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Materials and elasticity
Dr. Jay Maron

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

Hooke's law and elasticity

x     =  Displacement of the spring when a force is applied
K     =  Spring constant
Force =  Force on the spring
=  K x      (Hooke's law)

Young's modulus

The Young's modulus characterizes the stiffness of a wire and it is proportional to the spring constant.

Force =  Tension force on the wire
X     =  Length of wire under zero tension force
x     =  Increase in length of the wire when a tension force is applied
Area  =  Cross-sectional area of the wire
Stress=  Force / Area          (Pressure, measured in Pascals or Newtons/meter^2)
Strain=  Fractional change in length of the wire
=  x/X
Young =  Tensile modulus or "Young's modulus" for the (Pascals)
Hooke's law in terms of stress and strain is
Stress  =  Young * Strain

Young   =  K X / Area
If the Young's modulus of a string is too high it is too stiff to play. This is why the core of a string is often made from an elastic material such as nylon or gut.

Tensile strength

Tensile yield strength  =  Maximum stress before the material irreversibly deforms
Tensile strength        =  Maximum stress before the material breaks

Tensile toughness

Toughness is a measure of the maximum energy density that a material can absorb before breaking. A tough material must be both strong and flexible.

Density  =  Density of the wire
Stress   =  Force/Area on the wire
Strain   =  Fractional change in length of the wire
StressMax=  Maximum stress before breaking
=  Tensile strength
StrainMax=  Tensile strain strength.  Maximum strain before breaking
Young    =  Young's modulus
=  Stress/Strain
Yield    =  Yield modulus
=  Stress for which the material irreversibly deforms
Toughness=  Energy/volume in the wire when at maximum stress
=  .5 * StressMax^2 / Young
U        =  Toughness/Density
=  Energy/kg in the wire when at maximum stress

Stress     =  Young * Strain

StressMax  =  Young * StrainMax

Toughness  =  .5 * Young * StrainMax^2
=  .5 * StressMax^2 / Young

The ability of a material to keep an edge is related to its Young's modulus and tensile strength.

Piano strings are "tough" in the sense that they are designed to absorb a large strain before breaking. The tradeoff is that they cannot hold an edge.

Wootz steel is a steel-carbon alloy designed to be both strong and flexible.

The following table shows the maximum strain of a material before breaking.

Tensile toughness

Toughness is a measure of the maximum energy density that a material can absorb before breaking. A tough material must be both strong and flexible.

Density  =  Density of the wire
Stress   =  Force/Area on the wire
Strain   =  Fractional change in length of the wire
StressMax=  Maximum stress before breaking
=  Tensile strength
StrainMax=  Tensile strain strength.  Maximum strain before breaking
Young    =  Young's modulus
=  Stress/Strain
Yield    =  Yield modulus
=  Stress for which the material irreversibly deforms
Toughness=  Energy/volume in the wire when at maximum stress
=  .5 * StressMax^2 / Young
U        =  Toughness/Density
=  Energy/kg in the wire when at maximum stress

Stress     =  Young * Strain

StressMax  =  Young * StrainMax

Toughness  =  .5 * Young * StrainMax^2
=  .5 * StressMax^2 / Young

The ability of a material to keep an edge is related to its Young's modulus and tensile strength.

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
Magnesium     45     .100   .232   .0052     .598     344       .26    1.74       .29
Aluminum      70     .020   .050   .00071    .018      15       .245   2.70       .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
Silver        83            .170   .0020     .174      17       .024  10.49       .37
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
Platinum     168            .165   .00098    .081       3.8     .32   21.45       .38
Gold          78            .127   .0016     .103       5.3     .24   19.30       .44
Lead          16            .012   .00075    .045     3.8     .44     11.34       .44

Al Alloy      70     .414   .483   .0069    1.666     595              2.8
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   13.10     4100              3.21

Rubber          .1          .016                                                 .5
Nylon          3     .045   .075   .025     .938     815              1.15
Carbon fiber 181           1.600   .0088   7.07     4040              1.75
Kevlar       100    3.62   3.76
Zylon        180    2.70   5.80                                       1.56
Colossal tube        -     7                                           .116              Carbon colossal nanotube
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
Carbyne    32100                                                                         Unstable

Air            0    0      0      0            0       0               .0012
Water          0    0      0      0            0       0              1.00
Ice                         .001                                                  .33
Cork                       low                                                   0
Glass         45            .033                                      2.53        .25
Concrete      30            .005                                      2.7         .2
Granite       70            .025                                      2.7         .2
Basalt                      .030
Marble        70            .015                                      2.6         .2

Skin                 .015   .020                                      2.2
Bone          14     .120   .130   .0093     604     377              1.6
Balsa
Pine                        .04
Oak           11
Bamboo                      .50                                        .4
Ironwood      21            .181   .0086     780     650              1.2
Tooth enamel  83
Human hair                  .380
Spider silk                1.0                                        1.3

Sapphire     345     .4    1.9     .0055    5232    1315              3.98        .28
Diamond     1220    1.6    2.8     .0023    3210     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)

The listed strain is for when the material is at the breaking point.

The maximum energy/mass of a spring is proportional to the toughness/density.

Energy/mass
(MJ/kg)

Fission        69500000
Diesel fuel          47
Lithium battery        .95
Supercapacitor         .018
Spring                 .0003

Compressive strength

Tension
Compression

Concrete and ceramics typically have much higher compressive strengths than tensile strengths. Composite materials, such as glass fiber epoxy matrix composite, tend to have higher tensile strengths than compressive strengths.

Deformation

The deformation of a solid is characterized by shear strain, tensile strain, and bulk compression.

Tensile strain
Shear strain
Bulk compression

Tensile strength relates to the strength of wires.

Two vices pull on a wire

Shear strength relates to the strength of beams and columns.

Bending of a beam
Buckling of a column
Human humerus

The maximum force on a beam is determined by the shear strength.

F  =  Maximum force applied to the center of a beam before it breaks
X  =  Beam length
Y  =  Beam thickness
Z  =  Beam height
x  =  Deflection of the beam at the center when under force "F"

ShearStrength  =  3 F X / (2 Y Z^2)
If a column is short then it squashes before it buckles and if it is long then it buckles before it squashes.

A column's resistance to squashing is determined by the Bulk strength.

A  =  Area of the column
B  =  Bulk strength of the column
F  =  Force required to squash a column
=  B A
A column's resistance to buckling is determined by the Young's modulus. Suppose a column is a hollow cylinder.
L  =  Length of the column
R  =  Outer radius of the column
r  =  Inner radius of the column   (r=0 if the cylinder is not hollow)
Y  =  Young's modulus
Q  =  Dimensionless effective length of the column
=   .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
F  =  Force required to buckle the column

F  =  .5 Pi$3$ Y (R$4$-r$4$) / (Q L)$2$
If a column's buckling limit is equal to its squashing limit then (assume r=0)
R/L  =  (Q/Pi) * (2B/Y)$1/2$

Shear strain
Z             =  Height of a beam
X             =  Length of a beam
Y             =  Width of a beam
Area          =  Surface area of the top face of the beam
=  X Y
Force         =  Transverse force on the top face of the beam in the X direction
x             =  Transverse displacement of the top face of the beam when a transvere
force is applied
ShearStress   =  Force / (X Y)
Shearstrain   =  x/Y
ShearModulus  =  ShearStress / ShearStrain
ShearYield    =  Shear stress for which the material deforms irreversibly
ShearStrength =  Maximum shear stress before breaking
ShearStrainMax=  Maximum strain before breaking
ShearToughness=  Energy/Volume absorbed by the material when at maximum strain
=  .5 ShearModulus * ShearStrainMaax^2
=  .5 ShearStrength^2 / ShearModulus

ShearStress   =  ShearModulus * ShearStrain

ShearStrength =  ShearModulus * ShearStrainMax

Tensile strain and shear strain have analogous elastic variables.

TensileStress     <-->  ShearStress
Tensilestrain     <-->  Shearstrain
Tensilemodulus    <-->  ShearModulus
TensileYield      <-->  ShearYield
TensileStrength   <-->  ShearStrength
TensileStrainMax  <-->  ShearStrainMax
TensileToughness  <-->  ShearToughness

The bulk modulus is analogous to the tensile modulus and the strain modulus.

If a material can be compressed indefinitely (until it turns into a neutron star) then it has no concept of a "bulk strength". This applies especially to metals and diamonds. Hence, the "bulk modulus" is usually the only meaningful variable for compression.

Shear
Imbalanced forces within the material
Kelvin-Helmholtz instability
Ring of Saturn

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.

2 * (1 +   PoissonRatio) * ShearModulus  =  TensileModulus

3 * (1 - 2*PoissonRatio) * BulkModulus   =  TensileModulus

The Poisson ratio for most metals is in the range of 1/3 and for rubber it is 1/2. If we assume the Poisson ratio is 1/3 then
(8/3) * ShearModulus  =  TensileModulus
Materials based on carbon chains differ from isotropic materials in that they tend to have a large tensile strength and a low shear strength. These materials are good for wires.

The following figure characterizes the relation between the shear and tensile moduli. The relationship applies well for most metals.

Vertical axis    =  ShearModulus
Horizontal axis  =  TensileModulus / (2 + 2*PoissonRatio)

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

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.

Vickers   Tensile  Tensile  Tensile  Tensile  Poisson
hardness  modulus  strength tough     strain
(GPa)     (GPa)    (GPa)    (kPa)    (GPa)
Hardened steel  .8      250       .55                         .29
Osmium                  590      1.00      893     .25
Cobalt         1.04     209       .76                         .31
Chromium       1.06     279       .282     143     .00101     .21
Tantalum       1.2      186       .90                         .34
Beryllium      1.67     287      .448                         .032
Iridium        2.2      528     2.0       3788     .0038      .26
Sapphire       2.3      345     1.9       5232     .0055      .28
Uranium        2.5      208      .625                         .23
Si-C           2.6      450     3.44     13100     .0076      .19
W-C            2.6      650      .345       92     .00053     .23
Molybdenum     2.74     330      .324      159     .00098     .31
Ti-N           3.0                                            .25
Ti-C           3.2      439      .119                         .19
Titanium       3.42     120      .37       570     .0031      .32
Tungsten       4.6      441     1.51      2585     .0037      .28
Boron          4.9      400     3.1
BN             6.0      100      .083                         .27
Rhenium        7.58     483     1.07      1298     .0024      .30
Rhodium        8.0      275      .951                         .26
Diamond       10.0     1220     2.8       3210     .0023      .069

All hard materials have a small Poisson ratio.

Elements

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

Water           1.0
Nylon           1.2
Gut             1.5
Synthetic       2.5
Aluminum        2.8      <.01
Titanium        4.5       .01
Steel           7.9      <.01
Nickel          8.9       .01
Silver         10.5       .6
Tungsten       19.2       .05
Gold           19.3     24
Rhenium        21.0      6
Platinum       21.4     88
Iridium        22.56    13
Osmium         22.59    12        Densest element
Tungsten is the only dense metal that is not expensive.

Previous to the discovery of tungsten, gold was an uncounterfeitable currency because no material existed that was both more dense and less expensive than gold.

Osmium is the densest element and it is also expensive, making it useful as a currency.

Heavy metal

Color     =  Shear Modulus.  Red is low, orange is medium, and white is high.
Dot size  =  Density
Shear data
Density data
Strength to weight ratio

Color      =  Shear Modulus / Density          A measure of a material's "strength to weight" ratio
Dot size   =  Density
The metals with the highest strength to weight ratio are Chromium, Ruthenium, and Beryllium.

Chromium is common in the Earth's crust and Ruthenium is rare.

Shear data    Density data

Strength and melting point

Dot size   = (Shear Modulus)^(1/3)           An indicator of a material's strength
Dot color  =  Melting point

White  =  Highest melting points
Red    =  Lowest melting points
Blue   =  Elements that are a gas or a liquid at room temperature and pressure.
Liquids and gases have a shear modulus of 0.
Rocket cones are made from materials with a high melting point, a high shear strength, and a high atomic mass. Tungsten is the element of choice, especially because it's vastly cheaper than Rhenium, Osmium, and Iridium.

For carbon, the values are given for diamond form.

Shear data    Melt data

Precious metal

Color    =  Price per kilogram       Red is low, orange is medium, white is high
Blue indicates the element is radioactive with a
short half life
Dot size = -log(SolarAbundance)      The smaller the dot, the more abundant the element
Price data

Xenon is the only expensive non-metal. It is a miraculous anaesthetic.

Size

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

Size data

Poisson ratio

Dot size  =  [Shear modulus / Density]**(1/2)
Color     =  Poisson ratio, ranging from red to white

Hardness

Dot size  =  [Brinell hardness / Density]**(1/2)
Color     =  Poisson ratio, ranging from red to white
Brinell hardness is difficult to define for diamond, although it is substantially larger than that for any other element. A nominal value is used in the table only to indicate that it is large.

History of metallurgy

Stone age
Copper age, 5000 BCE
Bronze age, 3200 BCE
Iron age, 1200 BCE
Carbon age, 1987

The carbon age began in 1987 when Jimmy Connors switched from a steel to a carbon racquet.

Discovery   Yield    Density
(year)    Strength  (g/cm3)
(GPa)

Gold     Ancient   .20     19.3
Silver   Ancient   .10     10.5
Copper   -5000     .12      9.0
Bronze   -3200     .20      9     Copper + Tin.   Stronger than copper
Brass    -2000     .20      9     Copper + Zinc.  Stronger than copper
Iron     -1200     .25      7.9   In the form of steel. Stronger than bronze and brass

Discovery of elements
Discovery  Strength  Density
(year)     (GPa)    (g/cm3)

Carbon     Ancient   1.0      3.5     Diamond
Gold       Ancient    .20    19.3     Pure form exists naturally
Silver     Ancient    .10    10.5     Pure form exists naturally
Sulfur     Ancient    0       2.0     Pure form exists naturally
Copper       -5000    .12     9.0
Bronze (As)  -4200    .20     9       Copper + Arsenic
Bronze (Sn)  -3200    .20     9       Copper + Tin
Tin          -3200    .14     7.3
Antimony     -3000    .10     6.7
Brass        -2000    .20     9       Copper + Zinc
Mercury      -2000    0      13.5
Iron         -1200    .25     7.9     In the form of steel
Zinc          1300    .19     7.1     Date when first produced in pure form
Arsenic       1649    .01     5.7
Phosphorus    1669    0       2.3
Cobalt        1735    .48     8.9     First metal discovered since iron
Platinum      1735    .18    21.4
Nickel        1751    .19     8.9
Manganese     1774    .26     7.2
Molybdenum    1781    .45    10.3
Tungsten      1783    .95    19.2
Chromium      1798    .20     7.2
Osmium        1803    .80    22.6
Iridium       1803   1.19    22.6
Rhodium       1804    .28    12.4
Magnesium     1808    .10     1.74
Aluminum      1827    .02     2.70
Beryllium     1828    .34     1.85
Uranium       1841    .34    19.1
Ruthenium     1844    .42    12.4
Rhenium       1908    .29    21.0
Titanium      1910    .22     4.5
Scandium      1937    .12     3.0

Strength:          Yield strength, an indicator of the material's strength.
Strength/Density:  An indicator of a material's strength-to-weight ratio.
Gold was the densest known element until the discovery of platinum in 1735. This made it difficult to counterfeit as a currency.

The table includes all metals up to zinc, plus a few heavier metals. Zinc was not recognized as an element until 1746.

Wood fires are 200 Celsius short of the copper smelting temperature.

Cobalt was the first metal discovered since iron, and it's discovery inspired people to smelt every known mineral in the hopes of yielding a new metal. By 1800 nearly all of the smeltable metals had been discovered.

Some elements can't be smelted and require electrochemistry to purify. Electrochemistry began in 1800 with the invention of the battery and most of the remaining metals were discovered soon after.

Metals known since antiquity

Horizontal axis:  Density
Vertical axis:    Shear modulus / Density       (Strength-to-weight ratio)

Metals

Horizontal axis:  Density
Vertical axis:    Shear modulus / Density       (Strength-to-weight ratio)
Beryllium is beyond the top of the plot.

Metals with a strength-to-weight ratio less than lead are not included, except for mercury.

Strength to weight ratio

For the strong metals, strength tends scale with density and the strength-to-weight ratio is constant. Beryllium is an outlier with a superlatively large ratio. For the strongest metals,

Shear    Density  Shear/Density   Carbon
strength  (g/cm3)   (GJoule/kg)   smeltable
(GPa)
Magnesium    17      1.74      9.8      no
Beryllium   132      1.85     71.4      no
Aluminum     26      2.70      9.6      no
Scandium     29      3.0       9.7      no
Titanium     44      4.5       9.8      no
Chromium    115      7.2      16.0      yes
Manganese    75      7.2      10.4      yes
Iron         82      7.9      10.4      yes
Cobalt       75      8.9       8.4      yes
Nickel       76      8.9       8.5      yes
Molybdenum  120     10.3      11.7      yes
Tungsten    161     19.2       8.4      yes
The elements that are not carbon smeltable were discovered by electrochemistry. This includes all the light metals.

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
Copper      80   C   -5000      68
Sulfur     200   *   Ancient   420
Nickel     500   C    1751      90
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
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

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

Gibbs energy

Let MO be a metal oxide for which the Gibbs energy of CO is larger than MO and the oxygen binds to the metal preferentially over carbon.

The entropies of most metal oxides are similar and so changing the temperature has little effect on their relative Gibbs energies. CO is special because it is a gas and hence has a larger entropy than the solid metal oxides. As temperature increases the Gibbs energy of CO decreases faster than that of MO and at the critical smelting temperature they are equal. Above this temperature the oxygen unbinds to the metal and binds to carbon.

For the smelting of cobalt,

Standard temperature                 =  T0  =  298 Kelvin  =  25 Celsius
Smelting temperature                 =  Tsmelt
Temperature change                   =  t   =  Tsmelt - T0
Gibbs energy at standard temperature =  G
Entropy at standard temperature      =  S
Gibbs energy at temperature Tsmelt    =  g   =  G - t S
CoO Gibbs energy per mole O2         =  GCoO =  -428.4   kJoule/mole
CO  Gibbs energy per mole O2         =  GCO  =  -274.4   kJoule/mole
CoO entropy per mole O2              =  SCoO =      .12  kJoule/mole
CO  entropy per mole O2              =  SCO  =      .396 kJoule/mole
At the smelting temperature, the Gibbs energies of CoO and CO are equal and the reaction is in equilibrium. Below this temperature oxygen binds to cobalt and above this temperature it binds to carbon. The calculation is approximate because it assumes entropy is a constant as a function of temperature. To calculate the smelting temperature,
gCoO      =      gCO
GCoO - t SCoO  =  GCO - t SCO

t  =  (GCoO - GCO) / (SCoO - SCO)
=  558 Celsius

Tsmelt  =  583 Celsius  =  t + 25 Celsius      (The actual smelting temperature is 525 Celsius)

Smelting thermodynamics
Gibbs       Gibbs            Entropy       Entropy
kJoule/mole  kJoule/mole(O2)  kJoule/mole  kJoule/mole(O2)

Li2O     -561.9    -1123.8
Na2O     -377       -754
K2O      -322.2     -644.4
Cu2O     -146.0     -292.0
Ag2O      -11.2      -22.4

BeO      -579.1    -1158.2
CO       -137.2     -274.4     .198      .396
MgO      -596.3    -1192.6     .0269     .0538
CaO      -533.0    -1066.0     .0398     .0769
VO       -404.2     -808.4
MnO      -362.9     -725.8     .0597     .1194
CoO      -214.2     -428.4
NiO      -211.7     -423.4
CuO      -129.7     -259.4     .0426     .0852
ZnO      -318.2     -636.4
CdO      -228.4     -456.8
HgO       -58.5     -117.0

Fe3O4   -1014       -507       .0146     .0073
Co3O4    -795.0     -397.5

B2O3    -1184       -789
Al2O3   -1582.3    -1054.9     .0509     .0339
Ti2O3   -1448       -965.3
V2O3    -1139.3     -759.5
Cr2O3   -1053.1     -702.1     .0812     .0541
Fe2O3    -741.0     -494.0     .0874     .0583

CO2      -394.4     -394.4     .214      .214
SO2                            .2481     .2481
SiO2     -856.4     -856.4     .0418     .0418
TiO2     -852.7     -852.7
MnO2     -465.2     -465.2     .0530     .0530
MoO2     -533.0     -533.0
WO2      -533.9     -533.9
PbO2     -219.0     -219.0

MoO3     -668.0     -445.3
WO3      -764.1     -509.4

V2O4    -1318.4     -659.2

Cu          0
C (gas)   672.8

Minerals

Spodumene: LiAl(SiO3)2
Beryl: Be3Al2(SiO3)6
Periclase: MgO
Magnesite: MgCO3
Dolomite: CaMg(CO3)2
Bauxite: Al(OH)3 and AlO(OH)

Quartz: SiO2
Rutile: TiO2
Chromite: FeCr2O4
Pyrolusite: MnO2
Hematite: Fe2O3

Hematite: Fe2O3
Pyrite: FeS2
Iron meteorite
Cobaltite: CoAsS
Millerite: NiS
Chalcocite: Cu2S

Chalcopyrite: CuFeS2
Sphalerite: ZnS
Germanite: Cu26Fe4Ge4S32
Zircon: ZrSiO4
Molybdenite: MoS2
Acanthite: Ag2S
Cassiterite: SnO2
Wolframite: FeWO4
Cinnabar: HgS
Platinum nugget
Gold nugget
Galena: PbS

Fluorite: CaF2
Volcanic sulfur
Alumstone: KAl3(SO4)2(OH)6

Ruby

Ruby in a green laser
Synthetic rubies

Emerald

Sapphire

Synthetic sapphire

Diamond

Raw diamond
Raw diamond
Synthetic diamond
Synthetic diamonds

Topaz

Quartz

Crystals
Crystal, polycrystal, and amorphous

Diamond
Diamond
Diamond
Diamond
Diamond

Diamond and graphite
Carbon phase diagram
Corundum (Al2O3)
Corundum
Corundum unit cell

Corundum
Tungsten Carbide
Metal lattice

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

Corundum is a crystalline form of aluminium oxide (Al2O3). It is transparent in its pure orm and can have different colors when metal impurities are present. Specimens are called rubies if red, padparadscha if pink-orange, and all other colors are called sapphire, e.g., "green sapphire" for a green specimen.

Metal impurity   Color

Chromium         Red
Iron             Blue
Titanium         Yellow
Copper           Orange
Magnesium        Green

Price
1 Carat                     =  .2 grams
Price of a 1 Carat diamond  =  C  ≈  1000 \$     (This varies according to quality)
Mass of diamond in Carats   =  M
Price of diamond in dollars =  C M2
Pure gold                   =  24 Karats
3/4 pure gold               =  18 Karats

1837  Gaudin produces the first synthetic ruby.
1905  Bridgman invents the diamond anvil, which reached a pressure of 10 GPa.
He was awarded the Nobel prize for this in 1946.
1910  Synthetic ruby begins to be mass produced.
1928  Sir Parsons produces the first synthetic diamonds.
1954  Hall produces the first commercially successful synthetic diamonds.
1970  First gem-quality synthetic diamonds produced.
2015  Synthetic diamonds reach 10 carats in size.

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  Crush    Compress    Compress
strength                  with grain  against
(g/cm^3)  (Gpa)     (Gpa)  (Gpa)                grain

Balsa         .12    .020      3.7     .012
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
Alder, red    .41                              5820         9800
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             1500  650   1553
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                      1200  715   1668
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     .093
Oak, live     .98    .130     13.8
Ironwood     1.1     .181     21.0
Lignum Vitae 1.26    .127     14.1
Fir, Douglas                                   1700   625   1668
Data #1     Data #2
Alloys

Copper
Orichalcum (gold + copper)
Gold

Alloy of gold, silver, and copper

Strongest alloys
Yield    Density   Yield/Density
strength  (g/cm3)    (GJoule/kg)
(GPa)
Magnesium  + Li        .14     1.43      .098
Magnesium  + Y2O3      .31     1.76      .177
Aluminum   + Be        .41     2.27      .181
LiMgAlScTi            1.97     2.67      .738
Titanium   + AlVCrMo  1.20     4.6       .261
AlCrFeCoNiTi          2.26     6.5       .377
AlCrFeCoNiMo          2.76     7.1       .394
Steel      + Co Ni    2.07     8.6       .241
VNbMoTaW              1.22    12.3       .099
Molybdenum + W Hf     1.8     14.3       .126

Sapphire               .4      3.98      .101
Diamond               1.6      3.5       .457
Magnesium              .10     1.74      .057
Beryllium              .34     1.85
Aluminum               .020    2.70
Titanium               .22     4.51
Chromium               .14     7.15
Iron                   .10     7.87
Cobalt                 .48     8.90
Nickel                 .19     8.91
Copper                 .12     8.96
Molybdenum             .25    10.28
Tin                    .014    7.26
Tungsten               .947   19.25
Rhenium                .290   21.02
Osmium                        22.59
Iridium                       22.56
Alloys can be vastly stronger than their constituent elements. Alloys such as "TiScAlLiMg" are "high entropy alloys", which are a mix of elements in approximately equal proportions.
For comparison, the table includes pure metals, diamond, and sapphire. Large synthetic sapphires and small synthetic diamonds can be constructed. The recently developed LiMgAlScTi alloy is the first metal to outpeform diamond.
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 cannot spark

Column buckling

If too much weight is placed on a column it buckles. Suppose a column is constructed with constant mass and varying density. The lower the density the wider and stronger the column.

Length            =  L
Density           =  D
Mass              =  M  =  π D L R2
Buckling constant =  C
Tensile modulus   =  K
Force             =  F  =  C K R4 L-2      Force requird to buckle the column
Quality           =  Q  =  F / M  =  K M D-2 L-4
The figure of merit for a material for columns is K D2. Balsa wood has a density of .16 g/cm3 and outperforms the strongest alloys.
Yield    Density  Yield/Density  Yield/Density2
strength  (g/cm3)    (GJoule m3/kg2)
(GPa)
Balsa                  .006     .16      .037    .234
Bamboo                 .0079    .35      .023    .064
Magnesium  + Li        .14     1.43      .098    .068
Magnesium  + Y2O3      .31     1.76      .177    .100
Aluminum   + Be        .41     2.27      .181    .080
LiMgAlScTi            1.97     2.67      .738    .276
Titanium   + AlVCrMo  1.20     4.6       .261    .057
AlCrFeCoNiTi          2.26     6.5       .377    .053
AlCrFeCoNiMo          2.76     7.1       .394    .055

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
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 supercede 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.

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

Full list of alloys
Primary  Added   Yield  Break  Stiff  Strain  Poi-  Density Vick  Elong  Yield/   Melt
metal    metals  (GPa)  (GPa)  (Gpa)          sson  (g/cm3)              density  (C)

Magnesium  Li              .16    45                 1.43             .098
Magnesium  Y2O3     .312   .318                      1.76             .177
Magnesium  Tube     .295   .39    49                 1.83        .05  .161
Beryllium           .345   .448  287  .0016  .032    1.85             .186
Aluminum   Be40     .41    .46   185                 2.27        .07  .181
Aluminum   Mg Li    .21    .35    75  .0047          2.51
Aluminum   Cu Li    .48    .53                       2.59             .185
Aluminum   Mg Sc    .433   .503                      2.64        .105 .164
LiMgAlScTi         1.97                              2.67  5.8        .738
Titanium   Be Al                                     3.91
Titanium   Al6V4    .89   1.03   114         .33     4.43   .34  .14          1660
Titanium   VCrMoAl 1.20   1.30                       4.6         .08  .261
Vit 1              1.9                               6.1   5.7
AlCoCrFeNiTih      2.26   3.14                       6.5         .23  .377
Zirconium  Liquid  1.52   1.52    93                 6.57   .56  .018 .231
AlCoCrFeNiMo       2.76                              7.1              .394
AlMohNbTahTiZr     2.0    2.37                       7.4              .270
Inconel 718                                          8.19
Copper     Be      1.2    1.48   130         .30     8.25             .145     866
CrFeNiV.5W                2.24                      ~8.5
Iron       Co Ni   2.07   2.38                       8.6         .11  .241
Iron       Cr Mo                                     9      .32
Nickel     Cr      1.2    2.3    245         .32     8.65  6.6
TiZrNbHfTa          .93                              9.94  3.83  .5
TiVNbMoTaW                                          11.70  4.95
VNbMoTaW                                            12.36
NbMoTaW                                             13.75
Molybdenum W45Hf1 ~1.8    2.14                     ~14.3   3.6   .126
Tungsten   MoNiFe   .62    .90   365                17.7         .10  .035

Yield:     Yield modulus
Break:     Tensile strength (breaking point)
Stiffness: Young's modulus
Strain:    Fractional strain at the breaking point
Poisson:   Poisson ratio
Many properties of alloys are approximately equal to a linear sum of the properties of its constituent elements. This applies for density, stiffness modulus, and Poisson's ratio.

Many properties of alloys can be dramatically different from those of its constiuent elements. This applies for the yield modulus, the tensile breaking modulus, and the hardness.

For aluminum alloys, density = 2.71 - .01 Mg - .079 Li.

Magnesium strengthens when alloyed with aluminum, nickel, copper, and neodymium.

Data:    TiVNbMoTaW    AlTiNbMo½Ta½Zr    Mg    Be+Al    Aluminum+Mg+Li    Table    Al+Be    Mg + Li    Mg alloys    Elasticity    Ti alloy    Ti alloy    Ti alloy textbook    Liquidmetal    Mg + tubes    Elasticity table    Al Cu Li    Al + tubes    Mg + tubes    W + Mo    Al Mg Sc    Fe + Co + Ni    Li2MgSc2Ti3Al2    Entropy survey    Entropy survey    Entropy survey *    CrFeNiV½W    Entropy rev 2014    Nickel Chromium    Copper textbook    TiZrNbHfTa

Vickers hardness
Min   Max

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

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
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

Conductivity

White: High conductivity
Red:   Low conductivity

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
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
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

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                                         |

Electrical and thermal conductivity of a wire
L  =  Length of wire            meters
A  =  Cross section of wire     meters^2
_______________________________________________________________________________________________________
|
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

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

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)

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
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
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

Dimond iso 10    40000
Diamond     e-16  2320                                     .509
Tube       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

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

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

Refractive index
Index
Vacuum         1
Air            1.000293
Water          1.333
Olive oil      1.47
Ice            1.309
Glass          1.5
Plexiglass     1.5
Cubic zirconia 2.15
Diamond        2.42

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
Niobium-Titanium     10        15       2   Cheaper than Niobium3-Tin.  Ductile
Niobium3-Aluminum

Technetium           11.2               2
Niobium               9.26       .82    2
Tantalum              4.48       .09    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
Protactinium          1.4
Thorium               1.4
Thallium              2.4
Molybdenum             .92
Zinc                   .85       .0054
Osmium                 .7
Zirconium              .55
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.

Superconducting critical current

The maximum current density decreases with temperature and magentic field.

Maximum current density in kAmps/mm2 for 4.2 Kelvin (liquid helium):

Teslas               16    12     8      4    2

Niobium3-Tin         1.05  3
Niobium3-Aluminum           .6   1.7
Niobium-Titanium            -    1.0    2.4   3
Magnesium-Boron2-C          .06   .6    2.5   4
Magnesium-Boron2            .007  .1    1.5   3

Maximum current density in Amps/mm2 for 20 Kelvin (liquid hydrogen):

Teslas               4     2

Magnesium-Boron2-C   .4   1.5
Magnesium-Boron2     .12  1.5

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

History of superconductivity
1898  Dewar liquefies hydrogen (20 Kelvin) using regenerative cooling and
his invention, the vacuum flask, which is now known as a "Dewar".
1908  Helium liquified by Onnes. His device reached a temperature of 1.5 K
1911  Superconductivity discovered by Onnes.  Mercury was the first superconductor
found
1935  Type 2 superconductivity discovered by Shubnikov
1953  Vanadium3-Silicon found to be superconducting, the first example of a
superconducting alloy with a 3:1 chemical ratio.  More were soon found
1954  Niobium3-Tin superconductivity discovered
1955  Yntema builds the first superconducting magnet using niobium wire, reaching
a field of .7 T at 4.2 K
1961  Niobium3-Tin found to be able to support a high current density and
magnetic field (Berlincourt & Hake). This was the first material capable of
producing a high-field superconducting magnet and paved the way for MRIs.
1962  Niobium-Titanium found to be able to support a high current density and
magnetic field.  (Berlincourt & Hake)
1965  Superconducting material found that could support a large
current density (1000 Amps/mm^2 at 8.8 Tesla)
(Kunzler, Buehler, Hsu, and Wernick)
1986  Superconductor with a high critical temperature discovered in a ceramic
(35 K) (Lanthanum Barium Copper Oxide) (Bednorz & Muller).
More ceramics are soon found to be superconducting at even higher temperatures.
1987  Nobel prize awarded to Bednorz & Muller, one year after the discovery of
high-temperature superconductivity.  Nobel prizes are rarely this fast.

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

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
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.

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
(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

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
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.

=  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
=  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 and calculated the speed of sound in air to be
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
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                Petyr Baelish
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.
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