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Materials science and elasticity
Dr. Jay Maron


Tensile stress

For a rod under tension,

Rod cross-sectional area = A         meter2
Tensile force            = F         Meter
Tensile stress           = P = F/A   Pascal

Strain

Tensile strain is stretching, and is measured as a fractional change in length. For a rod under strain,

Rod length            =  X         Meter
Rod change in length  =  x         Meter
Strain                =  S  = x/X  Dimensionless

Tensile modulus

The tensile modulus is resistance to strain, and can be thought of as stiffness. This is a form of Hooke's law. For a rod under stress,

Stress on the rod     = P        Pascals
Rod tensile modulus   = T        Pascals
Rod strain            = S = P/T  Dimensionless


Tensile strength

A material's "tensile strength" is the maximum tensile stress it can take before breaking.

Tensile yield strength =  tbreak              Newton/meter2   (Pascals)
Cross-sectional area   =  A                  meter2
Breaking force         =  Fbreak = tbreak A    Newton

Yield strength

A material's "tensile yield strength" is the maximum stress it can take before deforming irreversibly. For metals, the yield strength is usually 3/4 of the tensile strength, and for wood, the yield strength is only slightly less than the tensile strength.

For engineering, we focus on yield strength rather than tensile strength.


Elasticity data


Yield strain

A material's "tensile yield strain" is the maximum strain it can take before yielding. For metals, the value varies widely.

Alloying a metal doesn't change the tensile modulus, but it improves yield strain.

Yield strength    =  t          Pascals
Tensile modulus   =  T          Pascals
Yield strain      =  s  =  t/T  Dimensionless


Variables

Density                =  ρ          kg/meter2

Cross-sectional area   =  A          meter2
Force                  =  F          Newton
Tensile stress         =  P  =  F/A  Pascals

Tensile modulus        =  T          Pascals
Tensile yield stress   =  t          Pascals
Tensile breaking stress=  tbreak      Pascals

Strain                 =  S  =  P/T  Dimensionless
Tensile yield strain   =  s  =  t/T  Dimensionless

Tensile energy/volume  =  evol =  ½ T S2     Joule/meter3
Tensile energy/mass    =  e   =  ½ T S2/ρ    Joule/kg

Textbook on elasticity


Strong metals

The table shows the "strong metals", the metals with good strength/density.


Modulus/Density

For bridges, what counts is strength/density.

For wood and for the strong metals,

Tensile modulus / Density  ~  25 MJoules/kg

Beryllium is an exception, with a value of 137 MJoules/kg.

The plot shows the tensile modulus divided by density.


Energy/mass

For many applications, the measure of the quality of a material is the elastic energy/mass it can take before yielding. For this, a material should have a large tensile modulus and a large yield strain.

Tensile modulus        =  T
Tensile yield strength =  t
Tensile yield strain   =  s
Density                =  ρ
Tensile energy/volume  =  evol =  ½ T s2       Joule/meter3
Tensile energy/mass    =  e    =  ½ T s2/ρ     Joule/kg

The best materials are polymers such as Kevlar.

Alloys outperform pure metals. The best alloy is titanium alloy.


Deformation

The types of deformation are tension, shear, and bulk compression.

Tensile
Shear
Bulk compression


Poisson ratio

A wire shortens when stretched and widens when compressed.

dX            =  Fractional increase in length of the wire
dY            =  Fractional decrease in diameter of the wire
PoissonRatio  =  dY / dX

For an isotropic material, the tensile, shear, and bulk moduli are related through the dimensionless Poisson ratio.

Tensile modulus        =  T          Pascals
Shear modulus          =  Tshear     Pascals
Bulk modulus           =  Tbulk      Pascals
Poisson ratio          =  U          dimensionless

T  =  2 (1 + U) Tshear  =  3 (1 - 2U) Tbulk

Beams

For beams, the types of stresses are:

Tension
Bending
Compression


Tension

Tensile strength relates to the strength of wires.

Two vices pull on a wire

Tensile yield strength =  t          Newton/meter2   (Pascals)
Cross-sectional area   =  A          meter2
Yield force            =  F  = tA    Newton

Bending

For bending, the yield force of a beam is determined by the shear yield strength.

Beam length            =  X   meter
Beam width             =  Y   meter
Beam height            =  Z   meter          The force is in the Z direction
Beam yield force       =  f   Newton
Tensile yield strength =  t   Pascal
Poisson ratio          =  U   dimensionless
Shear yield strength   =  tshear = ½ t / (1+U)   Pascal
Bending yield force    =  f  =  ⅔ tshear Y Z2 / X

Z matters more than Y. If you have a beam with a 2x4 cross section, it's best to align the beam so that Z=4 and Y=2, rather than with Z=2 and Y=4.


Column crushing

Short columns fail by crushing and long columns fail by buckling.

Crushing strength is determined by bulk yield strength.

Bulk yield strength  =  tbulk        Newton/meter2   (Pascals)
Cross-sectional area =  A            meter2
Crushing force       =  F  = bbulkA  Newton

Column buckling

Long columns fail by buckling, and strength is determined by tensile yield strength.

For a column that is cylindrical and hollow,

Column length          =  L
Column outer radius    =  R
Column inner radius    =  r
Column boundary factor =  K        dimensionless
                       =   .5      if both ends are fixed
                       =  2        if one end is fixed and the other end is free to move laterally
                       =  1        if both ends are pinned  (hinged and free to rotate)
                       =   .699    if one end is fixed and the other is pinned
Tensile yield strength =  t
Buckling force         =  f  =  ½ π3 t (R4-r4) / (K L)2

Buckling threshold

If a column's buckling limit is equal to its squashing limit, and if r=0,

R/L  =  (K/π) (tbulk/t)1/2

Beam scalings

Beam length            = L
Beam radius            = R
Beam tension strength  ~  R2
Beam bending strength  ~  R3/L
Beam buckling strength ~  R4/L2

Beams and density

For beams and columns, the lower the density, the better.

For a square beam with Y=Z,

Density           =  ρ
Mass              =  M  =  X Y Z ρ
Beam yield force  =  F  =  ⅔ s Y Z2 / X  =  ⅔ S M3/2 ρ-3/2 / X5/2

At fixed length and mass, the measure of quality is t/ρ3/2.


Columns and density

For a cylindrical column,

Density               =  ρ
Mass                  =  M  =  π R2 L ρ
Column buckling force =  F  =  ½ π3 t R4 / (Q L)2  =  ½ π3 t M2 ρ-2 / (Q2 L4)

At fixed length and mass, the measure of quality is t/ρ2.


Quality

The measure of merit depends on the application. If force/mass is what counts, then the measure of merit is

Tensile yield strength / Density         Beam under tension
Tensile yield strength / Density3/2      Beam under shear
Tensile yield strength / Density2        Beam under compression

Energy/Mass

For many applications, the measure of merit for a material is energy/mass, where "energy" is the maximum elastic energy the material can take before breaking. This applies to things like racquets, aircraft, and swords. The cases are:

Case            Measure of merit

Tension         Energy / Mass
Shear           Energy / Mass / Density1/2
Compression     Energy / Mass / Density

Strong woods

The strongest woods are:

              Density   Tensile   Tensile  Hardness
                        strength  modulus
              gram/cm3  GPascal   GPascal  kNewton

 Balsa            .12    .020     3.7      .31
 Cedar, white     .32    .046     5.7
 Cedar, red       .34    .054     8.2
 Pine, white      .37    .063     9.0     1.9
 Spruce, red      .41    .072    11.7
 Redwood          .44    .076     9.6
 Ash, black       .53    .090    11.3
 Walnut, black    .56    .104    11.8     4.5
 Ash, white       .64    .110    12.5     5.9
 Mahogany         .67    .124    10.8
 Locust, black    .71    .136    14.5     7.6
 Hickory          .81    .144    15.2
 Bamboo           .85    .15     20.0     7.2
 Oak, live        .98    .13     13.8
 Ironwood        1.1     .181    21.0    14.5
 Verawood        1.19    .178    15.7    16.5
 Quebracho       1.24    .14     16.6    20.3
 Lignum vitae    1.26    .127    14.1    19.5
 Ironwood, black 1.36    .125    20.5    16.3

Wood Strength/Density

Tensile yield strength  =  t   Pascal
Density                 =  ρ   kg/meter3

For most woods, t/ρ has a similar value. For t/ρ2, balsa wins. We plot t/ρ, t/ρ3/2, and t/ρ2.


Wood Energy/Density


Materials

To compare wood to other materials,


Wood grain

For a vertical tree trunk, "longitudinal" is the vertical direction. "Radial" is the direction from the tree center axis, going outward in the horizonal plane. "Tangential" is the direction along a tree ring, in the horizontal plane.

Poisson numbers:

                 Longitudinal  Radial  Tangential

Wood, low density    .4         .25       .2
Wood, high density   .43        .35       .18

The strongest direction is the longitudinal direction and the weakest direction is the radial direction. For a beam under bending stress, you should align the longitudinal grain with the long axis of the beam, and you should align the tangential grain with the direction of the force.


Bridge design

Truss

Brown truss
Pratt truss
Pratt truss
Howe truss
Bowstring truss

A hollow beam is weaker than a solid beam, but it has a better stength/mass ratio. This is the point of a truss. A truss consists of a set of upper and lower beams connected by struts. The struts deliver forces between the beams and they resist warp. Struts are arranged as triangles because triangles resist warp better than squares.

In a Pratt truss, diagonal beams are under tension and vertical beams are under compression.

A Howe truss is like a Pratt truss except that the diagonals slant the opposite way. In a Howe truss, diagonal beams are under compression and vertical beams are under tension.

Beams under compression should be wider than beams under tension. Compression is harder than tension.

Bridges from centuries ago tended to use wood for compression elements and steel rods for tension elements.


Triangles

Triangles are stronger than squares. A structure needs triangles to resist warping.


3D truss

Diamond lattice
Diamond lattice


Suspension bridge

Tension is easy. If you can use pure tension, do it.


Catenary

Roman bridge

A cable hangs as a catenary, and the ideal form for an arch is a catenary.

A hanging catenary transforms the load into pure tension.

An arch transforms the load into pure compression.

For a suspension bridge supporting a road, if the cable is heavier than the road, then the cable hangs as a catenary. If the road is heavier than the cable, the cable hangs as a parabola.


Tower

If you want height, use a concave catenary. If you want volume, use a convex catenary.


Arch bridge

The arch can go above or below.


Arch and truss

You can combine an arch and a truss.


Cantor design


Compressive strength

Tension
Compression

Concrete and ceramics typically have much higher compressive strengths than tensile strengths. Concrete is typically mixed with steel bars to improve tensile strength.


Hardness

Brinell hardness test
Vickers hardness test

Diamond indenter for a Vickers hardness test
Indentation left in steel by a diamond indenter

Brinell  =  A measure of a material's resistance to dents, measured in Pascals
Mohs     =  A dimensionless measure of a material's resistance to dents
The Mohs scale of mineral hardness reflect's a material's ability to resist scratching. If two materials are scraped together then the material with the lower Mohs value will be scratched more. Diamond has the largest Mohs value of any material.
                Mohs

Diamond          10
RhB2              9.5
Silicon carbide   9.5
Corundum          9
Tungsten carbide  9
Chromium          8.5
Emerald           8
Topaz             8
Tungsten          8
Hardened steel    8
Quartz            7
Osmium            7
Rhenium           7
Vanadium          7

If a material has a large Brinell hardness then it has a large Mohs hardness. The reverse is not necessarily true. Materials exist with a large Mohs hardness and a small Brinell hardness.



The Brinell hardness is related to the tensile modulus and tensile strength.





Elements


Size

Dot size  =  Atomic radius
          =  (AtomicMass / Density)**(1/3)
For gases, the density at boiling point is used.

Size data


Wave speed

Pressure wave
Shear wave
Rayleigh surface wave

Wave type            Wave speed squared

Sound in air         1.4*Pressure / Density
String wave          Tension      / Density / CrossSection
Longitudinal wave    BulkModulus  / Density
Shear wave           ShearModulus / Density
Torsion wave         ShearModulus / Density
Deep water wave      GravityConstant * Wavelength / (2 Pi)    (If Depth > .5 Wavelength)
hallow water wave    GravityConstant * Depth                  (If Depth < .5 Wavelength)

GravityConstant = 9.8 meters/second^2

In rock, pressure waves propagate at ~ 5 km/s and shear waves propagate at ~ 3 km/s. You can estimate the distance from the center of an earthquake by measuring the time difference between the arrival of the pressure and shear waves.

If the wave speed doesn't depend on the frequency then waves propagate without changing shape (without dispersion). This applies to all waves except for deep water waves.

           Bulk      Density   Sound speed
          modulus    (g/cm^3)    (km/s)
           (GPa)
Air           .00014   .0012      .34
Water        2.2      1.0        1.43
Wood        13        1          3.6
Granite    100        2.75       6.0
Steel      170        7.9        6.1
Diamond    443        3.5       12.0
Beryllium  130        1.85      12.9         Fastest sound speed of any material

Mohs hardness

Carbide

Carbides are the hardest metallic materials.

10     Diamond
 9.5   BN, B4C, B, TiB2, ReB2
 9.25  TiC, SiC
 9.0   Corundum, WC, TiN
 8.5   Cr, TaC, Si3N4
 8     Topaz, Cubic zirconia
 7.5   Hardened steel, tungsten, emerald, spinel
 7     Osmium, Rhenium, Vanadium, Quartz

Vickers hardness
                       Min   Max

Valence compounds     1000  4000     carbides, borides, silicides
Intermetallic          650  1300
BCC lattice            300   700
FCC lattice            100   300

Ancient metallurgy

Stone
Copper
Bronze
Iron
Carbon

The earliest metals were gold and silver, the only ones that occur naturally in pure form. Iron can occasionally be found as iron meteorites.

Gold nugget
Silver nugget
Iron meteorite

Copper was discovered around 7000 BCE by smelting copper minerals in a wood fire. Around 3200 BCE it was found that copper is strenghened by tin, and this is called bronze. Around 2000 BCE it was found that copper is also strengthed by zinc, and this is called brass.

The earliest metals were smeltable with a wood fire and they consist of copper, lead, silver, tin, zinc, and mercury. They come from the following minerals:

Gold and silver were known since antiquity, but gold mining didn't start until 6000 BC, and silver smelting didn't start until 4000 BC.

The minerals that were used by ancient civilizations to smelt metal are:

Lead. Galena. PbS
Copper. Chalcocite. Cu2S
Silver. Acanthite. Ag2S
Tin. Cassiterite. SnO2
Zinc. Sphalerite. ZnS
Mercury. Cinnabar. HgS

The next metal to be discovered was iron (c. 1200 BC), which requires a bellows-fed coal fire to smelt.

Iron. Hematite. Fe2O3
Iron. Pyrite. FeS2

No new metals were discovered until cobalt in 1735. Once cobalt was discovered, it was realized that new minerals may have new metals, and the race was on to find new minerals. This yielded nickel, chromium, manganese, molybdenum, and tungsten.

Cobalt. Cobaltite. CoAsS
Nickel. Millerite. NiS
Chromium. Chromite. FeCr2O4
Manganese. Pyrolusite. MnO2
Molybdenum. Molybdenite. MoS2
Tungsten. Wolframite. FeWO4

Chromium is lighter and stronger than steel and it was discovered in 1797. It satisfies the properties of "Valyrian steel" from Game of Thrones. There's no reason chromium couldn't have been discovered earlier.

Coal smelting can't produce the metals lighter than chromium. For these you need electrolysis. The battery was invented in 1799, enabling electrolysis, and the lighter metals were discovered shortly after. These include aluminum, magnesium, titanium, and beryllium.

Aluminum. Bauxite. Al(OH)3 and AlO(OH)
Mangesium. Magnesite. MgCO3
Titanium. Rutile. TiO2
Beryllium. Beryl. Be3Al2(SiO3)6

Carbon fiber eclipses metals. The present age could be called the carbon age. The carbon age became mature in 1987 when Jimmy Connors switched from a wood to a carbon racket.

The plot shows the strength of materials.

Alloys can be much stronger than pure metals.

Wood rivals alloys for strength.


Currency

Gold was the densest element known until the discovery of platinun in 1735. It was useful as an uncounterfeitable currency until the discovery of tungsten in 1783, which has the same density as gold. Today, we could use iridium, platinum, or rhenium as an uncounterfeitable currency.


Modern chemistry and the discovery of elements

Prior to 1800, metals were obtained by smelting minerals, and the known metals were gold, silver, copper, iron, tin, zinc, mercury, cobalt, manganese, chromium, molybdenum, and tungsten. Elements to the left of chromium titanium and scandium cant's be obtained by smelting, and neither can aluminum, magnesium, and beryllium. They require electrolysis, which was enabled by Volta's invention of the battery in 1799.

Prior to 1800, few elements were known in pure form. Electrolyis enabled the isolation of most of the rest of the elements. The periodic table then became obvious and was discovered by Mendeleev 1871. The battery launched modern chemistry, and the battery could potentially have been invented much earlier.

Electrolysis enabled the isolation of sodium and potassium in 1807, and these were used to smelt metals that can't be smelted with carbon.

         Discovery   Method of             Source
          (year)     discovery

Carbon     Ancient   Naturally occuring
Gold       Ancient   Naturally occuring
Silver     Ancient   Naturally occuring
Sulfur     Ancient   Naturally occuring
Lead         -6500   Smelt with carbon     Galena       PbS
Copper       -5000   Smelt with carbon     Chalcocite   Cu2S
Bronze (As)  -4200   Copper + Arsenic      Realgar      As4S4
Tin          -3200   Smelt with carbon     Calamine     ZnCO3
Bronze (Sn)  -3200   Copper + Tin
Brass        -2000   Copper + Zinc         Sphalerite   ZnS
Mercury      -2000   Heat the sulfide      Cinnabar     HgS
Iron         -1200   Smelt with carbon     Hematite     Fe2O3
Arsenic       1250   Heat the sulfide      Orpiment     As2S3
Zinc          1300   Smelt with wool       Calamine     ZnCO3 (smithsonite) & Zn4Si2O7(OH)2·H2O (hemimorphite)
Antimony      1540   Smelt with iron       Stibnite     Sb2S3
Phosphorus    1669   Heat NaPO3 Excrement
Cobalt        1735   Smelt with carbon     Cobaltite    CoAsS
Platinum      1735   Naturally occuring
Nickel        1751   Smelt with carbon     Nickeline    NiAs
Bismuth       1753   Isolated from lead
Hydrogen      1766   Hot iron + steam      Water
Oxygen        1771   Heat HgO
Nitrogen      1772   Isolated from air
Manganese     1774   Smelt with carbon     Pyrolusite   MnO2
Molybdenum    1781   Smelt with carbon     Molybdenite  MoS2
Tungsten      1783   Smelt with carbon     Wolframite   (Fe,Mn)WO4
Chromium      1797   Smelt with carbon     Crocoite     PbCrO4
Palladium     1802   Isolated from Pt
Osmium        1803   Isolated from Pt
Iridium       1803   Isolated from Pt
Rhodium       1804   Isolated from Pt
Sodium        1807   Electrolysis
Potassium     1807   Electrolysis
Magnesium     1808   Electrolysis          Magnesia     MgCO3
Cadmium       1817   Isolated from zinc
Lithium       1821   Electrolysis of LiO2  Petalite     LiAlSi4O10
Zirconium     1824   Smelt with potassium  Zircon       ZrSiO4
Aluminum      1827   Smelt with potassium
Silicon       1823   Smelt with potassium
Beryllium     1828   Smelt with potassium  Beryl        Be3Al2Si6O18
Thorium       1929   Smelt with potassium  Gadolinite   (Ce,La,Nd,Y)2FeBe2Si2O10
Vanadium      1831   Smelt VCl2 with H2    Vanadinite   Pb5(VO4)3Cl
Uranium       1841   Smelt with potassium  Uranite      UO2
Ruthenium     1844   Isolated from Pt
Tantalum      1864   Smelt with hydrogen   Tantalite    [(Fe,Mn)Ta2O6]
Niobium       1864   Smelt with hydrogen   Tantalite    [(Fe,Mn)Ta2O6]
Fluorine      1886   Electrolysis
Helium        1895   From uranium ore
Titanium      1910   Smelt with sodium     Ilmenite     FeTiO3
Hafnium       1924   Isolated from zirconium
Rhenium       1928   Isolated from Pt
Scandium      1937   Electrolysis          Gadolinite   FeTiO3

History of mineralogy

 -384  -322   Aristotle. Wrote "Meteorology"
 -370  -285   Theophrastus. Wrote "De Mineralibus"
         77   Pliny the Elder publishes "Natural History"
  973  1050   Al Biruni. Published "Gems"
       1546   Georgius Agricola publishes "On the Nature of Rocks"
       1556   Georgius Agricola publishes "On Metals"
       1609   de Boodt publishes a catalog of minerals
       1669   Brand: Discovery of phosphorus
       1714   John Woodward publishes "Naturalis historia telluris illustrata & aucta", a mineral catalog
       1735   Brandt: Discovery of cobalt
       1777   Lavoisier: Discovery of sulfur
       1778   Lavoisier: Discovery of oxygen and prediction of silicon
       1783   Lavoisier: Discovery of hydrogen
       1784   T. Olof Bergman publishes "Manuel du mineralogiste, ou sciagraphie du regne mineral",
              and founds analytical chemistry
       1778   Lavoisier: Discovery of oxygen
       1801   Rene Just Huay publishes "Traite de Mineralogie", founding crystallography
       1811   Avogadro publishes "Avogadro's law"
       1860   The Karlsruhe Congress publishes a table of atomic weights
       1869   Mendeleev publishes the periodic table

Metals known since antiquity

For a metal, the stiffness is characterized by the "shear strength" and the sword worthiness is characterized by the shear strength over the density (the "strength to weight ratio"). For example for iron,

Shear modulus    =  S         =   82 GJoules/meter3
Density          =  D         = 7900 kg/meter3
Sword worthiness =  Q  = S/D  = 10.4 MJoules/kg

Metals

This plot includes all metals with a strength/density at least as large as lead, plus mercury. Beryllium is beyond the top of the plot.


Wootz steel

-600  Wootz steel developed in India and is renowned as the finest steel in the world.
1700  The technique for making Wootz steel is lost.
1790  Wootz steel begins to be studied by the British Royal Society.
1838  Anosov replicates Wootz steel.
Wootz steel is a mix of two phases: martensite (crystalline iron with .5% carbon), and cementite (iron carbide, Fe, 6.7% carbon).

Iron meteorites

In prehistoric times iron meteorites were the only source of metallic iron. They consist of 90% iron and 10% nickel.


Metal smelting

Prehistoric-style smelter

Most metals are in oxidized form. The only metals that can be found in pure form are gold, silver, copper, platinum, palladium, osmium, and iridium.

Smelting is a process for removing the oxygen to produce pure metal. The ore is heated in a coal furnace and the carbon seizes the oxygen from the metal. For copper,

Cu2O + C  →  2 Cu + CO
At low temperature copper stays in the form of Cu2O and at high temperature it gives the oxygen to carbon and becomes pure copper.

For iron, the oxidation state is reduced in 3 stages until the pure iron is left behind.

3 Fe2O3 + C  →  2 Fe3O4 + CO
Fe3O4   + C  →  3 FeO   + CO
FeO     + C  →    Fe   + CO
Oxidation state  =  Number of electrons each iron atom gives to oxygen

       Oxidation state
CuO          2
Cu2O         1
Cu           0
Fe2O3        3
Fe3O4       8/3
FeO          2
Fe           0

Smelting temperature

The following table gives the temperature required to smelt each element with carbon.

        Smelt  Method  Year  Abundance
         (C)                   (ppm)

Gold        <0   *   Ancient      .0031
Silver      <0   *   Ancient      .08
Platinum    <0   *    1735        .0037
Mercury     <0  heat -2000        .067
Palladium   <0  chem  1802        .0063
Copper      80   C   -5000      68
Sulfur     200   *   Ancient   420
Lead       350   C   -6500      10
Nickel     500   C    1751      90
Cadmium    500   C    1817        .15
Cobalt     525   ?    1735      30
Tin        725   C   -3200       2.2
Iron       750   C   -1000   63000
Phosphorus 750  heat  1669   10000
Tungsten   850   C    1783    1100
Potassium  850   e-   1807   15000
Zinc       975   C    1746      79
Sodium    1000   e-   1807   23000
Chromium  1250   C    1797     140
Niobium   1300   H    1864      17
Manganese 1450   C    1774    1120
Vanadium  1550   ?    1831     190
Silicon   1575   K    1823  270000
Titanium  1650   Na   1910   66000
Magnesium 1875   e-   1808   29000
Lithium   1900   e-   1821      17
Aluminum  2000   K    1827   82000
Uranium   2000   K    1841       1.8
Beryllium 2350   K    1828       1.9

Smelt:      Temperature required to smelt with carbon
Method:     Method used to purify the metal when it was first discovered
            *:  The element occurs in its pure form naturally
            C:  Smelt with carbon
            K:  Smelt with potassium
            Na: Smelt with sodium
            H:  Smelt with hydrogen
            e-: Electrolysis
            heat:  Heat causes the oxide to decompose into pure metal. No carbon required.
            chem:  Chemical separation
Discovery:  Year the element was first obtained in pure form
Abundance:  Abundance in the Earth's crust in parts per million
Elements with a low carbon smelting temperature were discovered in ancient times unless the element was rare. Cobalt was discovered in 1735, the first new metal since antiquity, and this inspired scientists to smelt every known mineral in the hope that it would yield a new metal. By 1800 all the rare elements that were carbon smeltable were discovered.

The farther to the right on the periodic table, the lower the smelting temperature, a consequence of "electronegativity".

The battery was invented in 1800, launching the field of electrochemistry and enabling the the isolation of non-carbon-smeltable elements. Davy used electrolysis in 1807 to isolate sodium and potassium and then he used these metals to smelt other metals. To smelt beryllium with potassium, BeO + 2 K ↔ Be + K2O.

Titanium can't be carbon smelted because it forms the carbide Ti3C.

Data

For an expanded discussion of smelting physics, see jaymaron.com/metallurgy.html.


Thermite

Thermite is smelting with aluminum. For example, to smelt iron with aluminum,

Fe2O3 + 2 Al  →  2 Fe + Al2O3

Smelting reactions

The following table shows reactions that change the oxidation state of a metal. "M" stands for an arbitrary metal and the magnitudes are scaled to one mole of O2. The last two columns give the oxidation state of the metal on the left and right side of the reaction. An oxidation state of "0" is the pure metal and "M2O" has an oxidation state of "1".

                            Oxidation state   Oxidation state
                                at left          at right
 2  M2O   ↔  4  M     + O2        1                0
 4  MO    ↔  2  M2O   + O2        2                1
 2  M3O4  ↔  6  MO    + O2       8/3               2
 6  M2O3  ↔  4  M3O4  + O2        3               8/3
 2  M2O3  ↔  4  MO    + O2        3                2
 2  MO    ↔  2  M     + O2        2                0
2/3 M2O3  ↔ 4/3 M     + O2        3                0
 1  MO2   ↔  1  M     + O2        4                0
 2  MO2   ↔  2  MO    + O2        4                2

Minerals

These elements are not necessarily on the Science Olympiad list.

We list minerals by element, with the most abundant mineral for each element listed first.

Lithium

Spodumene: LiAl(SiO3)2
Stilbite: LiAlSi2O6
Tourmaline: (Ca,Na,K,)(Li,Mg,Fe+2,Fe+3,Mn+2,Al,Cr+3,V+3)3(Mg,Al,Fe+3,V+3,Cr+3)6((Si,Al,B)6O18)(BO3)3(OH,O)3(OH,F,O)

Beryllium

Beryl: Be3Al2(SiO3)6
Morganite: Be3Al2(SiO3)6
Emerald

Carbon

Diamond: C

Sodium

Halite: NaCl

Magnesium

Periclase: MgO
Magnesite: MgCO3
Dolomite: CaMg(CO3)2
Peridot: (Mg,Fe)2SiO4
Spinel: MgAl2O4
Spinel: MgAl2O4

Aluminum

Bauxite: Al(OH)3 and AlO(OH)
Alumstone: KAl3(SO4)2(OH)6
Muscovite mica: KAl2(AlSi3O10)(F,OH)2 or KF2(Al2O3)3(SiO2)6(H2O)
Corundum: Al2O3
Topaz: Al2SiO4(F,OH)2

Epidote: Ca2(Al2,Fe)(SiO4)(Si2O7)O(OH)
Jadeite: NaAlSi2O6
Albite: NaAlSi3O8
Amazonite: KAlSi3O8
Labradorite: (Na,Ca)(Al,Si)4O8

Silicon

Amethyst: SiO2
Quartz: SiO2
Citrine: SiO2
Opal: SiO2·nH2O
Agate: SiO2

Sulfur

Volcanic sulfur

Calcium

Fluorite: CaF2
Calcite: CaCO3
Satin Spar: CaSO4 · 2H2O
Selenite: CaSO4 · 2H2O
Aragonite: CaCO3
Pearl: CaCO3
Calcite: CaCO3

Titanium, vanadium, chomium, and manganese

Rutile: TiO2
Vanadinite: Pb5(VO4)3Cl
Chromite: FeCr2O4
Pyrolusite: MnO2
Rhodonite: MnSiO3
Rhodochrosite: MnCO3

Iron

Hematite: Fe2O3
Hematite: Fe2O3
Pyrite: FeS2
Iron meteorite
Goethite: FeO(OH)

Cobalt and nickel

Cobaltite: CoAsS
Millerite: NiS

Copper

Chalcocite: Cu2S
Chalcopyrite: CuFeS2
Malachite: Cu2CO3(OH)l2
Azurite: Cu3(CO3)2(OH)2
Bornite: Cu5FeS4
Turquoise: CuAl6(PO4)4(OH)8•4(H2O)

Zinc and germanium

Sphalerite: ZnS
Germanite: Cu26Fe4Ge4S32

Strontium, zirconium, molybdenum

Celestine: SrSO4
Strontianite: SrCO3
Zircon: ZrSiO4
Molybdenite: MoS2

Silver

Argentite: Ag2S
Acanthite: Ag2S
Silver nugget

Tin

Cassiterite: SnO2

Caesium, barium, rare-earths

Pollucite: (Cs,Na)2Al2Si4O12·2H2O
Barite: BaSO4
Monazite: (Ce,La,Nd,Th)PO4

Tungsten

Wolframite: FeWO4
Scheelite: WCaO4
Hubnerite: WMnO4

Platinum, gold, mercury, lead

Sperrylite: PtAs2
Platinum nugget
Gold nugget
Cinnabar: HgS
Galena: PbS
Anglesite: PbSO4
Thorite: (Th,U)SiO4


Gems

Ruby
Diamond
Topaz
Zircon: ZrSiO4
Spinel: MgAl2O4

Sapphire
Sapphire
Sapphire

Emerald
Beryl: Be3Al2(SiO3)6
Morganite

Quartz
Amethyst: SiO2
Amethyst: SiO2
Citrine: SiO2

Garnet: [Mg,Fe,Mn]3Al2(SiO4)3 & Ca3[Cr,Al,Fe]2(SiO4)3
Peridot: (Mg,Fe)2SiO4
Opal: SiO2·nH2O
Jadeite: NaAlSi2O6
Pearl: CaCO3
Amber: Resin

Corundum is a crystalline form of aluminium oxide (Al2O3). It is transparent in its pure form and can have different colors when metal impurities are present.

             Color    Colorant  carat ($)

Painite                          55000  CaZrAl9O15(BO3)
Diamond      Clear                1400  C
Ruby         Red      Chromium   15000  Al2O3
Sapphire     Blue     Iron         650  Al2O3
Sapphire     yellow   Titanium          Al2O3
Sapphire     Orange   Copper            Al2O3
Sapphire     Green    Magnesium         Al2O3
Emerald      Green    Chromium          Be3Al2(SiO3)6
Beryl        Aqua     Iron              Be3Al2(SiO3)6   AKA "aquamarine"
Morganite    Orange   Manganese    300  Be3Al2(SiO3)6
Topaz        Topaz                      Al2SiO4(F,OH)2
Spinel       Red      Red               MgAl2O4
Quartz       Clear                      SiO2
Amethyst     Purple   Iron              SiO2
Citrine      Yellow                     SiO2
Zircon       Red                        ZrSiO4
Garnet       Orange                     [Mg,Fe,Mn]3Al2(SiO4)3 & Ca3[Cr,Al,Fe]2(SiO4)3
Garnet       Blue                 1500  [Mg,Fe,Mn]3Al2(SiO4)3 & Ca3[Cr,Al,Fe]2(SiO4)3
Opal                                    SiO2·nH2O
Opal         Black               11000  SiO2·nH2O
Jet          Black                      Lignite
Peridot      Green                      (Mg,Fe)2SiO4
Pearl        White                      CaCO3
Jade         Green                      NaAlSi2O6
Amber        Orange                     Resin

Crystals
Crystal, polycrystal, and amorphous

Diamond
Carbon phase diagram

Corundum (Al2O3)
Corundum unit cell
Corundum

Metal lattice
Salt (NaCl)
Tungsten Carbide

Alpha quartz (SiO2)
Beta quartz
Glass (SiO2)
Ice


Fullerines

Buckyball with 540 atoms
Buckyball with 60 atoms
Buckyballs in the liquid phase

Nanotube

Buckyballs in a nanotube
Graphene


Polymers

Zylon
Vectran
Aramid (Kevlar)
Polyethylene

Aramid
Nylon
Hydrogen bonds in Nylon

Spider silk
Lignin

Lignin comprises 30 percent of wood and it is the principal structural element.


Rope

               Year   Young  Tensile  Strain  Density   Common
                      (GPa)  strength         (g/cm3)   name
                              (GPa)
Gut           Ancient           .2
Cotton        Ancient                   .1       1.5
Hemp          Ancient   10      .3      .023
Duct tape                       .015
Gorilla tape                    .030
Polyamide      1939      5     1.0      .2       1.14    Nylon, Perlon
Polyethylene   1939    117                       1.4     Dacron
Polyester      1941     15     1.0      .067     1.38
Polypropylene  1957                               .91
Carbon fiber   1968            3.0               1.75
Aramid         1973    135     3.0      .022     1.43    Kevlar
HMPE           1975    100     2.4      .024      .97    Dyneema, Spectra
PBO            1985    280     5.8      .021     1.52    Zylon
LCAP           1990     65     3.8      .058     1.4     Vectran
Vectran HT              75     3.2      .043     1.41    Vectran
Vectran NT              52     1.1      .021     1.41    Vectran
Vectran UM             103     3.0      .029     1.41    Vectran
Nanorope             ~1000     3.6      .0036    1.3
Nanotube              1000    63        .063     1.34
Graphene              1050   160        .152     1.0


Strain  =  Strength / Young
Carbon fiber is not useful as a rope.

A string ideally has both large strength and large strain, which favors Vectran.

Suppose Batman has a rope made out of Zylon, the strongest known polymer.

Batman mass            =  M         =    100 kg               (includes suit and gear)
Gravity constant       =  g         =     10 meters/second2
Batman weight          =  F         =   1000 Newtons
Zylon density          =  D         =   1520 kg/meter3
Zylon tensile strength =  Pz        = 5.8⋅109 Newtons/meter2
Rope load              =  P         = 1.0⋅109 Newtons/meter2   (safety margin)
Rope length            =  L              100 meters
Rope cross section     =  A  = F/P  =1.0⋅10-6 meters2
Rope radius            =  R  =(A/π)½=     .56 mm
Rope mass              =  Mr = DAL  =     .15 kg

Wood

         Density   Tensile   Young
                   strength
         (g/cm^3)  (Gpa)     (Gpa)

Balsa         .12    .020      3.7
Corkwood      .21
Cedar         .32    .046      5.7     Northern white
Poplar        .33    .048      7.2     Balsam
Cedar         .34    .054      8.2     Western red
Pine          .37    .063      9.0     Eastern white
Buckeye       .38    .054      8.3     Yellow
Butternut     .40    .057      8.3
Basswood      .40    .061     10.3
Spruce, red   .41    .072     10.7
Aspen         .41    .064     10.0
Fir, silver   .42    .067     10.8
Hemlock       .43    .061      8.5     Eastern
Redwood       .44    .076      9.6
Ash, black    .53    .090     11.3
Birch, gray   .55    .069      8.0
Walnut, black .56    .104     11.8
Ash, green    .61    .100     11.7
Ash, white    .64    .110     12.5
Oak, red      .66    .100     12.7
Elm, rock     .66    .106     10.9
Beech         .66    .102     11.8
Birch, yellow .67    .119
Mahogany      .67    .124     10.8     West Africa
Locust        .71    .136     14.5     Black or Yellow
Persimmon     .78    .127     14.4
Oak, swamp    .79    .124     14.5     Swamp white
Gum, blue     .80    .118     16.8
Hickory       .81    .144     15.2     Shagbark
Eucalyptus    .83    .122     18.8
Bamboo        .85    .169     20.0
Oak, live     .98    .130     13.8
Ironwood     1.1     .181     21.0
Lignum Vitae 1.26    .127     14.1
Data #1     Data #2
Plastic

           Density   Tensile   Young
                     strength
           (g/cm^3)  (Gpa)     (Gpa)


Polyamide            .11       4.5
Polyimide            .085      2.5
Acrylic              .07       3.2
Polycarbonate        .07       2.6
Acetyl copoly        .06       2.7
ABS                  .04       2.3
Polypropylene  .91   .04       1.9
Polystyrene          .04       3.0
Polyethylene   .95   .015       .8

Alloys

Copper
Orichalcum (gold + copper)
Gold

Alloy of gold, silver, and copper


Superstrong amorphous alloys

Crystal, polycrystal, amorphous

New alloys have been discovered that are stronger and ligher than diamond. These alloys have an amorphous structure rather than the crystalline structure of conventional alloys. A crystaline alloy tends to be weak at the boundaries between crystals and this limits its strength. Amorphous alloys don't have these weaknesses and can be stronger.

Pure metals and alloys consisting of 2 or 3 different metals tend to be crystaline while alloys with 5 or more metals tend to be amorphous. The new superalloys are mixes of at least 5 different metals.

A material's strength is characterized by the "yield strength" and the quality is the ratio of the yield strength to the density. This is often referred to as the "strength to weight ratio".

Yield strength  =  Y            (Pascals)
Density         =  D            (kg/meter3)
Quality         =  Q  =  Y/D    (Joules/kg)
The strongest allyos are:
       Yield strength   Density   Quality
       (GPa)        (g/cm3)    (MJoule/kg)

Magnesium + Lithium             .14        1.43        98
Magnesium + Y2O3                .31        1.76       177
Aluminum  + Beryllium           .41        2.27       181
Amorphous LiMgAlScTi           1.97        2.67       738
Diamond                        1.6         3.5        457
Titanium  + AlVCrMo            1.20        4.6        261
Amorphous AlCrFeCoNiTi         2.26        6.5        377
Steel     + Cobalt, Nickel     2.07        8.6        241
Amorphous VNbMoTaW             1.22       12.3         99
Molybdenum+ Tungsten, Hafnium  1.8        14.3        126
The strongest pure metals are weaker than the strongest alloys.
       Yield strength   Density   Quality
       (GPa)        (g/cm3)    (MJoule/kg)

Magnesium                        .10       1.74        57
Beryllium                        .34       1.85       184
Aluminum                         .02       2.70         7
Titanium                         .22       4.51        49
Chromium                         .14       7.15        20
Iron                             .10       7.87        13
Cobalt                           .48       8.90        54
Molybdenum                       .25      10.28        24
Tungsten                         .95      19.25        49

Alloy types
Beryllium + Li           →  Doesn't exist. The atoms don't mix
Beryllium + Al           →  Improves strength
Magnesium + Li           →  Weaker and lighter than pure Mg. Lightest existing alloy
Magnesium + Be           →  Only tiny amounts of beryllium can be added to magnesium
Magnesium + Carbon tubes →  Improves strength, with an optimal tube fraction of 1%
Aluminum  + Li,Mg,Be,Sc  →  Stronger and lighter than aluminum
Titanium  + Li,Mg,Sc     →  Stronger and lighter than titanium
Steel     + Cr,Mo        →  Stronger and more uncorrodable than steel. "Chromoly"
Copper    + Be           →  Stronger than beryllium and is unsparkable

High-temperature metals (refractory metals)
          Melting point (Celsius)

Tungsten    3422
Rhenium     3186
Osmium      3033
Tantalum    3017
Molybdenum  2623
Niobium     2477
Iridium     2446
Ruthenium   2334
Hafnium     2233
Technetium  2157
Rhodium     1964
Vanadium    1910
Chromium    1907

High-temperature superalloys

Most alloys weaken with increasing temperature except for a small subset called "superalloys" that strengthen with temperature, such as Ni3Al and Co3Al. This is called the "yield strength anomaly".

Nickel alloys in jet engines have a surface temperature of 1150 Celsius and a bulk temperature of 980 Celsius. This is the limiting element for jet engine performance. Half the mass of a jet engine is superalloy.

Current engines use Nickel superalloys and Cobalt superalloys are under development that will perform even better.

Yield strength in GPa as a function of Celsius temperature.

                   20   600   800  900  1000  1100 1200  1400  1600 1800  1900  Celsius

VNbMoTaW          1.22         .84        .82       .75  .66   .48   .4
AlMohNbTahTiZr    2.0   1.87  1.60  1.2   .74  .7   .25
Nickel superalloy 1.05        1.20   .90  .60  .38  .15
Tungsten           .95   .42   .39        .34  .31  .28  .25   .10   .08  .04
Below 1100 Celsius AlMohNbTahTiZr has the best strength-to-mass ratio and above this VNbMoTaW has the best ratio. Both alloys supersede nickel superalloy and both outperform tungsten, the metal with the highest melting point. Data:   
Entropy, nickel superalloy
Copper alloys
                  Yield strength (GPa)

Copper                  .27
Brass                   .41     30% zinc
Bronze                  .30     5% tin
Phosphor bronze         .69     10% tin, .25% phosphorus
Copper + beryllium     1.2      2% beryllium, .3% cobalt
Copper + nickel + zinc  .48     18% nickel, 17% zinc
Copper + nickel         .40     10% nickel, 1.25% iron, .4% manganese
Copper + aluminum       .17     8% aluminum

Bells and cymbals

Bells and cymbals are made from bell bronze, 4 parts copper and 1 part tin.


High-performance materials

Below is a list of the elastic variables for a material, with examples for large and small values for each variable.


                      Small value        Large value

Tensile modulus       Climbing rope      Bicycle spokes
Tensile yield
Tensile strength                         Elevator cable
Tensile max strain                       Climbing rope
Tensile toughness                        Elevator cable
Shear modulus         Golf driver        Bike frame
Shear yield                              Spring
Shear strength                           Beam
Shear toughness                          Sword interior
Shear max strain                         Golf driver
Bulk modulus          Pillow
Brinell hardness      Machining metal    Sword edge
Mohr hardness         Pencil lead        Cutting tools
Poisson ratio
Density               Airplane frame     Bullet

Some examples of materials used for high-performance applications are

Rocket cones       Tungsten
Wire               Zylon, Sapphire, Carbon nanorope
Sword inerior      Aluminum alloy, Sapphire
Sword edge         Diamond
Clock escapement   Diamond,  Tungsten carbide
Formula-1 brakes   Carbon
Bullet             Lead, Tungsten, Uranium, Osmium
Lubricants         Fullerines

Brass is a useful machining metal because it is easy to cut.

Not all of of the above variables are independent. The toughness and maximum strain are determined from the strength and modulus. The most important variables for high-performance materials are


                      Example
Tensile strength      Maximum stress on a wire
Shear modulus         Stiffness of a beam or column
Shear strength        Maximum stress on a beam or column
Brinell hardness      Sword edge
Mohr hardness         Cutting tools
Density               Bullets
Below is a table of the stoutest engineering materials.

           Young's  Yield  Tensile Tensile  Tough  Tough/  Brinell  Density  Poisson
           modulus         strengh strain          density (GPa)    (g/cm^3)
            (GPa)   (Gpa)  (GPa)            (MPa)  (J/kg)
Beryllium    287     .345   .448   .0016    .350     189      .6     1.85     .032
Mg alloy      45     .100   .232   .0052    .598     344      .26    1.74     .29
Al alloy      70     .414   .483   .0069   1.666     595      .245   2.8      .35
Titanium     120     .225   .37    .0031    .570      54      .72    4.51     .32
Chromium     279            .282   .00101   .143     199     1.12    7.15     .21
Iron         211     .100   .35    .0017    .290      37      .49    7.87     .29
Cobalt       209     .485   .760   .0036   1.382     155      .7     8.90     .31
Nickel       170     .190   .195   .0011    .112      12.5    .7     8.91     .31
Copper       130     .117   .210   .0016    .170      19      .87    8.96     .34
Molybdenum   330            .324   .00098   .159      15     1.5    10.28     .31
Tin           47     .014   .200   .0043    .426      59      .005   7.26     .36
Tungsten     441     .947  1.51    .0037   2.585     134     2.57   19.25     .28
Rhenium      483     .290  1.07    .0024   1.298      62     1.32   21.02     .30
Osmium       590           1.00    .0018    .893      40     3.92   22.59     .25
Iridium      528           2.0     .0038   3.788     168     1.67   22.56     .26

Brass        125     .20    .55    .0044   1.210     139             8.73
Bronze       120            .800   .0067   2.667     300             8.9
Steel        250     .250   .55    .0022    .605      77             7.9      .30
Stainless    250     .52    .86    .0034   1.479     185             8.0      .30
W-C          650            .345   .00053   .092       5.9          15.63
Si-C         450     -     3.44    .0076  13.100    4100             3.21

Carbon fiber 181           1.600   .0088   7.070    4040             1.75
Kevlar              3.62   3.76
Zylon        180    2.70   5.80                                      1.56

Nanorope   ~1000     -     3.6     .0036    6.5      4980            1.3                Rope made from nanotubes
Nanotube    1000     -    63       .063    1980   1480000            1.34               Carbon nanotube
Graphene    1050     -   160       .152   12190  12190000            1.0
ColossalTube low     -     7                                          .116

Balsa
Bamboo                      .500                                      .4
Ironwood      21            .181   .0086    .780     650             1.2

Sapphire     345     .4    1.9     .0055   5.232    1315             3.98        .28
Diamond     1220    1.6    2.8     .0023   3.210     920    1200     3.5         .069

           Young's  Yield  Tensile  Tensile  Tough    Tough/  Brinell  Density  Poisson
           modulus         strengh  strain            density (GPa)    (g/cm^3)
            (GPa)   (Gpa)  (GPa)             (kPa)    (J/kg)

Sports

There is an analogy between elasticity and sports. An athlete should have both agility and endurance and there tends to be a tradeoff between the two. For materials, the tradeoff is between tensile strength and tensile modulus.

Endurance    =  Energy  / Mass  =  Strength / Density
Agility      =  Power   / Mass
Stiffness    =                  =  Modulus  / Density

Wrestleness  =  Agility   * Endurance  =  Power   * Energy   / Mass2
Swordness    =  Stiffness * Endurance  =  Modulus * Strength / Density2
"Wrestleness" reflects a synthesis of endurance and agility, and "swordness" reflects a synthesis of tensile modulus and tensile strength. Both give emphasis to being lightweight.
Mass         =  M
Volume       =  Υ
Density      =  D  =  M / Υ
Strength     =  Ρ                          Tensile strength. Pressure to break the material
Modulus      =  ΡM                         Young's modulus
Endurance    =  Ε  =  Ρ  / D  =  E / M
Stiffness    =  ΕM =  ΡM / D
Energy       =  E  =  Ε M

Time         =  T                           Wave time for the lowest overtone
Power        =  P  =  E / T
Agility      =  Ξ  =  P / M
Wrestleness  =  Ω  =  Ξ  Ε    =  P  E / M2
Swordness    =  Ψ  =  ΕM Ε    =  ΡM Ρ / D2
Time is analogous to malleability.
Frequency    =  F
Time         =  T  =  E / P  =  F-1
Malleability =  ε  =  Ρ / ΡM                Fractional elongation at the breaking point
Rigidity     =  ξ  =  ε-1
Wrestleness  =  Ω  =  Ξ  Ε  =  Ξ2 T  =  Ε2 F
Swordness    =  Ψ  =  ΕM Ε  =  ΕM2 ε  =  Ε2 ξ

High-temperature metals

This table shows the elements with the highest melting points.

Element   Density Melt  Boil  Young Young/   $/kg  ppm in metallic
         (g/cm^3) (K)   (K)    GPa  Density        asteroid

Tungsten   19.25  3693  5828   411   21.4      50     ~ 1
Rhenium    21.0   3459  5869   463   22.0    4600     ~ 1
Osmium     22.59  3306  5285   550   24.3   12000       2
Tantalum   16.7   3290  5731   186   11.1     400     ~  .5
Molybdenum 10.28  2896  4912   329   31.0      21     ~10
Niobium     8.75  2750  5017   105   12.0      40     ~ 3
Iridium    22.4   2739  4701   528   23.6   14000       2
Ruthenium  12.45  2607  4423   447   35.9    5500       5

Magnets
               Composition  Teslas  kJoules/  kAmps/     Max      Density  Tensile
                                    meter^3   meter  Temperature  (g/cm^3) strength
                                                         (C)               (GPa)
Neodymium       Nd2Fe14B     1.4      440      2000      400       7.4      .075
Samarium-Cobalt SmCo5        1.15     240      1300      800       8.3      .035
Alnico                       1.4       88       275      860
The composition of alnico alloys is typically 8-12% Al, 15-26% Ni, 5-24% Co, up to 6% Cu, up to 1% Ti, and the balance is Fe.

"Temperature" refers to the maximum temperature before the magnet loses its magnetism.


Glue
1942  Cyanoacrylate glue discovered, by accident
1958  Cyanoacrylate becomes commercially available

              Shear strength   Density
                 (MPa)         (g/cm^3)
Cyanoacrylate      3            1.1
Epoxy              6
Cyanoacrylate polymerizes when it comes into contact with water vapor from the air.

Glues typically fail by shear.


Resonant Q
Silicon Carbide   64
Sapphire          11.3
Diamond            3.7
Quartz             3.2
Silicon            2.3

Engineering

Roman bridge
Incan bridge


Strings


Characteristic string tension

For a given instrument there is a characteristic ideal tension for the strings. If the tension is too low or high the string becomes unplayable. The tension can be varied to suit the performer's taste but it can't be changed by an extreme degree.


           String                      Height of   Height of
           length   Tension (Newtons)  top string  bottom string
           (mm)     E   A   D   G   C    (mm)       (mm)
            
Violin      320    80  50  45  45         3.2        5.2
Viola       388        65  55  55  55     4.8        6.2
Cello       690       160 130 130 130     5.2        8.2
Bass       1060       160 160 160 160
Guitar      650       120 120 120 120
Bass guitar 860       160 160 160 160
The height of the string is the distance from the fingerboard.
Waves on a string

The frequency of a string and the speed of a wave on the string are related by:

                                              Values for a violin A-string

L  =  Length of a string                   =  .32 meters
F  =  Vibration frequency of the string    =  440 Hertz
V  =  Speed of a wave on the string        =  281.6 meters/second
   =  2 F L
For a given instrument and string frequency, the wavespeed is fixed.

The speed of a wave on a string is

WaveSpeed^2  =  Tension / (Density * Pi * Radius^2)
The variables you can vary for a string are {Tension, Density, Radius}. Once you have chosen the frequency and length of the string then these variables are related by
Tension = Constant * Density * Radius^2

The larger the radius the more difficult the string is to play and the more impure the overtones. The radius can be minimized by using a material with a high density. This is why cello, bass, and bass guitar strings are often made of tungsten.

High-density strings are only appropriate for low-frequency strings because they have a low wavespeed. High-frequency strings require a material with low density.

String manufacturers almost never state the density and radius of the string. You can infer the density from the type of metal used, with numbers given the table below.

The speed of sound in air has an analogous form as the speed of a wave on a string.

SoundSpeed^2  =  (7/5) Pressure / Density

String tensile strength

If the tension force on a string exceeds the "Tensile strength" then the string breaks.

Force  =  Force on the string
A      =  Area of the string
S      =  Stress on the string
       =  Force / A
Smax   =  Tensile strength
       =  Maximum string stress before breaking
Z      =  Strength to weight ratio
Z      =  Smax / Density


                 Tensile   Density   Z/10^6   Young's
                 strength                     modulus
                   (GPa)   (g/cm^3)  (J/kg)   (GPa)

Carbon nanotube    7          .116   60.3              Technology not yet developed
Nylon               .045     1.15      .04      5
Kevlar             3.6       1.44     2.5
Zylon              5.8       1.5      3.9
Gut                 .2       1.5       .13      6
Magnesium alloy     .4       1.8       .22
Aluminum            .05      2.7
Titanium alloy      .94      4.5       .21
Nickel              .20      8.9
Chromium            .28      7.2
Steel alloy        2.0       7.9       .25    220
Brass               .55      8.7
Silver              .17     10.5
Tungsten            .55     19.2       .029
Gold                .13     19.3
Osmium             1.0      22.6
Iridium

Maximum frequency of a string

F    =  String frequency
R    =  String radius
A    =  String cross-sectional area
     =  Pi R^2
D    =  String density
L    =  String length
Force=  String tension force (Newtons)
S    =  Tensile stress (Pascals)
     =  Force / A
Smax =  Maximum string tensile stress before breaking
     =  Tensile strength
V    =  Speed of a wave on the string
     =  SquareRoot(P/D)
Z    =  String strength-to-weight ratio
     =  S/D
Fmax =  Maximum frequency of a string
The maximum frequency of a string happens when S=Smax.
Fmax  =  V / (2L)
      =  SquareRoot(Smax/D) / (2L)
      =  SquareRoot(Z) / (2L)
The maximum frequency of a string depends on the strength-to-weight ratio Z. Values for Z for various string materials are given in the table above. Steel alloy is often used for the highest-frequency strings on a violin or piano.

A space elevator requires a material with Z > 100.


Maximum frequency of a string for various materials
            Gut   Steel   Zylon   Carbon    Tungsten
                                 nanotube

Violin      563    781    2960    12100       266
Viola       465    644    2440     3160       220
Cello       261    362    1370     5620       123
Bass        170    236     895     3660        80
Guitar      277    385    1519     5973       131
Bass guitar 209    291    1148     4514        99
Frequencies are in Hertz.

Gut was usually used in the Baroque age because steel alloys hadn't been perfected. A-strings were tuned to a frequency of around 420 Hertz. Modern steel made possible the 660 Hertz E-string and the high-frequency strings on a piano.

You can use zylon to make a bass sound like a violin.


Low-frequency strings

Tungsten is a high-density metal that can be used to make low-frequency strings ("Darth Vader" strings). You can make a violin sound like a bass.

The larger the diameter of a string the more difficult it is to play. Diameter sets the lower limit of the frequency of a string.

Frequency = Constant * SquareRoot(Smax/D) / R
String frequency is inversely proportional to radius. A string can be made an octave lower by doubling the radius.

If a string is made of tungsten with a density of 19.25 g/cm^3 then the diameter of the lowest string on each instrument is

-
              Freq   Length  Diameter
              (Hz)    (mm)    (mm)

Violin G      196     320     .46
Viola C       130     388     .62
Cello C        65     690    1.07
Bass E         41    1060    1.18
Guitar E       82     650     .90
Bass guitar E  41     860    1.7

String diameter

The "Tungsten" lines are string diameters for tungsten and the "Zylon" lines are string diameters for zylon. Tungsten diameters assume a density of 19.3 g/cm^3 and zylon diameters assume a density of 1.5 g/cm^3. The zylon lines cut off at the right at the frequency where the string breaks.


String price

The price is for strings made of gold with a density 19.3 g/cm^3, the same as for tungsten. If the strings are made from iridium or osmium then the metal price is half this. For tungsten strings the price of the tungsten is negligible.

Even though iridium is half the price of gold, gold wire may be cheaper because gold is easier to forge.


              Density   Price
              (g/cm^3)  ($/g)

Zylon           1.5     Cheap
Tungsten       19.2       .05
Gold           19.3     40
Rhenium        21.0     10
Platinum       21.4     80
Iridium        22.4     20
Osmium         22.6     20

String stiffness

When a beam is bent it exerts a restoring force. If a string is too stiff it acts like a beam and becomes impossible to play. The stiffness is inversely proportional to the Young's modulus. This is why metal strings are usually wound around a flexible core.

Examples of beam vibrations.


String winding

Strings typically have a flexible core with a low Young's modulus and a high-density metallic winding.


String inharmonicity

The overtones of an ideal string are exact integer ratios. If the string is non-ideal then the overtones can change. The principal source of non-ideality is the finite thickness of the string. String stiffness also contributes non-ideality.

R    =  String radius
L    =  String length
D    =  String density
Y    =  Young's modulus for the string
Force=  Tension force on the string
N    =  An integer greater than or equal to 1
Fn   =  Frequency of overtone N
     =  N F (1 + C N^2)
C    =  Constant of inharmonicity
     =  Pi^3 R^4 Y / (8 L^2 Force)
If C=0 then there is no inharmonicity and the overtones are exact integer multiples of the fundamental mode. If the string has finite thickness then the frequencies of the overtones shift.

Plucked strings exhibit inharmonicity. Bowed strings are "mode-locked" so that the harmonics are exact integer ratios. Reed instruments and the human voice are also mode locked.

The coefficient of inharmonicity can be expressed in terms of density as

C  =  Pi Force Y / (128 D^2 F^4 L^6)
Increasing the density decreases the inharmonicity.

Low strings are more inharmonic than high strings.

The higher the note you play on a string, the smaller the effective string length and the more inharmonic the note. This is what prevents you from playing notes of arbitrarily high frequency.

The following is a table of inharmonicity coefficients for various instruments. We have assumed standard values for the string tension and we assume the string has the density of steel.


               String   Tension  Frequency  Density  Radius  Young's   C
               length  (Newtons)  (Hertz)   (g/cm^3)  (mm)   modulus
                (mm)                                         (GPa)
            
Violin E gut     320      80        660      1.5     .31       6     .000026
Violin E steel   320      80        660      7.9     .13     220     .000033
Violin G steel   320      45        196      7.9     .34     220     .00012
Viola C steel    388      55        130      7.9     .47     220     .00019
Cello C steel    690     130         65      7.9     .81     220     .000098
Bass E steel    1060     160         41      7.9     .92     220     .000047
Guitar E steel   650     140         82      7.9     .70     220     .000058
Bass guitar E    860     220         41      7.9    1.34     220     .00017

If we set the frequency shift from inharmonicity equal to the frequency resolution for human hearing,
N^2 C = 1/170

If C=.0001 then N=7.7       (The inharmonicity appears at the 8th overtone)

Instrument size and inharmonicity

The lower the frequency of a string, the more inharmonic it is. Low-frequency strings typically consist of a synthetic core (for elasticity) and an outer metallic winding (for density). You can't use metal for the entire string because metal is too stiff (the Young's modulus is too high.

An ideal core material has a high tensile strengh, so that you can use a small core diameter, and a low Young's modulus, to minimize inharmonicity. The synthetic material that is best suited for this is Vectran (see the table above).

L    =  Length of the string
R    =  Outer radius of the string
r    =  Radius of the inner core
     =  K R               where K is a dimensionless constant
Y    =  Young's modulus of the core material
D    =  Density of the outer winding
Force=  Force on the string
     =  k L               where k is a constant
Y    =  Young's modulus of the core
S    =  Stress on the inner core
     =  Force / (Pi r^2)
s    =  Strain on the inner core
     =  S/Y
C    =  Constant of inharmonicity
     =  Pi^3 R^4 Y / (8 L^2 Force)
     =  Pi Force Y / (128 D^2 F^4 L^6)
The strain should be as large as possible to minimize the Young's modulus, but if it is too large then the string loses functionality. We assume that the strain is a constant value.

For constant string length the ideal force doesn't depend on frequency.

Force / (Pi r^2) = Y s
The larger the value of "r" the lower the value of "Y" and the lower the inharmonicity.

If "r" is too large compared to "R" then the string loses density. We assume that r is is a fixed fraction of R and that r/R ~ 2/5.

Using

Force  =  Pi R^2 4 D F^2 L^2
       =  Pi r^2 Y s
We have
4 D F^2 L^2  =  K^2 Y s
The inharmonicity is
C  =  Pi Force Y / (128 D^2 F^4 L^6)
   =  Pi Force 4 D F^2 L^2 / (128 K^2 s D^2 F^4 L^6)
   =  Pi Force / (32 K^2 D F^2 s L^4)
   =  Constant * Force / (F^2 L^4)
If we assume that
Force = Constant * L
then
C  =  Constant / (F^2 L^3)
The lowest practical frequency of an instrument scales as L^(-3/2).

Let

Relative inharmonicity  =  1/(Freq^2 Length^3)
The relative inharmonicity of the lowest string for various instruments is given by the following table. The value is similar for all instruments.
              Freq   Length    Relative inharmonicity
              (Hz)    (mm)     = 1/(Freq^2 Length^3)

Violin G      196     320      .00079
Viola C       130     388      .00101
Cello C        65     690      .00072
Bass E         41    1060      .00050
Guitar E       82     650      .00054
Bass guitar E  41     860      .00094

Parameters for low-frequency strings

The following table shows a set of example parameters for low-frequency strings. We assume a core of Vectran (density=1400 kg/m^3) and a winding of osmium (density=22600 kg/m^3).

          Note   Freq   Tension  Core    Core    Outer   Core  Core
                (Hertz)          stress  radius  radius  Young strain
                          (N)    (GPa)   (mm)            (GPa)

Viola       C   130.4     50      .2      .28     .70     70    .0155
Viola       C    65.2     50      .2      .28    1.30     70    .0155
Viola       C    65.2     50     1.0      .126   1.25     70    .0155

Conductivity

White: High conductivity
Red:   Low conductivity

Electric and thermal conductivity
         Electric  Thermal  Density   Electric   C/Ct     Heat   Heat      Melt   $/kg  Young  Tensile Poisson  Brinell
         conduct   conduct            conduct/            cap    cap                                   number   hardness
        (e7 A/V/m) (W/K/m)  (g/cm^3)  Density   (AK/VW)  (J/g/K) (J/cm^3K)  (K)         (GPa)  (GPa)             (GPa)

Silver      6.30   429      10.49       .60      147       .235   2.47     1235    590    83   .17      .37      .024
Copper      5.96   401       8.96       .67      147       .385   3.21     1358      6   130   .21      .34      .87
Gold        4.52   318      19.30       .234     142       .129   2.49     1337  24000    78   .124     .44      .24
Aluminum    3.50   237       2.70      1.30      148       .897   2.42      933      2    70   .05      .35      .245
Beryllium   2.5    200       1.85      1.35      125      1.825   3.38     1560    850   287   .448     .032     .6
Magnesium   2.3    156       1.74      1.32      147      1.023   1.78      923      3    45   .22      .29      .26
Iridium     2.12   147      22.56       .094     144       .131   2.96     2917  13000   528  1.32      .26     1.67
Rhodium     2.0    150      12.41       .161     133       .243   3.02     2237  13000   275   .95      .26     1.1
Tungsten    1.89   173      19.25       .098     137       .132   2.54     3695     50   441  1.51      .28     2.57
Molybdenum  1.87   138      10.28       .182     136       .251            2896     24   330   .55      .31     1.5
Cobalt      1.7    100       8.90       .170               .421            1768     30   209   .76      .31      .7
Zinc        1.69   116       7.14                          .388             693      2   108   .2       .25      .41
Nickel      1.4     90.9     8.91                          .444            1728     15
Ruthenium   1.25   117      12.45                                          2607   5600
Cadmium     1.25    96.6     8.65                                           594      2    50   .078     .30      .20
Osmium      1.23    87.6    22.59                          .130            3306  12000
Indium      1.19    81.8     7.31                                           430    750    11   .004     .45      .009
Iron        1.0     80.4     7.87                          .449            1811          211   .35      .29      .49
Palladium    .95    71.8                                                   1828
Tin          .83    66.8                                                    505     22    47   .20      .36      .005
Chromium     .79    93.9                                   .449            2180
Platinum     .95                                           .133            2041
Tantalum     .76                                           .140            3290
Gallium      .74                                                            303
Thorium      .68
Niobium      .55    53.7                                                   2750
Rhenium      .52                                           .137            3459
Vanadium     .5     30.7                                                   2183
Uranium      .35
Titanium     .25    21.9                                   .523            1941
Scandium     .18    15.8                                                   1814
Neodymium    .156                                                          1297
Mercury      .10     8.30                                  .140             234
Manganese    .062    7.81                                                  1519
Germanium    .00019                                                        1211

Diamondiso 10     3320
Diamond     e-16  2200                                     .509
Nanotube   10     3500                                                Carbon nanotube. Electric conductivity = e-16 laterally
Tube bulk          200                                                Carbon nanotubes in bulk
Graphene   10     5000
Graphite    2      400                                     .709       Natural graphite
Al Nitride  e-11   180
Brass       1.5    120
Steel               45                                                Carbon steel
Bronze       .65    40
Steel Cr     .15    20                                                Stainless steel (usually 10% chromium)
Quartz (C)          12                                                Crystalline quartz.  Thermal conductivity is anisotropic
Quartz (F)  e-16     2                                                Fused quartz
Granite              2.5
Marble               2.2
Ice                  2
Concrete             1.5
Limestone            1.3
Soil                 1
Glass       e-12      .85
Water       e-4       .6
Seawater    1         .6
Brick                 .5
Plastic               .5
Wood                  .2
Wood (dry)            .1
Plexiglass  e-14      .18
Rubber      e-13      .16
Snow                  .15
Paper                 .05
Plastic foam          .03
Air        5e-15      .025
Nitrogen              .025                                1.04
Oxygen                .025                                 .92
Silica aerogel        .01

Siemens:    Amperes^2 Seconds^3 / kg / meters^2     =   1 Ohm^-1
For most metals,
Electric conductivity / Thermal conductivity  ~  140  J/g/K

Magnetic field magnitudes
                                     Teslas

Field generated by brain             10-12
Wire carrying 1 Amp                  .00002     1 cm from the wire
Earth magnetic field                 .0000305   at the equator
Neodymium magnet                    1.4
Magnetic resonance imaging machine  8
Large Hadron Collider magnets       8.3
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       1010
Magnetar neutron star              1014

Dielectric strength

The critical electric field for electric breakdown for the following materials is:


              MVolt/meter
Air                3
Glass             12
Polystyrene       20
Rubber            20
Distilled water   68
Vacuum            30        Depends on electrode shape
Diamond         2000

Relative permittivity

Relative permittivity is the factor by which the electric field between charges is decreased relative to vacuum. Relative permittivity is dimensionless. Large permittivity is desirable for capacitors.

             Relative permittivity
Vacuum            1                   (Exact)
Air               1.00059
Polyethylene      2.5
Sapphire         10
Concrete         4.5
Glass          ~ 6
Rubber           7
Diamond        ~ 8
Graphite       ~12
Silicon         11.7
Water (0 C)     88
Water (20 C)    80
Water (100 C)   55
TiO2         ~ 150
SrTiO3         310
BaSrTiO3       500
Ba TiO3     ~ 5000
CaCuTiO3    250000

Magnetic permeability

A ferromagnetic material amplifies a magnetic field by a factor called the "relative permeability".

                Relative    Magnetic   Maximum    Critical
              permeability  moment     frequency  temperature
                                       (kHz)      (K)
Metglas 2714A    1000000                100               Rapidly-cooled metal
Iron              200000      2.2                 1043
Iron + nickel     100000                                  Mu-metal or permalloy
Cobalt + iron      18000
Nickel               600       .606                627
Cobalt               250      1.72                1388
Carbon steel         100
Neodymium magnet       1.05
Manganese              1.001
Air                    1.000
Superconductor         0
Dysprosium                   10.2                   88
Gadolinium                    7.63                 292
EuO                           6.8                   69
Y3Fe5O12                      5.0                  560
MnBi                          3.52                 630
MnAs                          3.4                  318
NiO + Fe                      2.4                  858
CrO2                          2.03                 386

Effect of temperature on conductivity

Resistivity in 10^-9 Ohm Meters

              293 K   300 K   500 K

Beryllium     35.6    37.6     99
Magnesium     43.9    45.1     78.6
Aluminum      26.5    27.33    49.9
Copper        16.78   17.25    30.9
Silver        15.87   16.29    28.7

Current density

Current density
Resistor

                  Electric quantities             |                Thermal quantities
                                                  |
Q  =  Charge                 Coulomb              |   Etherm=  Thermal energy          Joule
I  =  Current                Amperes              |   Itherm=  Thermal current         Watts
E  =  Electric field         Volts/meter          |   Etherm=  Thermal field           Kelvins/meter
C  =  Electric conductivity  Amperes/Volt/meter   |   Ctherm=  Thermal conductivity    Watts/meter/Kelvin
A  =  Area                   meter^2              |   A     =  Area                    meter^2
Z  =  Distance               meter                |   Z     =  Distance                meter^2
J  =  Current flux           Amperes/meter^2      |   Jtherm=  Thermal flux            Watts/meter^2
   =  I / A                                       |         =  Ittherm / A
   =  C * E                                       |         =  Ctherm * Etherm
V  =  Voltage                Volts                |   Temp  =  Temperature difference  Kelvin
   =  E Z                                         |         =  Etherm Z
   =  I R                                         |         =  Itherm Rtherm
R  =  Resistance             Volts/Ampere = Ohms  |   Rtherm=  Thermal resistance      Kelvins/Watt
   =  Z / (A C)                                   |         =  Z / (A Ct)
H  =  Current heating        Watts/meter^3        |
   =  E J                                         |
P  =  Current heating power  Watts                |
   =  E J Z A                                     |
   =  V I                                         |

Continuum
Continuum quantity       Macroscopic quantity

     E             <->      V
     C             <->      R = L / (A C)
     J = C E       <->      I = V / R
     H = E J       <->      P = V I

Superconductors

                 Critical    Critical  Type
                temperature  field
                 (Kelvin)    (Teslas)

Magnesium-Boron2     39        55       2   MRI machines
Niobium3-Germanium   23.2      37       2   Field for thin films.  Not widely used
Magnesium-Boron2-C   34        36           Doped with 5% carbon
Niobium3-Tin         18.3      30       2   High-performance magnets.  Brittle
Vanadium3-Gallium    14.2      19       2
Niobium-Titanium     10        15       2   Cheaper than Niobium3-Tin.  Ductile
Niobium3-Aluminum

Technetium           11.2               2
Niobium               9.26       .82    2
Vanadium              5.03      1       2
Tantalum              4.48       .09    1
Lead                  7.19       .08    1
Lanthanum             6.3               1
Mercury               4.15       .04    1
Tungsten              4                 1    Not BCS
Tin                   3.72       .03    1
Indium                3.4        .028
Rhenium               2.4        .03    1
Thallium              2.4        .018
Thallium              2.39       .02    1
Aluminum              1.2        .01    1
Gallium               1.1
Gadolinium            1.1
Protactinium          1.4
Thorium               1.4
Thallium              2.4
Molybdenum             .92
Zinc                   .85       .0054
Osmium                 .7
Zirconium              .55
Cadmium                .52       .0028
Ruthenium              .5
Titanium               .4        .0056
Iridium                .1
Lutetium               .1
Hafnium                .1
Uranium                .2
Beryllium              .026
Tungsten               .015

HgBa2Ca2Cu3O8       134                 2
HgBa2Ca Cu2O6       128                 2
YBa2Cu3O7            92                 2
C60Cs2Rb             33                 2
C60Rb                28         2       2
C60K3                19.8        .013   2
C6Ca                 11.5        .95    2    Not BCS
Diamond:B            11.4       4       2    Diamond doped with boron
In2O3                 3.3       3       2
The critical fields for Niobium-Titanium, Niobium3-Tin, and Vanadium3-Gallium are for 4.2 Kelvin.

All superconductors are described by the BCS theory unless stated otherwise.

         Boiling point (Kelvin)

Water      273
Ammonia    248
Freon R12  243
Freon R22  231
Propane    230
Acetylene  189
Ethane     185
Xenon      165.1
Krypton    119.7
Oxygen      90.2
Argon       87.3
Nitrogen    77.4     Threshold for cheap superconductivity
Neon        27.1
Hydrogen    20.3     Cheap MRI machines
Helium-4     4.23    High-performance magnets
Helium-3     3.19
The record for Niobium3-Tin is 2643 Amps/mm^2 at 12 T and 4.2 K.

Titan has a temperature of 94 Kelvin, allowing for superconducting equipment. The temperature of Mars is too high at 210 Kelvin.


Gases

Ideal gas law

Molecules in a gas
Brownian motion

Pressure                          =  P             (Pascals or Newtons/meter2 or Joules/meter3)
Temperature                       =  T             (Kelvin)
Volume                            =  Vol           (meters3)
Total gas kinetic energy          =  E             (Joules)
Kinetic energy per volume         =  e  =  E/Vol   (Joules/meter3)
Number of gas molecules           =  N
Mass of a gas molecule            =  M
Gas molecules per volume          =  n  =   N / Vol
Gas density                       =  D  = N M / Vol
Avogadro number                   =  Avo=  6.022⋅1023  moles-1
Moles of gas molecules            =  Mol=  N / Avo
Boltzmann constant                =  k  =  1.38⋅10-23 Joules/Kelvin
Gas constant                      =  R  =  k Avo  =  8.31 Joules/Kelvin/mole
Gas molecule thermal speed        =  Vth
Mean kinetic energy / gas molecule=  ε  =  E / n  =  ½ M Vth2     (Definition of the mean thermal speed)
Gas pressure arises from the kinetic energy of gas molecules and has units of energy/volume.
The ideal gas law can be written in the following forms:
P  =  23 e                    Form used in physics
   =  R Mol T / Vol            Form used in chemistry
   =  k N   T / Vol
   =  13 N M Vth2/ Vol
   =  13 D Vth2
   =  k T D / M
Gas simulation at phet.colorado.edu
Derivation of the ideal gas law
History

Boyle's law
Charles' law

1660  Boyle law          P Vol     = Constant          at fixed T
1802  Charles law        T Vol     = Constant          at fixed P
1802  Gay-Lussac law     T P       = Constant          at fixed Vol
1811  Avogadro law       Vol / N   = Constant          at fixed T and P
1834  Clapeyron law      P Vol / T = Constant          combined ideal gas law

Boltzmann constant

For a system in thermodynamic equilibrium each degree of freedom has a mean energy of ½ k T. This is the definition of temperature.

Molecule mass                =  M
Thermal speed                =  Vth
Boltzmann constant           =  k  =  1.38⋅10-23 Joules/Kelvin
Molecule mean kinetic energy =  ε
A gas molecule moving in N dimensions has N degrees of freedom. In 3D the mean energy of a gas molecule is
ε  =  32 k T  =  ½ M V2th

Speed of sound

The sound speed is proportional to the thermal speed of gas molecules. The thermal speed of a gas molecule is defined in terms of the mean energy per molecule.

Adiabatic constant  =  γ
                    =  5/3 for monatomic molecules such as helium, neon, krypton, argon, and xenon
                    =  7/5 for diatomic molecules such as H2, O2, and N2
                    =  7/5 for air, which is 21% O2, 78% N2, and 1% Ar
                    ≈  1.31 for a triatomic gas such as CO2
Pressure            =  P
Density             =  D
Sound speed         =  Vsound
Mean thermal speed  =  Vth
K.E. per molecule   =  ε  =  ½ M Vth2

V2sound  =  γ  P / D  =  13  γ  V2th
The sound speed depends on temperature and not on density or pressure.

For air, γ = 7/5 and

Vsound  =  .68  Vth
These laws are derived in the appendix.

We can change the sound speed by using a gas with a different value of M.

                   M in atomic mass units

Helium atom                4
Neon atom                 20
Nitrogen molecule         28
Oxygen molecule           32
Argon atom                40
Krypton atom              84
Xenon atom               131
A helium atom has a smaller mass than a nitrogen molecule and hence has a higher sound speed. This is why the pitch of your voice increases if you inhale helium. Inhaling xenon makes you sound like Darth Vader. Then you pass out because Xenon is an anaesthetic.

In a gas, some of the energy is in motion of the molecule and some is in rotations and vibrations. This determines the adiabatic constant.

Ethane
Molecule with thermal vibrations


History of the speed of sound
1635  Gassendi measures the speed of sound to be 478 m/s with 25% error.
1660  Viviani and Borelli produce the first accurate measurement of the speed of
      sound, giving a value of 350 m/s.
1660  Hooke's law published.  The force on a spring is proportional to the change
      in length.
1662  Boyle discovers that for air at fixed temperature,
      Pressure * Volume = Constant
1687  Newton publishes the Principia Mathematica, which contains the first analytic
      calculation of the speed of sound.  The calculated value was 290 m/s.
Newton's calculation was correct if one assumes that a gas behaves like Boyle's law and Hooke's law.

The fact that Newton's calculation differed from the measured speed is due to the fact that air consists of diatomic molecules (nitrogen and oxygen). This was the first solid clue for the existence of atoms, and it also contained a clue for quantum mechanics.

In Newton's time it was not known that changing the volume of a gas changes its temperature, which modifies the relationship between density and pressure. This was discovered by Charles in 1802 (Charles' law).


Gas data
       Melt   Boil  Solid    Liquid   Gas      Mass   Sound speed
       (K)    (K)   density  density  density  (AMU)  at 20 C
                    g/cm3    g/cm3    g/cm3            (m/s)

He        .95   4.2            .125   .000179    4.00  1007
Ne      24.6   27.1           1.21    .000900   20.18
Ar      83.8   87.3           1.40    .00178    39.95   319
Kr     115.8  119.9           2.41    .00375    83.80   221
Xe     161.4  165.1           2.94    .00589   131.29   178
H2      14     20              .070   .000090    2.02  1270
N2      63     77              .81    .00125    28.01   349
O2      54     90             1.14    .00143    32.00   326
Air                                   .0013     29.2    344     79% N2, 21% O2, 1% Ar
H2O    273    373     .917    1.00    .00080    18.02
CO2    n/a    195    1.56      n/a    .00198    44.00   267
CH4     91    112              .42    .00070    16.04   446
CH5OH  159    352              .79    .00152    34.07           Alcohol
Gas density is for 0 Celsius and 1 Bar. Liquid density is for the boiling point, except for water, which is for 4 Celsius.

Carbon dioxide doesn't have a liquid state at standard temperature and pressure. It sublimes directly from a solid to a vapor.


Height of an atmosphere

M  =  Mass of a gas molecule
V  =  Thermal speed
E  =  Mean energy of a gas molecule
   =  1/2 M V^2
H  =  Characteristic height of an atmosphere
g  =  Gravitational acceleration
Suppose a molecule at the surface of the Earth is moving upward with speed V and suppose it doesn't collide with other air molecules. It will reach a height of
M H g  =  1/2  M  V^2
This height H is the characteristic height of an atmosphere.
Pressure of air at sea level      =  1   Bar
Pressure of air in Denver         = .85  Bar      One mile high
Pressure of air at Mount Everest  = 1/4  Bar      10 km high
The density of the atmosphere scales as
Density ~ (Density At Sea Level) * exp(-E/E0)
where E is the gravitational potential energy of a gas molecule and E0 is the characteristic thermal energy given by
E0 = M H g = 1/2 M V^2
Expressed in terms of altitude h,
Density ~ Density At Sea Level * exp(-h/H)
For oxygen,
E0  =  3/2 * Boltzmann_Constant * Temperature
E0 is the same for all molecules regardless of mass, and H depends on the molecule's mass. H scales as
H  ~  Mass^-1

Atmospheric escape
S = Escape speed
T = Temperature
B = Boltzmann constant
  = 1.38e-23 Joules/Kelvin
g = Planet gravity at the surface

M = Mass of heavy molecule                    m = Mass of light molecule
V = Thermal speed of heavy molecule           v = Thermal speed of light molecule
E = Mean energy of heavy molecule             e = Mean energy of light molecule
H = Characteristic height of heavy molecule   h = Characteristic height of light molecule
  = E / (M g)                                   = e / (m g)
Z = Energy of heavy molecule / escape energy  z = Energy of light molecule / escape energy
  = .5 M V^2 / .5 M S^2                         = .5 m v^2 / .5 m S^2
  = V^2 / S^2                                   = v^2 / S^2


For an ideal gas, all molecules have the same mean kinetic energy.

    E     =     e      =  1.5 B T

.5 M V^2  =  .5 m v^2  =  1.5 B T
The light molecules tend to move faster than the heavy ones. This is why your voice increases in pitch when you breathe helium. Breathing a heavy gas such as Xenon makes you sound like Darth Vader.

For an object to have an atmosphere, the thermal energy must be much less than the escape energy.

V^2 << S^2        <->        Z << 1


          Escape  Atmos    Temp    H2     N2      Z        Z
          speed   density  (K)    km/s   km/s    (H2)     (N2)
          km/s    (kg/m^3)
Jupiter   59.5             112   1.18    .45   .00039   .000056
Saturn    35.5              84   1.02    .39   .00083   .00012
Neptune   23.5              55    .83    .31   .0012    .00018
Uranus    21.3              53    .81    .31   .0014    .00021
Earth     11.2     1.2     287   1.89    .71   .028     .0041
Venus     10.4    67       735   3.02   1.14   .084     .012
Mars       5.03     .020   210   1.61    .61   .103     .015
Titan      2.64    5.3      94   1.08    .41   .167     .024
Europa     2.02    0       102   1.12    .42   .31      .044
Moon       2.38    0       390   2.20    .83   .85      .12
Ceres       .51    0       168   1.44    .55  8.0      1.14
Even if an object has enough gravity to capture an atmosphere, it can still lose it to the solar wind. Also, the upper atmosphere tends to be hotter than at the surface, increasing the loss rate.

The threshold for capturing an atmosphere appears to be around Z = 1/25, or

Thermal Speed  <  1/5 Escape speed

Heating by gravitational collapse

When an object collapses by gravity, its temperature increases such that

Thermal speed of molecules  ~  Escape speed
In the gas simulation at phet.colorado.edu, you can move the wall and watch the gas change temperature.

For an ideal gas,

3 * Boltzmann_Constant * Temperature  ~  MassOfMolecules * Escape_Speed^2
For the sun, what is the temperature of a proton moving at the escape speed? This sets the scale of the temperature of the core of the sun. The minimum temperature for hydrogen fusion is 4 million Kelvin.

The Earth's core is composed chiefly of iron. What is the temperature of an iron atom moving at the Earth's escape speed?

      Escape speed (km/s)   Core composition
Sun        618.             Protons, electrons, helium
Earth       11.2            Iron
Mars         5.03           Iron
Moon         2.38           Iron
Ceres         .51           Iron

Derivation of the ideal gas law

We first derive the law for a 1D gas and then extend it to 3D.

Suppose a gas molecule bounces back and forth between two walls separated by a distance L.

M  = Mass of molecule
V  = Speed of the molecule
L  = Space between the walls
With each collision, the momentum change = 2 M V

Time between collisions = 2 L / V

The average force on a wall is

Force  =  Change in momentum  /  Time between collisions  =  M  V^2  /  L
Suppose a gas molecule is in a cube of volume L^3 and a molecule bounces back and forth between two opposite walls (never touching the other four walls). The pressure on these walls is
Pressure  =  Force  /  Area
          =  M  V^2  /  L^3
          =  M  V^2  /  Volume

Pressure * Volume  =  M  V^2
This is the ideal gas law in one dimension. For a molecule moving in 3D,
Velocity^2  = (Velocity in X direction)^2
            + (Velocity in Y direction)^2
            + (Velocity in Z direction)^2

Characteristic thermal speed in 3D  =  3  *  Characteristic thermal speed in 1D.
To produce the 3D ideal gas law, replace V^2 with 1/3 V^2 in the 1D equation.
Pressure * Volume  =  1/3  M  V^2        Where V is the characteristic thermal speed of the gas
This is the pressure for a gas with one molecule. If there are n molecules,
Pressure  Volume  =  n  1/3  M  V^2            Ideal gas law in 3D
If a gas consists of molecules with a mix of speeds, the thermal speed is defined as
Kinetic dnergy density of gas molecules  =  E  =  (n / Volume) 1/2 M V^2
Using this, the ideal gas law can be written as
Pressure  =  2/3  E
          =  1/3  Density  V^2
          =  8.3  Moles  Temperature  /  Volume
The last form comes from the law of thermodynamics:
M V^2 = 3 B T

Virial theorem

A typical globular cluster consists of millions of stars. If you measure the total gravitational and kinetic energy of the stars, you will find that

Total gravitational energy  =  -2 * Total kinetic energy
just like for a single satellite on a circular orbit.

Suppose a system consists of a set of objects interacting by a potential. If the system has reached a long-term equilibrium then the above statement about energies is true, no matter how chaotic the orbits of the objects. This is the "Virial theorem". It also applies if additional forces are involved. For example, the protons in the sun interact by both gravity and collisions and the virial theorem holds.

Gravitational energy of the sun  =  -2 * Kinetic energy of protons in the sun

Newton's calculation for the speed of sound

Hooke's law for a spring
Wave in a continuum
Gas molecules


Because of Hooke's law, springs oscillate with a constant frequency.

X = Displacement of a spring
V = Velocity of the spring
A = Acceleration of the spring
F = Force on the spring
M = Spring mass
Q = Spring constant
q = (K/M)^(1/2)
t = time
T = Spring oscillation period
Hooke's law and Newton's law:
F  =  - Q X  =  M A

A  =  - (Q/M) X  =  - q^2 X
This equation is solved with
X  =      sin(q t)
V  =  q   cos(q t)
A  = -q^2 sin(q t)  =  - q^2 X
The oscillation period of the spring is
T  =  2 Pi / q
   =  2 Pi (M/Q)^(1/2)

According to Boyle's law, a gas functions like a spring and hence a gas oscillates like a spring. An oscillation in a gas is a sound wave.

For a gas,

P   =  Pressure
dP  =  Change in pressure
Vol =  Volume
dVol=  Change in volume
If you change the volume of a gas according to Boyle's law,
P Vol            =  Constant
P dVol + Vol dP  =  0

dP = - (P/Vol) dVol
The change in pressure is proportional to the change in volume. This is equivalent to Hooke's law, where pressure takes the role of force and the change in volume takes the role of displacement of the spring. This is the mechanism behind sound waves.


In Boyle's law, the change in volume is assumed to be slow so the gas has time to equilibrate temperature with its surroundings. In this case the temperature is constant as the volume changes and the change is "isothermal".

P Vol = Constant
If the change in volume is fast then the walls do work on the molecules, changing their temperature. If there isn't enough time to equilibrate temperature with the surroundings then the change is "adiabatic". You can see this in action with the "Gas" simulation at phet.colorado.edu. Moving the wall changes the thermal speed of molecules and hence the temperature.


If a gas consists of pointlike particles then

Vol =  Volume of the gas
Ek  =  Total kinetic energy of gas molecules within the volume
E   =  Total energy of gas molecules within the volume
    =  Kinetic energy plus the energy from molecular rotation and vibration
dE  =  Change in energy as the volume changes
P   =  Pressure
dP  =  Change in pressure as the volume changes
D   =  Density
C   =  Speed of sound in the gas
d   =  Number of degrees of freedom of a gas molecule
    =  3 for a monotomic gas such as Helium
    =  5 for a diatomic gas such as nitrogen
G   =  Adiabatic constant
    =  1 + 2/d
    =  5/3 for a monatomic gas
    =  7/5 for a diatomic gas
k   =  Boltzmann constant
T   =  Temperature
The ideal gas law is
P Vol =  (2/3) Ek                    (Derived in www.jaymaron.com/gas/gas.html)
This law is equivalent to the formula that appears in chemistry.
P Vol = Moles R T
For a gas in thermal equilibrium each degree of freedom has a mean energy of .5 k T. For a gas of pointlike particles (monotomic) there are three degrees of freedom, one each for motion in the X, Y, and Z direction. In this case d=3. The mean kinetic energy of each gas molecule is 3 * (.5 k T). The total mean energy of each gas molecule is also 3 * (.5 k T).

For a diatomic gas there are also two rotational degrees of freedom. In this case d=5.

In general,

Ek  =  3 * (.5 k T)
E   =  d * (.5 k T)

Ek  =  (3/d) E
If you change the volume of a gas adiabatically, the walls change the kinetic and rotational energy of the gas molecules.
dE  =  -P dVol
The ideal gas law in terms of E instead of Ek is
P Vol =  (2/d) E

dP  =  (2/d) (dE/Vol - E dVol/Vol^2)
    =  (2/d) [-P dVol/Vol - (d/2) P dVol/Vol]
    = -(1+2/d) P dVol/Vol
    = - G P dVol/Vol
This equation determines the speed of sound in a gas.
C^2  =  G P / D
For air,
P = 1.01e5 Newtons/meter^2
D = 1.2    kg/meter^3
Newton assumed G=1 from Boyle's law, yielding a sound speed of
C  =  290 m/s
The correct value for air is G=7/5, which gives a sound speed of
C = 343 m/s
which is in accord with the measurement.


For a gas, G can be measured by measuring the sound speed. The results are

Helium     5/3    Monatomic molecule
Argon      5/3    Monatonic molecule
Air        7/5    4/5 Nitrogen and 1/5 Oxygen
Oxygen     7/5    Diatomic molecule
Nitrogen   7/5    Diatomic molecule
The fact that G is not equal to 1 was the first solid evidence for the existence of atoms and it also contained a clue for quantum mechanics. If a gas is a continuum (like Hooke's law) it has G=1 and if it consists of pointlike particles (monatonic) it has G=5/3. This explains helium and argon but not nitrogen and oxygen. Nitrogen and oxygen are diatomic molecules and their rotational degrees of freedom change Gamma.
                             Kinetic degrees   Rotational degrees    Gamma
                                of freedom         of freedom
Monatonic gas                      3                  0               5/3
Diatomic gas  T < 1000 K           3                  2               7/5
Diatomic gas, T > 1000 K           3                  3               4/3
Quantum mechanics freezes out one of the rotation modes at low temperature. Without quantum mechanics, diatomic molecules would have Gamma=4/3 at room temperature.

The fact that Gamma=7/5 for air was a clue for the existence of both atoms, molecules, and quantum mechanics.


Dark energy

For dark energy,

E  =  Energy
dE =  Change in energy
e  =  Energy density
Vol=  Volume
P  =  Pressure
The volume expands as the universe expands.

As a substance expands it does work on its surroundings according to its pressure.

dE = - P dVol
For dark energy, the energy density "e" is constant in space and so
dE = e dVol
Hence,
P = - e
Dark energy has a negative pressure, which means that it behaves differently from a continuum and from particles.

Dark matter consists of pointlike particles but they rarely interact with other particles and so they exert no pressure.


Valyrian steel

"Ice" is the sword with the red handle

Valyrian steel is a fictional substance from "Game of Thrones" that is stronger, lighter, and harder than steel. The only elements that qualify are beryllium, titanium, and vanadium, none of which were known in Earth history until the 18th century. Valyrian steel could be of these elements, an alloy, or a magical substance. According to George Martin, magic is involved.

The fact that it is less dense than steel means that it can't be a fancy form of steel such as Damascus steel or Wootz steel. Also, fancy steel loses its special properties if melted and hence cannot be reforged, whereas Valyrian steel swords can be reforged.

In Earth history, the first metal discovered since iron was cobalt in 1735. This launched a frenzy to smelt all known minerals and most of the smeltable metals were discovered by 1800. Then the battery and electrochemstry were discovered in 1800 and these were used to obtain the unsmeltable metals, which are lithium, beryllium, magnesium, aluminum, titanium, vanadium, niobium, and Uranium. Almost all of the strong alloys use these metals, and so the Valyrians must have used either electrochemistry or magic to make Valyrian steel.


Candidates for Valyrian steel

The following metals and alloys are both stronger and lighter than steel and could hypothetically be Valyrian steel.

                Yield     Density  Strength/Density
                strength  (g/cm3)   (GJoule/kg)
                (GPascal)
Beryllium            .34     1.85     .186
Aluminum + Be        .41     2.27     .181
LiMgAlScTi          1.97     2.67     .738
Titanium             .22     4.51     .050
Titanium + AlVCrMo  1.20     4.6      .261
Vanadium             .53     6.0      .076
AlCrFeCoNiTi        2.26     6.5      .377
AlCrFeCoNiMo        2.76     7.1      .394
Steel                .25     7.9      .032     Iron plus carbon
Copper               .12     9.0      .013
"Yield strength" is the maximum pressure a material can sustain before deforming. "Strength/Density" is the strength-to-weight ratio. Steel is stronger and lighter than copper.
Lore

Petyr Baelish: Nothing holds an edge like Valyrian steel.

Tyrion Lannister: Valyrian steel blades were scarce and costly, yet thousands remained in the world, perhaps two hundred in the Seven Kingdoms alone.

George Martin: Valyrian steel is a fantasy metal. Which means it has magical characteristics, and magic plays a role in its forging.

George Martin: Valyrian steel was always costly, but it became considerably more so when there was no more Valyria, and the secret of its making were lost.

Ned Stark's stord "Ice" is melted down and reforged into two smaller swords, "Oathkeeper" and "Widow's Wail". This rules out Valyrian steel being Wootz steel because Wootz steel loses its special properties when reforged.

Appearances of Valyrian steel in Game of Thrones:

        Name          Owner

Sword   Longclaw      Jon Snow
Sword   Heartsbane    Samwell Tarly
Dagger                Arya
Sword   Ice           Eddard Stark         Reforged into Oathkeeper and Widow's Wail
Sword   Oathkeeper    Brienne of Tarth
Sword   Widow's Wail  The Crown
Sword   Lady Forlorn  Ser Lyn Corbray
Sword   Nightfall     Ser Harras Harlow
Sword   Red Rain      Lord Dunstan Drumm
Arakh                 Caggo
Armor                 Euron Greyjoy
Horn    Dragonbinder  The Citadel of The Maesters
Some Maesters carry links of Valyrian steel, a symbol of mastery of the highest arts.
Black materials

Vantablack is the blackest known substance, composed of carbon nanotubes and invented in 2014. "Vanta" stands for Vertically Aligned Nano tube Arrays.

               Reflectivity
Black paint       .025
Super black       .004        Nickel-phosphorus alloy
Vantablack        .00035      Carbon nanotubes

Viscosity

Viscosity is analogous to electrical conductivity and thermal conductivity.

Quantity                    Electricity            Thermal               Viscosity

Stuff                       Coulomb                Joule                 Momentum
Stuff/volume                Coulomb/m^3            Joule/m^3             Momentum/m^3
Flow = Stuff/time           Coulomb/second         Joule/s               Momentum/s
Potential                   Volts                  Kelvin                Momentum/m^3
Field                       Volts/meter            Kelvins/meter         Momentum/m^3/m
Flow density = Flow/m^2     Amperes/meter^2        Watts/meter^2         Momentum/s/m^2
Conductivity                Amperes/Volt/meter     Watts/meter/Kelvin    m^2/s
Resistance                  Volts/Ampere           Kelvins/Watt          s/m^3


Flow density  =  Conductivity  *  Field

Flow          =  Potential  /  Resistance

Field         =  -Gradient(Potential)

Kinematic and dynamic viscosity
Fluid density          =  ρ              (kg/meter3)
Fluid velocity         =  V
Fluid momentum density =  U  =  D V
Kinematic viscosity    =  νk             (meters2 / second)
Dynamic viscosity      =  νd  =  ρ νk    (Pascal seconds)
Lagrangian time deriv. =  Dt

Dt U =  ∇⋅(νd∇U)
Dt V =  ∇⋅(νk∇V)

Viscosity
               Dynamic       Kinematic  Density
               viscosity     viscosity  (kg/m3)
                 (Pa s)      (m2/s)
Hydrogen            .00000876
Nitrogen            .0000178
Air                 .0000183  .0000150     1.22
Helium              .000019
Oxygen              .0000202
Xenon               .0000212
Acetone             .00031
Benzine             .00061
Water at   2 C      .00167
Water at  10 C      .00131    .0000010  1000
Water at  20 C      .00100              1000
Water at  30 C      .000798             1000
Water at 100 C      .000282             1000
Mercury             .00153    .00000012
Blood               .0035
Motor oil           .065
Olive oil           .081
Honey              6
Peanut butter    250
Asthenosphere   7e19         Weak layer between the curst and mantle
Upper mantle   .8e21
Lower mantle  1.5e21
1 Stokes = 1 cm2/s = 10-4 m2/s
Fluid mechanics
Schmidt number           = Momentum diffusivity / Mass diffusivity
Prandtl number           = Momentum diffusivity / Thermal diffusivity
Magnetic Prandtl number  = Momentum diffusivity / Magnetic diffusivity

                Prandtl   Schmidt
Air                .7       .7
Water             7
Liquid metals  << 1
Oils           >> 1

Plasma physics
n       =  Electron density
M       =  Electron mass
V       =  Electron thermal velocity
Q       =  Proton charge
k       =  Boltzmann constant
Temp    =  Temperature
Xdebye  =  Debye length                   (k*Temp/n/Q^2/(4 Pi Ke))^.5
Xgyro   =  Electron gyro radius           M V / Q B
Fgyro   =  Electron gyrofrequency


               Electron  Temp  Debye   Magnetic
               density   (K)    (m)    field (T)
               (m^-3)
Solar core       e32     e7    e-11    -
ITER          1.0e20     e8    e-4     5.3
Laser fusion  6.0e32     e8            -    National Ignition Facility.  density=1000 g/cm^3
Gas discharge    e16     e4    e-4     -
Ionosphere       e12     e3    e-3    e-5
Magnetosphere    e7      e7    e2     e-8
Solar wind       e6      e5    e1     e-9
Interstellar     e5      e4    e1     e-10
Intergalactic    e0      e6    e5      -


ITER ion temperature      = 8.0 keV
ITER electron temperature = 8.8 keV
ITER confinement time     = 400 seconds

Inertial confinement fusion
                                 Compression  Heating     Fusion   Heating  Density    Year
                                 laser (MJ)   laser (MJ)  energy   time     (kg/m^3)
                                                          (MJ)     (s)
NOVA                                                                          .3    1984.  LLNL
National Ignition Facility (NIF)   330           -          20                .9       2010
HiPER                                 .2        .07         30      e-11      .3       Future

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