burn you! However, a glass phial of Cs can be used to demonstrate
this effect! Hg is, of course, a liquid at room temperature, the only metal that is *assuming a temperate room
temperature of 25 C. (Francium probably melts at around 27 C, but this metal is very hard and expensive to
obtain!). Let us compare these to the highest melting metal, tungsten (W) with a melting point of 3410 C. The
electron configurations and energy bands of these metals are shown below. Note that for each the valence
electrons come from the 6s and 5d sub-shells (though in Hg the 5d shell is full and so not strictly a valency
sub-shell). These sub-shells are so close in energy that they overlap to form a single hybrid valence energy
band.

In caesium the band is almost energy and since the electrons tend to occupy the lowest energy levels, this
means that caesium is in a lower energy state when it forms molecular bands than in the original atomic state.
This energy drop, in forming molecular bands, is the cohesive or bonding energy in the metallic bonds between
the caesium atoms. However, only one electron benefits from this energy drop and so the total cohesive energy
is low.

In tungsten, all six valence electrons are at a lower energy in the molecular band than in atomic orbitals and so all
six contribute to bonding and the cohesive energy, giving tungsten a very high melting point.

In mercury, the valence band is full and just as many electrons have gained energy as those that have lost
energy in forming molecular orbitals, meaning that the cohesive energy is near zero! mercury does not form
strong bonds and so exists as a liquid at room temperature! (Note, there are other explanations given as to why
mercury is a liquid at room temperature, but some of these are erroneous and others are jusrt simplifications, the
explanation given here is the best I have seen).
Above: energy bands, the darker shaded regions show the levels occupied by electrons. Note that when
separate, the energies of equivalent atomic orbitals on the different atoms are the same - the energy level is
degenerate (has the same energy) across our population of atoms. When atoms come together to form
molecules, this degeneracy is removed, as some of the energy levels drop in energy, whilst others increase in
energy by the same amount. This lowering of energy is the basis of attractive forces between particles in
physics. What we have done is form molecular orbitals across the entire metal crystal!

Why does mercury not use the next vacant subshell (6p) and promote electrons into it? The 6s subshell is
significantly lowered in energy, which is why it forms a band with the 5d subshell. This lowering is due to a
relativistic correction to the energy level. (Schrodinger's wave-equation is non-relativistic, so we have to add
relativistic corrections back). This correction causes a lowering in energy of the s-orbitals, such that the 6s has a
similar energy to the 5d. This increases the jump between the 6s and 6p orbitals. Electrons need energy to make
that jump and possibly do so when mercury conducts electricity (explaining its electrical conductivity) but it will
not adopt this high-energy state as a matter of choice, under normal conditions, and so will not form metallic
bonds by using the 6p subshell. Note, that the relativistic shift is not sufficient by itself to explain the liquid state
of mercury, we need band theory too.

A note of caution should be added here. The relativistic correction, due, at least conceptually, to relativistic
speeds of the electron does not mean that 6s electrons are orbiting around the nucleus very fast! The 6s is an
eigenstate or stationary state and the quantum mechanics predict NO time-evolution of this state! (Other than a
wavefunction phase change which has no observable consequence). This means that electrons in any one
subshell, s,p,d, etc. do NOT orbit the nucleus since they do NOT follow classical trajectories according to
quantum theory! This point is subtle and very hard to understand! Electrons in atoms can only follow a trajectory
if we allow 'hidden variables', or variables that exist but can not be measured directly or observed. It may that
such variables do exist, however, current experimental evidence (of a subtle nature we may discuss in a
separate article) suggests that there are probably no hidden variables! An electron can move if its state is not an
eigenstate, such as a superposition of an s and a p-state, however, once a measurement is performed it must
collapse into an eigenstate - we can never observe the electron in an intermediate state, so it can never interact
with energy/matter and remain in such a state. (Hybrid orbitals and molecular orbitals are another matter).
Further, our measurements of position and velocity, etc. of the electron, are the values the electron has AFTER
the measurement - the measurement itself perturbs and changes the system!

Energy band theory also explains why electrons in metals conduct electricity and heat so well. The electrons are
free to move up and down the various energy levels within an energy band, since the levels are closely spaced,
carrying energy with them around the crystal. As electrons move around the crystal, the 'ions' in our metal lattice
must vibrate or oscillate about their fixed positions, so as to conserve total angular momentum. these oscillations
are thus due to the thermal energy of the crystal.
Another common metal structure seen is body-centred cubic (bcc). In this case the coordination number is 8 and
the packing is not close, with only 68% of the volume being occupied by the spheres. this structure is seen in metals
such as sodium (Na) and the other Group I alkali metals, vanadium (V), chromium (Cr), tungsten (W), yttrium (Y)
and iron (Fe). This arrangement gives these metals lower density than they would have if they were close-packed.
Above: finding the height between (the close-packed) layers of close-packed spheres. A) consider 4 spheres, 3
from the bottom layer and one above - these form a tetrahedron (of 4 equilateral triangles). B) The height of the
tetrahedron is equal to the distance between the centres of the spheres in adjacent layers, which is what we want
to find. To find the height we need to find length
a on the bottom triangle of the tetrahedron. C) We do this using
the circumcircle around the equilateral triangle, whose radius is also a and then use the formula for the length of a
chord of a segment of a circle. D) using our value of length
a we can now find the height of the tetrahedron by
Pythagoras' Theorem. Note: d is the diameter of the spheres (distance between adjacent spheres within a layer).

s block metals

Colourful complexes

f-block metals: lanthanides, actinides, uranium and the trans-uranics

Metal halides and sulphides

Molecular bonding and molecular orbitals
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