The constants, functions, and relationships provided here are accessable from anywhere in the world.

Adiabatic Process

A thermodynamic process during which no heat is allowed to enter or escape.

Angstrom (A^o)

This very small unit of length is usually associated with wavelengths, and is equal to 1 x 10^-8 cm.

ASCII Code

Astronomical Unit (AU)

A unit of length used in astronomy equal to the mean distance of the earth from the sun or about 93 million miles (~ 150 million km).

AU = 1.496 x 10⁸ km

Atmosphere (atm.) = 1.01325 x 10⁵Pa = (1.01325 x 10⁵ N/m²) = 76 cm Hg

Atomic mass unit (amu)

amu = m_u = 1.660 538 73 x 10^-27 kg

Avogadro's number

The number of molecules in a mole or in a mass in grams of substance equal numerically to its molecular weight, i.e., 6.0235 x 10²³ molecules/mole

Babylonian Numbers

Babylonian Numbers in Cuniform format

Black body

If, for all values of wavelength of the incident radiant energy, all of the energy is absorbed the body is called a black body.

Boltzmann's constant (k)

Thermal energy coefficient = 1.380 6503 x 10^-23 Joule K^-1(SI Units)

Thermal energy coefficient = 1.380 6503 x 10^-16 erg K^-1(cgs Units)

Boyle's Law

An empirical law, exact only for an ideal gas, which states that the volume of a gas is inversely proportional to its pressure at constant temperature. PV = constant

P₁V₁ = P₂V₂

British Thermal Unit (BTU)

The amount of heat required to raise the temperature of 1 lb. of water through 1^oF. Usually the temperature range is from 60^o to 61^oF. It is written BTU. 1 BTU = 252 calories = 778.3 ft-lb. = 1,055 joules.

Cardioid

Cardioid curve

Catenary

Catenary curve of hanging string or rope

A catenary curve is traced out by a perfectly flexible rope or chain when suspended between two points. The equation of a catenary curve is y = a cosh(x/a). The radius of curvature at any point along the catenary is y²/a.

Celsius scale

Also previously known as Centigrade scale. On the Celsius scale, the temperature of freezing water is defined as 0^oC, and the temperature of water boiling under normal conditions is defined as 100^oC. To convert Fahrenheit readings to Celsius use the following equation: C = (5/9)(F - 32).

Temperature Conversions

Charles' Law

An empirical law, exact only for an ideal gas, which states that the volume of a gas is directly proportional to its temperature at constant pressure. P/T = constant

V₁ / T₁ = V₂ / T₂

Classification of Animals and Plants

In taxonomy, organisms are classified into categories called taxa (singular, taxon). A species is given a name consisting of a species name and a genus (plural, genera) name. For example, the domesticated dog is categorized into the genus Canis and is given the name Canis familiaris. Closely related animals are grouped in the same genus. Thus, the wolf, Canis lupis, and the coyote, Canis latrans, share the same genus with the domesticated dog. Genera that share related features are grouped in a family. Related families, in turn, are grouped in orders, which are grouped successively in classes, phyla (singular, phylum) (or divisions for fungi and plants), and finally, kingdoms. A good way to remember the classification order of taxa is to recall the phrase "Kings Play Chess On Fine Green Sand," in which each word gives the first letter of each taxon from kingdom to species.

Coulomb constant

K = 1/(4p e_o) = 8.988 x 10⁹N-m² C²

Curie point (magnetic)

All ferro-magnetic substances (alloys) have a definite temperature of transition at which the phenomena of ferro-magnetism disappear and the substances becomes merely paramagnetic. This transition temperature is called the "Curie Point" and is usually lower than the melting point. Above the Curie temperature, a ferromagnetic solid is paramagnetic.

Cycloid

Cycloid curve

Dalton's law of partial pressures

The pressure exerted by a mixture of gases is equal to the sum of the separate pressures which each gas would exert if it alone occupied the whole volume. This fact is expressed in the following formula:

PV = V(p₁ + p₂ + p₃, etc.)

Degrees of Freedom

The number of variables (temperature, pressure, and composition) which can be changed independently without changing the phase or phases of the system.

Diamagnetic Substances

Diamagnetic substancs are those which tend to move away from a stronger magnetic field (weakly repelled). The permeability of a diamagnetic substance is less than unity, which tends to disperse lines of magnetic flux. i.e., bismuth, beryllium, carbon, silicon, sulfur, zinc, and others are diamagnetic. (See also, paramagnetic and ferromagnetic substances.)

Diffraction

A change that light undergoes in passing by the edges of opaque bodies, or through narrow slits, or in being reflected from ruled surfaces and in which the rays appear to be deflected and to produce patterns of fringes of parallel light and dark or colored bands. Similar effects can be produced by other waves such as sound waves.

Diffusion

The transfer of matter within another solid, liquid or gas. The process has to do with particles of solids, liquids, or gases intermingling as the result of their spontaneous movement caused by thermal agitation and in the case of dissolved substances movement from a region of higher concentration to one of lower concentration. Diffusion in solids is important for doping semiconductors, surface hardening metals, annealing, and envirnmental embrittlement. Solid state mechanisms include: vacancy diffusion and interstitial diffusion.

Diffusion, Steady State

Fick's 1st Law: J = -D(dC/dx), where J is the flux in g/m²sec, D is the diffusion coefficient in cm/sec, C is the concentration of diffusing species in g/cm³ and x is the diffusion path length in cm.

Diffusion, Transient (Non-Steady State)

Fick's 2nd Law: dC/dt=d(D*dC/dx)/ dx

Doppler Effect

Named for Christian J. Doppler (~ 1905). It has to do with the change in frequency with which waves, such as sound or light, from a given source reach an observer when the source and the observer are in motion with respect to each other. The frequency increases or decreases according to the speed at which the distance inversely decreases or increases between the source and the observer.

e

e = 2.71828 18284 59045 23536

eccentricity (of an ellipse or astronomical orbit)

Ratio of the distance between the foci to the length of the major axis. It is used to measure an ellipse's deviation from circularity and is by definition less than one. The orbital eccentricity of earth is 0.017 (a perfect circle approaches zero).

Electron charge

e = -1.602 176 462 x 10^-19 Coulomb (SI Units)

Electron mass

m_e = 9.109 381 88 x 10^-31 kg

Eutectic

A term applied to a mixture of two or more substances which has the lowest melting point.

Fatigue

Tendency to fracture under cyclic stresses.

Ferromagnectic substances

Ferromagnetic substances are strongly attracted by magnetic fields. The permeability of a ferromagnetic substance is much greater than unity (i.e., 5,000), which tends to greatly concentrate the lines of flux. Cobalt, Iron and Nickel are examples of ferromagnetic metals. (See also, diamagnetic and paramagnetic substances.)

Fibonacci Numbers

An infinite sequence of integers 1,1,2,3,5,8,13,21,34,... of which the first term is 1 and each succeeding term is the sum of the two immediately preceding terms. This series was discovered by the medieval Italian mathematician, Leonardo ("Fibonacci") da Pisa about 1250. It commonly is found in naturally occurring spiral-shaped patterns of flowers and sea shells.

The daisy has a double spiraling pattern of florets in it's head. There are 21 spirals in a clockwise direction and 34 counterclockwise. The daisy's spiral ratio of 21:34 corresponds to two adjacent Fibonacci numbers, as do the pine cone's 5:8 and the pineapple's 8:13 -- and the same is true of many other plants with a spiral leaf-growth pattern.

Fibonacci numbers, besides bearing a curious relationship to botany, also appear to exert a strange influence on art and architecture. The ratio between any two adjacent Fibonacci numbers after 3 is 1:1.618. This is the so-called Golden Ratio, or Golden Section, which has intrigued experts for centuries because of its connection with esthetics. The ratio occurs in pentagons, circles and decagons -- but is most notable in the Golden Rectangle, a figure whose two sides bear the magic relationship to each other.

Rules for generating Fibonacci Numbers: F₀ = 0, F₁ = 1, F_{n + 2} = F _{n + 1} + F_n, n >= 0

 

Fraunhofer Lines

The dark absorption lines in the solar spectrum.

Gas constant

R = 8.31 J/mole K = 1.99 cal/mole K

Gibbs energy (G)

An important function in chemical thermodynamics, defined by G = H - TS, where H is the enthalpy, S is entropy, and T the thermodynamic temperature, or referred to simply as "free energy".

Gibbs phase rule

The statement that at equilibrium, F = C - P + 2, where F = Degrees of Freedom, C = the number of components, and P = the number of phases.

Golden Rectangle

The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms; for years experts have been finding examples in everything from the edifices of ancient Greece to art master pieces.

The dimensions of the sides of a Golden Rectangle are based on the ratio of two adjacent Fibonacci numbers -- 1:1.618. This is sometimes called the Golden Ratio, or Golden Section.

Golden Rectangle Construction

 

The geometric construction of a Golden Rectangle begins with a square (in green blue), which is then divided in two equal parts by the dotted line EF. Point F now serves as the center whose radius is the diagonal FC. An arc of the circle is drawn (CG) and the base line AD is extended to intersect it. This becomes the base of the rectangle. The new side HG is now drawn at right angle to the new base, with the line BH brought out to meet it. The resultant Golden Rectangle has one unusal property: if the original square is taken away, what remains will still be a Golden Rectangle CDGH. Once again, the ratio of the different sides of the Golden Rectangle is always, 1 to 1.618. The Golden Rectangle was used in the Parthenon design as seen in the following figures.

Graham's law

The relative ratios of diffusion of gases under the same conditions are inversely proportional to the square roots of the densities of those gases.

Gram mole, gram formula weight, gram equivalent

A mass in grams numerically equal to the molecular weight, formula weight, or chemical equivalent, respectively. Gram mole = gram molecular weight or gram molecular.

Gravity

Gravitational constant: G = 6.670 x 10^-11 N-m³/kg²

Acceleration due to gravity at sea level, lat. 45^o: g = 9.806 m/s²

Greek alphabet

A; a;   alpha   (a)

B; b;   beta   (b)

G; g;   gamma   (g)

D; d;   delta   (d)

E; e;   epsilon   (e)

Z; z;   zeta   (z)

H; h;   eta   (h)

Q; q;   theta   (q)

I; i;   iota   (i)

K; k;   kappa   (k)

L; l;   lambda   (l)

M; m;   mu   (m)

N; n;   nu   (n)

X; x;   xi   (x)

O; o;   omicron   (o)

P; p;   pi   (p)

R; r;   rho   (r)

S; s;   sigma   (s)

T; t;   tau   (t)

U; u;   upsilon   (u)

F; f;   phi   (f)

C; c;   chi   (c)

Y; y;   psi   (y)

W; w;   omega   (w)

Hall effect

When a steady current is flowing in a steady magnetic field, electromotive forces are developed which are at right angles both to the magnetic force and to the current and are proportional to the product of the intensity of the current, the magnetic force and the sine of the angle between the directions of these quantities.

Hardness Conversion Chart

Helmholz energy (A)

A thermodynamic function defined as, A = E - TS, where E is the internal energy, S the entropy, and T the thermodynamic temperature.

Hyperbolic functions

sinh x = (e^x - e^-x)/2

cosh x = (e^x + e^-x)/2

tanh x = sinh x / cosh x

ctnh x = 1/tanh x

sech x = 1/cosh x

csch x = 1/sinh x

cosh²x - sinh²x = 1

Index of refraction

For any substance, this is the ratio of the velocity of light in a vacuum to its velocity in the substance. It is also the ratio of the sine of the angle of incidence to the sine of the angle of refraction. In general, the index of refraction for any substance varies with the wavelength of the refracted light.

Isothermal Process

A thermodynamic process in which the temperature is maintained constant.

Joule is a unit of energy or work. Joule = newton-meter = watt-second = 10⁷ ergs.

Kepler's Laws

1. The orbits of the planets are ellipses with the sun at one focus.

2. An imaginery line from a planet to the sun sweeps out equal areas in equal time intervals whether the planet is close to or far from the sun.

3. The ratio of the average radius of a planet's orbit about the sun, r, cubed and the planet's period, T, (the time for it to travel around the sun once) squared, is a constant for all planets. This law can be expressed as, r³/T² = k.     Therefore, the ratio of the squares of the periods of revolution of two planets is the same as the ratio of the cubes of their average distances from the sun, as T₁²/ T₂² = r₁³/ r₂³.

Kip

A unit of load equalling 1000 pounds, or 453.59 kilograms.

ksi

kips per square inch = 1000 psi = 6.894 MPa. It is a common English unit of pressure.

Leaf Identification Forms

Light

The speed of light in a vacuum (c) = 2.997 925 0 x 10⁸ m s^-1(SI Units)

The speed of light in a vacuum (c) = 2.997 925 0 x 10¹⁰ cm s^-1(cgs Units)

Line, Straight

The straight line equation at data point (x, y) is: y = mx + b

(where, m = slope, b = intercept)

Logarithm conversion

ln x = 2.3026 log x

log x = 0.4343 ln x

Mechanical/Heat equivalence

Mechanical equivalent of heat = 4.185 J/cal

MPa

megapascal = 1,000,000 pascals = 145.03 psi. It is a common metric unit of pressure.

Mho

Unit of conductance; reciprocal of ohm

Moh

An old hardness scale used mainly for testing the hardness of minerals. It has a range of 10:

1. Talc (softest)

2. Gypsum

3. Calcite

4. Fluorite

5. Apatite

6. Orthoclase

7. Quartz

8. Topaz

9. Corundum

10. Diamond (hardest)

You can remember these ten by using the following odd sentence as a memory device: "The Girls Can Flirt And Other Queer Things Can Do." Some other common references: fingernail ~ 2.5, penny ~ 3, window glass or a knife blade ~ 5.5, steel file ~ 6.5

Molal solution

Contains one mole per 1000 grams of solvent.

Molar solution

Contains one mole or gram molecular weight of the solute in one liter of solution.

Mole

A mass numerically equal to the molecular weight.

Momentum (p): p = mv

Moore's Law

In 1965, Intel's Gordon Moore created a "law" that became shorthand for the rapid growth of technology. He predicted that the number of transistors on a chip would grow exponentially every year.

Nanostructure

Assemblages of tens to hundreds of atoms or molecules in a space with a diameter of less than 50 nanometers (a nanometer is about the size of two large atoms or four small ones).

Newton's Law of Cooling

The rate of cooling of a body under given conditions is proportional to the temperature difference between the body and and its surroundings.

Newton's Laws of Motion

1. Inertia (Unless acted upon by an outside force, an object at rest will remain at rest and an object in motion will remain in motion at the same speed and in the same direction.)

2. F = ma (When an outside force acts on an object, the object will accelerate, and the rate of acceleration will be proportional to the force and inversely proportional to the moving mass.)

3. Force = -Force (For every action force there is a reaction force equal in magnitude and opposite in direction.)

Occam's Razor

When choosing between two competing theories to describe a phenomenon, medieval English scholastic philosopher William Occam (ca. 1337)said, the simplest explanation is the best. Sure, maybe dachshunds exist on Earth not because of selective breeding but because aliens brought them here, but why make more assumptions than necessary? (a.k.a. William of Ockham)

 

Ohm's Law:

Current flowing in a conductor varies diectly with the potential difference (voltage) and inversely with the resistance. (I = V/R) First attributed to Georg Simon Ohm (1863)

Paramagnetic Substances

Paramagnetic substances tend to move into a stronger magnetic field (weakly attracked). The permeability of a paramagnetic substance is greater than unity, which tends to concentrate the lines of flux. Aluminum, Chromium, Manganese, Molybdenum, Sodium are some examples of paramagnetic metals. (See also, diamagnetic and ferromagnetic substances.)

Pareto Chart

Histogram chart useful for categorizing a sample or population and identifying the 20% of the categories that are causing 80% of the problems or activity.

Parsec

pc = 3.0856 x 10¹³ km = 3.2615 LY = 2.06265 x 10⁵ AU

Peltier Effect

In 1834, the French physicist, Jean C. Peltier (1785-1845), observed that heat energy was absorbed at one junction and liberated at the other when a electrical current passed around a circuit of two dissimilar metal conductors in series. This is basis of the operation of non-mechanical refrigeration modules. (See also the complementary Seebeck effect.)

Periodic table

An arrangement of chemical elements based on the periodic law. Ref. WebElements.

Permeability of free space

m_o = 4p x 10^-7 N/A² (henry/meter)

Permittivity of free space

e_o = 8.85 x 10^-12 C² /N m² (farad meter)

pH and Acid/Base Concentration

pH = - log₁₀[H⁺]; molar concentration

                                                                         pH Scale

1         2         3         4       5       6       7       8     9     10     11      12     13   14

Strong Acid           Weak Acid           Neutral             Weak Base             Strong Base

Bases turn red-litmus >> blue

Acids turn blue-litmus >> red

Acid salt: NH₄Cl (results from weak base + strong acid)

Basic salt: NaCN (results from strong base + weak acid)

Neutral salt: NaCl (results from strong base + strong acid)

Amphoteric: Capable of reacting chemically either as an acid or as a base.

pi

p = 3.14159 26535 89793 23846

Pilot's Phonetic Alphabet

A ~ Alpha

B ~ Bravo

C ~ Charlie

D ~ Delta

E ~ Echo

F ~ Foxtrot

G ~ Golf

H ~ Hotel

I ~ India

J ~ Juliet

K ~ Kilo

L ~ Lima

M ~ Mike

N ~ November

O ~ Oscar

P ~ Papa

Q ~ Quebec

R ~ Romeo

S ~ Sierra

T ~ Tango

U ~ Uniform

V ~ Victor

W ~ Whiskey

X ~ X-ray

Y ~ Yankee

Z ~ Zulu

Planck's constant (h)

h = 6.626 196 x 10^-34 joule sec (SI Units) = 4.136 x 10^-15 eV sec

h = 6.626 196 x 10^-27 erg sec (cgs Units)

Pressure

1 atm = 760 mm Hg = 14.70 lb/in = 101,325 Pa = 1.01325 bar

Prime Numbers

Prime numbers are numbers which cannot be evenly divided except by themselves or by 1. They are useful

in number theory. The following is a list of 168 prime numbers that can be found between 1 and 1000:

    2       3       5       7    11     13     17     19     23     29     31     37     41     43     47     53     59     61     67     71

  73     79     83     89     97   101   103   107   109   113   127   131   137   139   149   151   157   163   167   173

179   181   191   193   197   199   211   223   227   229   233   239   241   251   257   263   269   271   277   281

283   293   307   311   313   317   331   337   347   349   353   359   367   373   379   383   389   397   401   409

419   421   431   433   439   443   449   457   461   463   467   479   487   491   499   503   509   521   523   541

547   557   563   569   571   577   587   593   599   601   607   613   617   619   631   641   643   647   653   659

661   673   677   683   691   701   709   719   727   733   739   743   751   757   761   769   773   787   797   809

811   821   823   827   829   839   853   857   859   863   877   881   883   887   907   911   919   929   937   941

947   953   967   971   977   983   991   997   (1009)

Proton mass

m_p = 1836m_e = 1.672 621 58 x 10^-27 kg

Quadratic equations

quark

Any of several elementary particles that are postulated to come in pairs (as in the up and down varieties)of similar mass with one member having a charge of + 2/3 and the other a charge of - 1/3 and are held to make up hadrons. In about 1964, the term quark was coined by Murray Gell-Mann.

radian

1 radian = 57.29577 95131 degrees

Rydberg constant

R_INFINITY = 10 973 731.568 549 m^-1

Seebeck Effect

The effect by which an electromotive force (voltage) is developed in a closed circuit consisting of conductors of dissimilar metals in series whose junction points are maintained at different temperatures. Such a device is known as a thermocouple, and the cold junction is usually maintained at 0^oC. Iron-constantan, chromel-alumel, and platinum-rhodium are typical metal pairings for thermocouples. The effect was discovered in 1822 by Thomas Johann Seebeck (1770-1831). (See also the complementary Peltier effect.)

Snell's Law

The relation between the angle of incidence i and the angle of reflection r of a light beam which passes from a medium of refractive index n_o to a medium of index n_i : sin i / sin r = n_i / n_o

Solar constant

Solar constant (radiation from sun at earth's mean distance) = 1.35 kw/m²

Sound

Speed of sound in air at STP = 331 m/s

Specific Heat

q = m c(T₂ - T₁)

where, q is the heat absorbed by the material object , m is the mass of the object, (T₂ - T₁) is the change in temperature, and c is the specific heat of the substance.

Spheres

Surface Area of sphere = 4pR² = 12.5663R²

Surface Area of sphere = pD²

Volume of sphere = (4/3)pR³ = 4.1888R³

Volume of sphere = (1/6)pD³

Standard Deviations

1s = 68.268% of area under normal curve

2s = 95.45% of area under normal curve

3s = 99.73% of area under normal curve

Stefan's Law of Radiation

The rate at which energy is radiated from a hot body is proportional to the fourth power of the body's absolute temperature.

Temperature Conversions

Thermodynamics

First Law of Thermodynamics: The total increase in the thermal energy of a system is the sum of the work done on it and the heat added to it.

Second Law of Thermodynamics: Heat flows from an area of high temperature to an area of low temperature: natural processes go in the direction that increases the total entropy of the universe. Where entropy is the disorder in a system.

Trigonometry Functions

tan x = opposite/adjacent = 1/cot x

cot x = adjacent/opposite = 1/tan x

sin x = opposite/hypotenuse = 1/csc x

cos x = adjacent/hypotenuse = 1/sec x

sec x = hypotenuse/adjacent = 1/cos x

csc x = hypotenuse/opposite = 1/sin x

 

Law of Cosines (any triangle): c² = a² + b² - 2ab cos C.

Law of Sines (any triangle): a/sin A = b/sin B = c/sin C.

 

sin²A + cos²A = 1

tan²A + 1 = sec²A

1 + cot²A = csc²A

sin(A ± B) = sin A cos B ± cos A sin B

cos(A +/- B) = cos A cos B -/+ sin A sin B

2sin A cos B = sin(A + B) + sin(A - B)

2sin A sin B = cos(A - B) - cos(A + B)

2cos A cos B = cos(A + B) + cos(A - B)

sin2A = 2sin A cos A

cos2A = cos²A - sin²A

sin x = (1/2i)(e^ix - e^-ix), i = +(-1)^1/2,

cos x = (1/2)(e^ix + e^-ix)

e^ix = cos x + i sin x

Ultimate strength

The maximum stress (tensile, compression, or shear) a material can sustain without fracture; determined by dividing maximum load by the original cross-sectional area of the specimen. Also known as nominal strength or maximum strength.

Volume per gram mole of ideal gas at STP = 22.4 liters

Water

Heat of fusion of ice = 79.7 cal/g = 3.34 x 10⁵ J/kg

Heat of vaporization of water at STP = 539.6 cal/g = 2.26 x 10⁶ J/kg

Wavelength

l = 1/frequency(Hz)

Widmanstätten structure

The structure is characterized by a geometrical pattern resembling a basketweave. It was first observed in meteorites, but is readily produced in many alloys, such as titanium, by appropriate heat treatment. The pattern results from the formation of a new phase along certain crystallographic planes of the parent solid solution. The orientation of the lattice in the new phase is related crystallographically to the orientation of the lattice in the parent phase.

Iron Meteorite with Widmanstätten Pattern

Year, Sidereal

The time in which the earth completes one revolution in its orbit around the sun measured with respect to the fixed stars: 365 days, 6 hours, 9 minutes, and 9.5 seconds of mean time.

Year, Tropical

Period of revolution of the earth about the sun with respect to the vernal equinox: 365.24 days = 3.1556925 x 10⁷ secs

Yield strength

The stress at which a material exhibits a specified deviation from proportionality of stress and strain. An offset of 0.2% is most commonly used when testing metals.