Units
Content
Units, constants and conversions
By Dr J Floor Anthoni (2001-?) www.seafriends.org.nz/books/units.htm Have you forgotten the physical constants of the physical world around us, and the formulas that define how things move? You are not alone. Have you never heard of them? Don't despair, because it is easy to understand how to apply them. The information on this page helps you making your own estimates and calculations (on the back of an envelope), while also furthering your understanding. Please note that this page is updated from time to time. . This page contains a collection of facts and rules, necessary for making ecological estimates and calculations. Print it out for your convenience when studying the topics on this web site.
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metric SI system: units, prefixes, standards dimension-less quantities: quantities that have no unit o ratios: %, ppt, ppm, billion, trillion, etc. o magic ratios and numbers: golden rule, natural logarithm, pi, aspects of paper, images, o angles: degrees, radians, minutes, seconds o density: water, air, Earth's crust, o averages: arithmetic and geometric averages o attenuation: Beer's Law, Lambert's Law, decay, absorption, attenuation o bel and decibel: measuring sound and expressing amplification and attenuation. o mathematical series: fibonacci, exponent, logarithm, scaling/ allometrics: how to scale effects up and down
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time/ frequency: second, minute, day, year, etc. o geological time: Geological Time Table o frequency: Hertz, Becquerel temperature: length, area, volume: oil measures: barrels, volume, mass biomass conversions: dry matter, carbon, energy o geometry: formulas for area and volume calculation o world dimensions: radius, area, volume, seas, etc. o cosmic dimensions: distances in space, etc. o speed, velocity, flow: knot, cfs, speed of sound, escape velocity, Poise mass, force, pressure: energy, work: energy statistics: energy use in 1975 energy conversion formulas: converting various mined fuels, coal, oil, gas. energy costs: the energetic cost of mining and manufacturing. o heat: specific heat (heat capacity), heat of fusion, heat of vaporisation
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radiation: energy per mass, Gray, Rad thermodynamics: the laws of thermodynamics electricity: voltage, current, charge, physical constants: frequencies: oscillation, electromagnetic spectrum biology: biological conversion factors and more NZ vital statistics: important environmental facts about New Zealand Mathematics refresher: how to solve those simple equations
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Jerrard H G (1963-1992): Dictionary of scientific units, including dimensionless numbers and scales. Chapman&Hall Rowlett, Russ (Internet): A dictionary of unitsof Measurements. www.unc.edu/~rowlett/units/ very extensive and accessible Harte, John (1939,1988): Consider a spherical cow, a course in environmental problem solving. Univ Science Books. Physics Fact Book: http://hypertextbook.com/facts/index-topics.shtml use of formula in sample problems, and interesting dimensions.
-- home -- Revised frequently: 20010412,20010830,20020110,20041110,20060901,
The metric system of units and measures
The international system of units (SI units) defines how physical properties are measured, previously known as the MKS (Metre, Kilogram mass and Seconds) system. Units consist of two parts, a prefix coding for magnitude (multiplication or division) and a unit of measure. The unit kg consists of the prefix k for 1000x and g for gram. As a rule, prefixes above 1000 are spelled with capital letters (M=mega=1,000,000x), and so are units derived from a person's name (W=Watt, J=Joule), but kilo k, hecto h and deka da are spelled in lower case. Units are always singular: An elephant has four feet. John is six foot tall. Mary travels to work about 20 kilometre. The bag contained 5 kilogram. Prefixes power of ten 24 prefix name Y 21 Z 18 15 E P 12 9 T G 6 M 3 k 2 h 1 da
yotta zetta exa peta tera giga mega kilo hecto deka
power of ten -24 prefix name y
-21 z
-18 -15 a f
-12 -9 p n
-6
µ
-3 m
-2 c
-1 d
yocto zepto atto femto pico nano micro milli centi deci
In computing, the K unit is used for 210=1024 instead of 1000. Here are the new conventions: power of two 10 20 approximate 3 power of ten prefix name 6 30 40 50 9 12 15 60 18 Ei
Ki Mi
Gi Ti Pi
kibi mebi gibi tebi pebi exbi
Basic SI units
Physical quantity length mass time temperature electric current luminous intensity amount of substance Derived SI units
Unit metre kilogram second Kelvin Ampere candela mole
Symbol m kg s ºK A cd mol
Physical quantity concentration density frequency energy, enthalpy force pressure power electric potential difference electric charge electric capacitance electric resistance electric conductance magnetic flux magnetic flux density (inductance) inductance luminous flux illumination activity
Unit mole per cubic metre kilogram per cubic metre Hertz Joule Newton Pascal Watt Volt Coulomb Faraday, farad Ohm Siemens Weber Tesla Henry lumen lux
Named after Heinrich Rudolf Hertz James Prescot Joule Isaac Newton Blaise Pascal James Watt Count Alessandro Volta Charles Augustin de Coulomb Michael Faraday Georg Simon Ohm Werner von Siemens Wilhelm Eduard Weber Nikola Tesla Joseph Henry Latin for giving light Latin for light Antoine-Henri
Symbol Definition mol /m3 kg /m3 Hz J N Pa W V C F (Ohm) S Wb T H lm lx Bq 1/s kg m2 /s2 kg m /s2 N /m2 J /s J /A/s (J/C) As C /V V /A A /V Vs WB /m2 V s /A cd s m lm /m2 1/s
absorbed dose dose equivalent
Becquerel Gray Sievert
Becquerel L H Gray Rolf Sievert
Gy Sv
J /kg J /kg
Russ Rowlett of the University of Carolina has a very extensive web site about this. The International Bureau of Standards in Paris also has an extensive web site. Bureau International des Poids et Mesure. A very precise unit converter converts one unit into another. Note! On the Seafriends web site, the powers of ten are often written as 1E6 rather than 10^6 or 106. 15000 = 15E3 = 1.5E4 Units are spelled the English way: metres rather than meters. Formulas are spelled out like e = m x c x c, rather than e=mc2. In scientific literature it is conventional to spell m/s as m s-1 , but this is not followed in this web site.
Dimension-less quantities Ratios
% = percent or procent = per hundred = 0.01 o /oo = promille = per thousand = 0.001 ppt = parts per thousand = .001 = 1E-3 ppm = parts per million = .000001 = 1E-6 ppb = parts per billion (Am) = 1E-9 1 billion (American) = 1,000,000,000 = 1 thousand millions = 1E9 1 trillion (American) = 1 million millions = 1E12 1 quadrillion (American) = 1 thousand millions = 1E15 1 billion (European) = 1 million millions = 1E12 1 trillion (European) = 1 million million millions = 1E18 1 order of magnitude = 10 x to decimate originally means, to take 10% (= x 0.9), but is often used as meaning to leave 10% (= x 0.1). Magic numbers and ratios One (1): One is perhaps the most beautiful number of all; the number of symmetry. Figures drawn with an aspect ratio of one, are all perfectly symmetric: the circle, square, polygons, stars and so on. It is the only number for which n / 1 = 1 / n and n = n0 = n1 = nn = nm. Square root of two: The international paper size A0, A1, A2, A3, A4, A5, ... series was designed such that an A0 sized sheet cut in half, would produce two A1 sheets of equal size and equal ratio between height and width (=aspect). There is only one mathematical number that has
this property: the square root of 2. SQR(2) = 1.4142. In this series of paper sizes, the height/width ratio is SQR(2). When cut in half, becomes SQR(2) / 2 = SQR(2) / (SQR(2) * SQR (2)) = 1 / SQR(2). Thus the ratio remains identical, although the cut sheets need to be turned ninety degrees. This is the only number/ratio which inverses when divided by two: 1 / n = n / 2 thus n x n = 2 or n = SQR( 2 ) Golden section: the division of a number so that the whole is to the greater part as that part is to the smaller part. It is also called the divine proportion, first defined by Euclid, and is used by architects in designing buildings (first used for the Parthenon). 1.61803 to 1 or 1 to 0.61803. A rectangle with these proportions is called a golden rectangle and is thought to be more pleasing to the eye. It is calculated by solving the equation 1 / n = n + 1 or n2 + n - 1 = 0 The golden section can directly be calculated as: (1 + SQR(5)) / 2 (see mathematics refresher below) Graphically it is drawn as follows: First draw a square ABCD. Then, find the midpoint M of side AB. Next, use a compass to extend AB to a point E so that ME = MC. Rectangle AEFD is a golden rectangle. To divide AB according to the golden section, use a compass to find a point G on AE so that EF = EG. Fibonacci series The Italian mathematician Leonardo Fibonacci defined the Fibonacci series 0,1,1,2,3,5,8,13,21,34,55,89,144, ..... where each following number is the sum of the two previous (2=1+1, 3=2+1, 144=89+55, ..). It is a series that has a constant progression as found in things that grow in nature. In fact, each following number can be calculated from the previous by multiplying with 1.618033988 (dividing by 0.618033988) and rounding the result. Note the remarkable mathematical quality 1 / n = 1 + n or the Golden Section above. Two (2): The lowest number that makes counting and arithmetic possible. The only number for which n x n = n + n = nn . It is the number that stands for duplication, replication and redundancy. A rectangle with aspect of two, becomes a smaller rectangle with the same ratio when folded or halved twice. Natural number or natural logarithm: e = 2.71828 . e is calculated by the series : e = 1 + 1/1 + 1/2 + 1/(2x3) + 1/(2x3x4) + .... + 1/(2x3x4x..n) ... to infinity. The natural logarithm is an important number for mathematics, especially when relating to physical quantities. Many formulas have been derived with it. e is also related to the most compact notation for information. Where we use the digital (0-9) number system and computers the binary (0-1) number system, the value system based on e expresses information in the most compact way, although it is continuous and not discrete (consisting of steps). Throughout the universe, e is an important quantity. Pi: the ratio of circumference to the diameter of a circle 3.14159... The number obtained by dividing 22 by 7 22/7=3.1428571.. is almost the same, but slightly larger. Pi can be calculated exactly as follows: pi / 2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x ..... x (n+1)/n x n/(n+1) . (John Wallis series), or by a
shorter series, which converges more slowly: pi / 4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 .... (Leibniz series (1673), also discovered by James Gregory (1671)) Aspect ratios of various images: 1.3333 or 4/3 the television frame, Super-8 film and 16mm film formats. Thought to be the ideal proportion matching our eyesight. Modern digital widescreen format is 4x4/3x3= 16/9. 1.5 or 3/2 the 35mm format for both movies (across) and still pictures (along, 36/24 mm). Other film formats: 6/4 cm, 9/6 cm, ? Cinemascope Perfect numbers A number is called perfect, if the sum of its factors adds up to the number itself, like: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. There are only four such perfect numbers: 6; 28; 496; 8128. (Nichomachus Gerasa, 100 AD). The 6th and 7th are: 8,859,869,056 and 137,438,691,328 (Pietro Cataldi, 1603) What is the fifth perfect number?
Angles A circle is 360 degrees = 2 x pi radians One degree = 60 minutes = 60 arcminutes One minute = 1 arcminute = 60 secondes = 60 arcseconds 1 minute over a meridian = 1 nautical mile = 1.852 km Density Density or specific weight is a ratio between weight and volume. Traditionally gram / cubic centimetre = kg / litre or ton/m3. Multiply by 27.7 to get lb/in3, and by 1.33 to get ton/yd3. The density of clean water is defined as 1.000. A rock with density 2.80 weighs 2800 kg / cubic metre = 2.8 t / m3 Density of air at the surface is 0.00013 Densest substance on Earth is iridium 22.5, followed by osmium 22.4, platinum 21.4, rhenium 21.0, neptunium 20.4, plutonium 19.8, gold 19.3, tungsten 19.3, uranium 18.9, ... mercury 13.53, ... lead 11.4, ... iron 7.86, .... Averages The arithmetic average of a, b, c = (a + b + c ) / 3 The geometric average of a,b,c = (a x b x c ) / 3 An arithmetic series is: a + (a+d) + (a+2d) + (a+3d) + .... A geometric series is: a + (a x r) + (a x r x r) + ... An exponential series is: ex = 1 + x + ( x2 ) / ( 1 x 2 ) + ( x3 )/ (1 x 2 x 3 ) + ... ( xn ) / n! A logarithmic series is: ln(1+x) = x - (x2) / 2 + (x3) / 4 - (x4) / 4 + ... Attenuation
Beer's Law and Lambert's Law When a quantity changes continually with a fixed rate of change, it follows either exponential growth or logarithmic decay, depending on whether the factor is larger or smaller than one. Example: 1,2,4,8,16,32,64,128,256,... is exponential growth with a fixed factor of 2. Example: 256,128,64,32,16,... is logarithmic decay with a fixed factor of 0.5. Examples from the natural world are: absorption of light/sound by a medium, radio active decay. The difference between Beer and Lambert is that the first was concerned with the density of the medium, whereas the second was concerned with the distance travelled through the medium. Both concerns are captured in the same formula: Id = I0 x e x x d or ln( Id / I0 ) = x x d Where I0 = starting intensity, Id = intensity at distance d and ex = rate of change for a distance equal to d = 1, and x = the decay constant. If x is larger than 0, there is exponential growth, but if it is less than zero, there is logarithmic decay. For absorption by a medium, x can be written as a constant times concentration : k x c To convert from the natural logarithm ln( to the decimal logarithm log(, correct by: e= 2.71828 Id = I0 x 10 ( x x d ) / e or log( Id / I0) = ( x x d ) / log( e ) = 2.3029 x x x d If the rate of change is known for a unit distance, then the exponent x can be calculated as: x = ln( unit rate of change) For the 1,2,4,8 series above, x can be calculated from the rate of change = 2: 2 = 1 x e x x1 or x = ln(2) = 0.693. Likewise, x = -0.693 for the 256,128,64,.. series. Bel and decibel Originally used for measuring sound amplitudes, the bel and decibel are also used for measuring amplification and abatement. The Bel unit is named after the inventor of the telephone, Alexander Graham Bell (1847-1922). One Bell louder means ten times louder, measured by amplitude. 1 bel = 1 B = log ( I / I0 ) = 10 decibels ; where I = current intensity and I0 is previous intensity, or a sound level of 2E-10 bar. 1 decibel = 1 dB = 10 x log( I / I0 ) The range of human hearing extends from 0 - 120 dB, at which level sound becomes painful, while also causing permanent deafness. The critical level for the start of ear damage is 85 dB. Conversation: 30-60 dB; jet plane 160 dB; Noise from trucks can exceed 80 dB measured at the roadside. People begin to complain when sound levels in their neighbourhood reach 35-40 dB. Mathematical series This section contains the various mathematical series important to nature.
Fibonacci: Leonardo Fibonacci (1175-1250), was an Italian mathematician who helped introduce the Hindu-Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) into western Europe, particularly the zero. He is also known as the originator of a special series of numbers, now called the Fibonacci sequence or the Fibonacci numbers. 1, (1+1=2), (1+2=3), (2+3=5), (3+5=8), (5+8=13), ..... Natural logarithm, e: e = 1 + 1/1 + 1/(1x2) + 1/(2x3x4) + 1/(2x3x4x5) + ....
Scaling / allometrics
Allometrics (Gk: allo=other; metron= measure) is the ability to scale size up and down according to the relationship between size and some other measure. Many of these relationships appear throughout all chapters of this page. For instance, the volume of a sphere is related to its radius as 4/3 x pi x r x r x r. With this formula, the volume of any sphere can be calculated. It is an exact and mathematically provable formula. However, a number of relationships have been established empirically (by experiment), and these are listed here. Note that empirical formulas are subject to some uncertainty. Organisms; weight m in kg. Basal Metabolic Rate of an organism, related to weight: P = 73.3 x m ^ 0.75 (kcal/day), where m= kg. Mammal brain weight, excluding apes & humans = 0.01 x m ^ 0.70 (kg) Ape brain weight = (0.03-0.04) x m ^ 0.66 (kg) Life span (not humans nor birds) = 11.8 x m ^ 0.20 (years) Life span for birds = 28.3 x m ^ 0.19 (years) Humans; weight m in kg. Heart weight = 0.005.8 * m ^ 0.98 (kg) Lung weight = 0.0113 x m ^ 0.98 (kg) about twice the weigh of the heart. Brain weight = 0.085 x m ^ 0.66 (kg) Tidal volume = 7.69 x m ^ 1.04 (ml) Vital capacity = 56.7 x m ^ 1.03 (ml) Blood volume = 65.6 x m ^ 1.02 (ml) Muscle mass = 0.40 x m ^ 1.00 (kg) Skeletal mass = 0.0608 x m ^ 1.08 (kg) Heart rate = 241 x m ^ -0.25 (beats/min) Note: heart rate decreases with weight Blood circulation time = 17.4 x m ^ 0.25 (seconds) Respiratory rate = 53.5 x m ^ -0.26 (per min) Respiratory rate decreases with weight. Oxygen consumption = 1.94 x m ^ 0.79 (ml/s) Glucose turnover = 5.59 x m ^ 0.75 Impact damage Nuclear weapon (power in Mt) area of destruction = 400 x (Mt) ^ 0.67 (km2). (Steel, 1995) Thus a bomb of 10 Mt TNT destroys 1870 km2, equal to an asteroid of 25m radius, density 3
and speed 20,000m/s. Asteroid impact energy for radius r (m), speed v (m/s), density d = (5.0E-13) x r ^3 x v ^2 x d (Mt TNT)
Time, frequency
1 day = 1440 minutes = 86400 sec. The day lengthens by 1 second every 62500 years. exact duration of a year = 365.25 days 1 yr = 8760 hr = 31,560,000 sec = 3.156E7 s 1 eon = 1E9 years. Life started 3.6 eons ago (3,600,000,000 years). The Big Bang 15 eons ago. Earth = 4.6 eons Geological time See Geologic Time Table. Frequency 1 hertz = 1 Hz = 1/s (per second) = 1 c/s (cycles/second) = 2 x pi rad/s = 6.28318 rad/s 1 revolution per minute (rpm) = 0.105 rad/s unit of (radioactive) activity = Becquerel = 1 Bq = 1/s (per second) 1 Curie = 1 Ci = 3.7E10 Bq
Temperature
Temperature is internal energy in a substance, the movement of its molecules. degrees Celsius = degrees Kelvin - 273.16 = (5 / 9) x (degrees Fahrenheit - 32) degrees Fahrenheit = (9 / 5) x degrees C + 32 0 ºK = -273.15 ºC Gas constant = 8.317 J/mol/ºK, amount of heat to warm up a gas Temperature of sun's surface = 5780ºK (as an ideal blackbody radiator); of its interior = 15,000,000 ºK If the sun was burning coal, it would have burnt up in 1500 years.
Length, area, volume
Length 1 m = 39.37 inches (in) = 3.281 feet (ft) = 1.094 yards (yd) 1 inch = 2.5400 cm
1 ft = 0.3048 m 1 km = 1000 m = 0.6214 statute mile (mi) = 0.5400 nautical mile 1 statute mile = 1.6093 km =1609.3 m 1 fathom = 6 ft = 1.829 m 1 furlong (fur) = 201 m 1 league = 1 nautical mile (n mile) = 1.852 km 1 chain = 66 ft = 20.1168m 1 Aengstrom (AE) = 1E-8 m = 0.1 nm = 100 pm (pico metre) Book printing: 1 inch = 6 pica = 72 point Area 1 m2 = 10.76 ft2 = 1.196 yd2 1 km2 = 1000000 m2 = 100 hectares (ha) = 247.1 acres = 0.3861 mi2 1 square nautical mile = 3.4299 km2 1 square mile = 2.59 km2 1 are = 100 m2 1 hectare (ha) = 10,000 m2 = 0.01 km2 1 square inch (in2) = 6.45 cm2 1 square foot (ft2) = 929 cm2 = 0.0929 m2 1 perch = 25.3 m2 1 acre = 0.405 ha = 4050 m2 1 rood (rd) = 0.101 ha Volume 1 litre (l) = 1000 cm3 = 33.81 fluid ounces = 1.057 US quarts 1 fluid ounce = 29.57353 ml (USA) = 28.4 ml (Imperial) 1 cu ft = 0.0283 m3; 1 in3 = 16.4 cm3; 1 yd3= 0.765 m3 1 Imperial pint = 0.5682 l 1 Imperial gallon = 1.2009 US gallons = 4.5461 litre 1 US gallon = 0.8327 Imperial gallon = 3.7854 litre 1 m3 = 1000 l = 35.31 ft3 = 264.2 US gallons = 6.290 barrels 1 acre foot = 1230 m3 =43,560 cu ft 1 bushel (US) = 1 bu = 8 (Imperial?) gallon = 35.23907 litre 1 wheat bushel (US) = 60 pound = 27.216 kg. Barley bushel = 48 pound = 21.772 kg. Oats bu = 32 pound = 14.515 kg 1 rye bu = 56 pound = 25.401 kg. Volume of one mole of a chemical substance at surface temp & pressure = 22.414 litre/mol = 6.0249E23 molecules
Oil quantities and measurements 1 m3 = 6.29 barrels (6.7 barrels/tonne for crude oil; 8.5 for motor spirits; avg 7.3) 1 tonne crude oil = 7.3 barrels average = 256 Imp gall = 301 US gall 1 barrel per day = 50 tonnes oil / yr (average)
1 barrel = 34.97 Imperial gallons = 42.00 US gallons = 159.0 l 1 km3 = 1E9 m3 = 0.2399 mi3 = 810600 acre-feet 1 mile-per-gallon US (mpg) = 0.4251 km/l kilometres-per-litre. 50mpg = 21.26 km/l 1 mile-per-gallon IMPERIAL (mpg) = 0.3541 km/l kilometres-per-litre. 50mpg = 17.71 km/l Sulfur content of gasoline is approximately 350 ppm. Gas quantities and measurements 1 Tcf= 1 Trillion-cubic-foot (USA)= 1E12 cuft = 2.83E10 m3 1 mole of an ideal gas = 22.4136 litre @ 0º C = 24.4651 l @ 25ºC
Geometry pi = 3.14159 Circumference of a circle = 2 x pi x r, where pi = 3.14159 and r = radius Area of a circle = pi x r x r. Area of an ellipse with axes a and b = pi x a x b Surface area of a sphere = 4 x pi x r x r Volume of a sphere = 4/3 x pi x r x r x r Volume of a cone with height h and base radius r = pi x h x r x r / 3 Volume of a pyramid with base area b and height h: = h x b / 3 Surface area of a triangle with base b and height h = b x h / 2 Area of a regular pentagon with side a, = 1.721 x a x a. Hexagon = 2.598 x a x a. Octagon = 4.828 x a x a Area of a trapezium with parallel sides a and b and height h = 0.5 x ( a + b ) x h The distance between two points (x1,y1) and (x2,y2) = SQR( (x2 - x1)2 + (y2 - y1)2 ) Equation or formula for a parabola: y2 = 4 x a x x, where a = focus Equation for an ellipse: x2 / a2 + y2 / b2 = 1 Equation for a hyperbola: x2 / a2- y2 / b2 = 1 World dimensions Radius: equatorial = 6378.163 km. polar = 6356.177 km Circumference: equatorial = 40074 km. polar = 39942 km Mean elevation of continents = 800 m (840m). Mean depth of oceans = 3800m (3729m*, 3730m, 3795m). Contains 97% of all water (not ice). Total surface = 510.1E6 km2. Surface of oceans = 361.3E6 km2 (70.8%) Mixed surface layer avg depth= 75m. Mixed layer volume= Surface area of ice-free land = 133E6 km2 Productive land 88E6 km2 = 8.8 Gha.(66%). Sea ice= 33E6km2. Continents & Islands: 148.8E6 km2. Eurasia= 0.536E14 m2. Africa= 0.298E14 m2. N-America + Central= 0.238E14 m2. S America= 0.179E14 m2. Antarctica= 0.149E14 m2. Oceania= 0.089E14m2. Volume of the earth = 1.083E12 km3 *) see formula above. Average density = 5.518. Internal temperature estimated at 5000ºK Mass of the earth = 5.976E24 kg. Mass of the oceans = 1.35E21 kg = 1.350E9 km3 *
Mass of ice caps and glaciers = 2.9E19 kg. Mass of lakes and streams = 1.3E17 kg. Mass of water in atmosphere= 1.3E16 kg. Mass of the atmosphere = 5.2E18 kg (5.14E18). Mass of N2 in atmosphere 3.95E18 kg. Mass of all organisms (wet weight) = 5E15 kg= 1.3E15 kg dry weight.
* Turekian, Karl K (1968): Oceans. Prentice-Hall.
Cosmic dimensions Mean distance to sun = 149.6E6 km = 497 s = 8.28 minutes Mean distance to moon = 384,000 km = 3.84E8 m = 1.3 light-seconds Geostationary orbit = 35,900 km from surface = 42,200 km from earth's centre (?) Mass of moon = 7.4E22 kg Mass of Earth = 600E22 kg. Diameter of the sun = 109 x Earth's =1,380,000 km = 864,000 km Mass of sun = 330,000 x mass of Earth = 200E26 kg ? Each second, the sun turns 700 million tonnes of hydrogen into 695 million tons of helium. The difference is radiated out. The sun rotates around its axis in 27 days. Sunspot cycles: 11 and 22 years. The more sunspots, the less energy reaches Earth (0.3% max variation; UV 30%). Year 2002 = year of maximum sunspots. In 220 million years, the sun completes an orbit around the centre of the milky way The moon does 12 cycles around the Earth in 354 days Velocity of light in vacuum (Einstein's constant), c = 2.9979E8 m/s (approx 300,000 km/s) 1 light year = 9.454E12 km Speed, velocity, flow 1 knot = 0.5144 m/s = 1.852 km/hr =1.151 mi/hr 1 mile per hour (mph) = 1.61 km/h = 0.447 m/s 1 cfs (cubic foot per second) = 448.9 gal/min = 8.931E5 m^3/yr = 724.0 acre-ft/yr Speed of sound in dry air = 331 m/s Speed of sound in the ocean is between 1450 and 1570 m/s. Speed of sound increases 1.3 m/s for each 0.1% increase in salinity and increases also 0.17 m/s for each bar or 10 m depth Escape velocity from Earth = 11.2 km/s. For orbital velocities see oceans/currents/atmosphere dynamic viscosity : 1 Poise = 100 cP = 100 m Pa s kinematic viscosity: 1 Stokes = 100 cSt = 1 cm2 /s = 1E-4 m2 /s Ocean currents: 1 Sv = 1E6 m3/s
Stokes' Law of settling velocities, as defined in the gram-cm metric system
[from Karl K Turekian (1968) Oceans. Prentice-Hall] How fast does a small particle sink? velocity v = [ D x D x ( dp - df ) x g ] / [ 18 x p ] cm/s where D= diameter cm, dp= density of particle g/cm3, df= density of fluid g/cm3, g= acceleration of gravity = 980cm/sec2, p=viscosity in poise g/sec cm. For a sphere of quartz (dp=2.6) falling through ocean water (df=1.0) at a temperature of 10ºC (p=0.0140 poise) and diameter D=0.2 mm=0.02 cm, the terminal velocity is 2.48 cm/s. Over this diameter, a correction (Oseen's equation) is necessary due to turbulence. This equation predicts a lower speed.
radius microns 1 10
velocity time to fall to cm/s ocean bottom 4000m 0.00025 51 years 0.025 185 days 1.8 days 2.7 days
100 (Stokes) 2.5 100 (Oseen's) 1.7
See also Heezen& Hollister's diagram of current velocities for erosion, transportation and sedimentation, oceano/soil51.gif For particles larger than 1mm their speed increases ten fold for a hundred fold increase in diameter thus following a square root function. A 1mm particle sinks at about 20cm/s. A one metre boulder would sink at 6.3 m/s or 22.7 km/h, at fast bicycling speed.
Mass, force, pressure
1 kg (weight/force) = g Newtons, where g = acceleration of gravity = 9.8066 m/s/s 1 gram (g) = 15.43 grains = 0.03527 ounces avoirdupois = 0.03215 troy ounces 1 ounce avoirdupois (oz) = 28.34952 g 1 kilogram (kg) = 2.205 pounds (lb) = 35.27 ounces avoirdupois (oz) 1 lb = 0.4536 kg; 1 lbf = 4.45 N 1 stone = 14 lb = 6.35 kg 1 metric ton (t, tonne) = 1000 kg = 1.102 short tons = 0.9842 long tons = 2204.6 lb 1 long tonne = 1016.047 kg, previously also used for deadweight tonne. 1 deadweight ton (dwt) = 1 t (the difference between an empty and fully laden ship, including fuel & passengers) 1 newton (N) = 1E5 dynes = 0.1020 kg force =0.2248 lb force 1 bar = 100 kPa = 1E5 Pa = 1E5 N/m2 = 10.20 t/m2 = 0.9869 atmosphere = 750 torr 1 atmosphere = 1.013 bar = 14.7 lb/in2 = 76.0 cm of mercury = 1.013E6 dyne/cm/cm = 1013 kPa 1 inch of mercury (inHg) = 33.9 mb = 3.390 Pa 1 torr = 1mm mercury = 1.333 mbar 1 ton/in2 = 15.4 MPa 1 pound/acre = 1.12085 kg/ha 1 pound/in2 (psi) = 1 lbf/in2 = 6895 Pa =6.805E-2 atm = 51.71 mm Hg = 6.859E-2 bar
1 atomic unit of mass = 1 u = 1.66043E-27 kg Mass of proton = 1.672614E-27 kg Mass of neutron = 1.674920E-27 kg Mass of electron = 9.109558E-31 kg Standard acceleration of free fall on Earth = 9.80665 m/s/s Universal gravitational constant G = 6.672E-11 N m2 /kg2 (in calculating the attraction between two masses) Atmosphere at sea level: mass density = 1.225 kg/m3. Pressure = 1.01325 bar Molecules/litre = 2.547E22. Average temperature = 15.0 ºC Molecular weight of air = 28.96 g/mole. Oxygen = 2 x 16 = 32. Nitrogen = 2 x 14 = 28. Centripetal acceleration c = - (m x v x v ) / r , where v = velocity, m = mass and r = radius. (m/s/s) Gravity attraction of Earth (free fall acceleration) c = ( G x mE ) / r x r , where G= gravitational constant, mE= mass of Earth c= (6.67E-11 x 6E24 ) / r x r = 400E12 / r x r (attraction of 1 kg by Earth's gravity) for each orbit, the two above must balance : 400E12 / (r x r) = v x v / r , thus v = SQR(400E12/ r)
Energy, work, power
1 Joule (J) = 1 N m = 2 kg m/s = 1E7 ergs = 1 watt-second (Ws) = 0.2390 calories (cal) One joule is the amount of energy used when a force of 1 newton moves an object 1 metre in the direction of the force. 1 erg = 1 g cm cm = 1E-7 J 1 food calorie = 1 kcal = 4184 J 1 kilojoule (kJ) = 1000 J = 0.9484 Btu (British thermal units) = 737.6 foot pounds (ft lb) = 2390 cal = 0.2778E-3 kWhr 1 Btu is the quantity of heat needed to raise the temperature of 1 pound of water by 1 °F. 1 British thermal unit (Btu) = 1055.056 J 1 therm = 100,000 Btu = 106 MJ 1 horsepower (hp) = 0.746 kW; 1 kW = 1.34 hp 1 kilowatt-hour = 3600 kJ = 3.6 MJ = 3413 Btu = 859.845 kilocalories (kcal) = 1.98E6 footpounds = 0.034 therms 1 kilowatt-year = 31.56E9 J 1 kilowatt (kW) - 1 kJ/s = 1.341 horsepower (Hp) 1 quad = 1 quadrillion BTU = 1E15 Btu = 1.055E18 J = 1.055 EJ = 2.930E11 kWh = 0.001 Q 1 langley = 1 cal/cm2 ; 1 Langley/minute = 0.6974 kW/m2 1 electronvolt = 1.602E-19 J (1.502???)
1 litre gasoline = 35.2E6 J = 8413 kcal. 1 kiloton of TNT explosives = 4.2E12 J; 1 ton TNT = 4.2E9J; 1kg TNT = 4.2E6 J (Ehrlich, 1977) 1 tonne of coal = 8000 kWh = 28.8 GJ (Average coal burns at 28.8 MJ/kg) Combustion of dry biomass = 15-30 kJ/g
A small car example: a small motor car's engine is about 40 kW energy output = 40 kJ/s = 40 x 1.341 = 53.6 Hp In one hour (=3600 s), this car can produce 3600 x 40 kJ = 144000 kJ = 144E6 J = 144/4.2E6, equal to 34.2 kg of TNT explosives. An internal combustion motor of this capacity uses about 10 litre fuel per hour (352E6 J), travelling at 120 km/hr (at 32 KW): An internal combustion motor runs at near 30% efficiency, requiring an amount of fuel equal to 3x its energy output. The energy content of gasoline is about 46 MJ/kg or about 37 MJ/litre. The car above would have consumed 144 x 3 MJ = 11.7 litres. Energy statistics relating to the airplane attack on the World Trade Centre in New York, 11 September 2001: Towers: 410m tall; total weight 1.25 Mt; Collapse energy = 0.5 x 410 x 1.25E9 x 9.8= 2.5E12 J = 597 t TNT (Sci-Am: 500t ). Fuel: Boeing 767 fuel capaciry 23,980 Gall (USA) = 90,800 litres = 2.7E12 J = 640 t TNT; Together 1280t TNT (Sci-Am: 1170 t) Impact: direct explosion 3000 gall = 180 t TNT; fire afterwards 990 t TNT. Takeoff weight of airplane 412,000 lb = 187 t. Typical cruising speed: 530 mph = 853 km/h = 237 m/s. Kinetic energy of each plane : 0.5 x 187000 x 237 x 237 = 5.2E9 J = 1.24t TNT; Together 2.5 ton TNT (Sci-Am: 2 t). By comparison: Tomahawk cruise missile = 0.5 t TNT; Hiroshima bomb = 20,000 t TNT; a typical tornado = 5,100 t TNT. (Source: Scientific American Nov 2001 p 10) Note that Sci-Am arrives at slightly different values for energy equivalents.
Solar energy striking top of atmosphere = 1.72E17 W. Solar energy reaching surface = 8.6E16 W Solar energy per m2 on outside of atmosphere: average 1.360 kW/m2 ; 1.406 in June solstice and 1.314 in December solstice. Ultraviolet 0-380nm = 10.0%; visible 380 - 760 nm = 44.8%; infrared > 760nm = 45.2%. Evaporation of water = 4.13E15 W. Gross photosynthesis = 2E14 W. Global gross primary production (V Smil) 1E14 W. A modern power station = 1 million kW = 1 GW = 1E9 J/s = 1 GJ/s = 1000 MJ/s This power station consumes 375 tonnes of coal per hour or 3 million kW. Total generating efficiency is about 30% Total electricity generating power of the USA = 8.3E11 W = 8.3E5 MW = 8.3E5 MJ/s. Per year = United Kingdom consumes an amount of energy equal to 330 million tonnes of coal/ year = 2.64E12 kWh = 9.5E18 J
Theoretical amount of energy from uranium U-235 burner-type fission = 79E12 J/kg (2,700,000 x coal), but these reactors run at 1-2% fuel utilisation (50,000 x coal). A breeder-type of fission can run at 40-70% fuel utilisation (1,500,000 x coal). Because Uranium ore contains about 0.2% U3O8 ore, 1 kg of uranium-containing ore produces as much energy as 100 kg of coal in a standard fission reactor.
Daily intake of a well-nourished Westerner = 3 kWh = 10.8 MJ Maximum daily output of a manual worker = 0.5 kWh = 1.8 MJ Work output of a horse in an 8-hour day = 12 kWh = 43.2 MJ Energy statistics for 1975 Energy to feed 1 person for 1 day (2400 food calories) Nonfood energy use per person per day, world average Nonfood energy use per person per day, USA average 1 tank of gasoline (15 USgal = 56.8 litre) 1 barrel of oil (42 USgal = 158.9 litre) 1 metric ton of coal (= 1000 kg), average Boeing 707 flight, San Francisco - New York 1 kg Uranium-235, completely fissioned (used-up) Summer thunderstorm Fuel input to a 1000 MW power plant for 1 day Hydrogen bomb (1 Megaton TNT equivalent) Total human non-food energy use per day Sunlight striking top of atmosphere, 1 day (From Ehrlich, 1977). Note! enormous discrepancies exist, JFA Energy densities of various fuels and food Substance MJ/kg Substance Hydrogen 114.0 Pure plant oils Gasolines 46-47 Butter Crude oils 42-44 Pure protein Natural gases 33-37 Pure carbohydrates Ethanol 29.6 Cereal grains Best bituminous coals 27-29 Lean meats Common steam coals 22-24 Fish Good lignites 18-20 Potatoes Air-dried wood 14-15 Fruits Cereal straw 12-15 Vegetatbles Human feces 1.8-3.0 Urine 0.1-0.2 Source: V. Smil: Energies Typical energy costs of common materials (MJ/Kg) Material MJ/kg Made from Material MJ/kg Aluminium 227Bauxite Polyethylene 87-115 Energy MJ 10 200 1,000 2,000 5,900 29,000 1,400,000 79,000,000 160,000,000 260,000,000 4,000,000,000 800,000,000,000 15,000,000,000,000
MJ/kg 37-38 29-30 23.0 17.0 15.2-15.4 5-10 2.9-9.3 3.2-4.8 1.5-4.0 0.6-1.8
Made from Crude oil
342 Clay Polystyrene 62-108 Crude oil 2-5 Clay and Polyvinylchloride 85-107 Crude oil 5-9 limestone Sand 0.08-0.1 Riverbed 60-125 Sulfide ore Silicon 230-235 Silica 18-35 Sand, etc. Steel 20-50 Iron 20-25 Iron ore Sulfuric acid 2-3 Sulfur 0.07- Sedimentary Titanium 900-940 Ore concentrate 0.1 rock Water 0.001Streams, 230-70 Ore concentrate Wood 0.01 reservoirs 87-115 Standing timber 3-7 Standing timber Source: V Smil: Energies. 1 tonne of coal = 0.585 tonnes of petroleum = 7400 kWh = 708 m3 natural gas = 1430 m3 town gas 1 tonne of petroleum = 1.7 tonne coal = 12500 kWh = 1189 m3 natural gas = 2450 m3 town gas 1 tonne coal = 1.0 t brown coal NZ = 2.0-3.0 t brown coal elsewhere = 2.0 t peat = 0.77 t crude oil = 0.66 t petroleum products = 750 t natural gas = 8000 kWh (United Nations statistics) 1 tonne oil = 1.5 t coal = 4.9 t lignite = 3.3 t peat = 1167 m3 natural gas = 12000 kWh (British Petroleum statistics) 1 tonne tropical topsoil @ 1.45% Cn = 14.5 kg carbon. 1 tonne loam soil @ 3-7% C = 30-70 kg C. About 735 joules of energy are required to lift 15 kg of oil 5 meters out of the ground just to overcome gravity.
Bricks Cement Copper Glass Iron Limestone Nickel Paper
Energy conversion Potential energy E = m x h x g (J) where m = mass (kg) and h = height (m) and g= 9.81 = acceleration of gravity Kinetic energy E = 0.5 x m x v x v where m= mass and v = velocity Converting mass to energy: E = m x c x c where m = mass and c = speed of light 1 g of matter = 9E13 J = 25 GWh 1 atomic unit of mass = 1 u = 1.49E-10 J= 931 MeV 1 electron = 0.000549 u = 0.510 MeV Electrical noise power P = k x T x B watt, where k = Boltzman's constant, T = temperature ºK and B = bandwidth (hz) To photosynthesise 1 kg of carbon, 4.77E8 J in potential energy is obtained, = 1.14E8 cal. per gram of carbohydrate: 15.9kJ= 3.8 kcal. To electrolyse 1 mole of a substance, 96,500 Coulombs of electricity are required for each of its valence. e.g. 96,500 Coulombs to electrolyse 1 mole of copper (Cu++) = 63.54 deposits 31.77 gram. Heat Heat is the energy passed from one substance to another. Heat capacity = specific heat = heat storage (per kg per degree)
Heat of fusion of water (melting energy) at 0 ºC = 79.71 cal/g = 333.52 J/g Heat of vaporization of water (evaporation energy) at 100 ºC = 539.55 cal/g = 2256 j/g Latent heat of evaporation at 10ºC ? = 2.5E6 J/kg Specific heat of water = 1.000 = calories to warm 1 g by 1 ºC = 1.000 cal = 4.184 J Specific heat of sea water at 3.5% salinity = 0.932 cal = 3899 J/kg/ºC (roughly 4000) [density = 1000kg/m3] Specific heat of rock = 800 J/kg/ºC (about 5 times less than water). [density is 3000kg/m3] Specific heat of air = 1030 J/kg/ºC [density is 1.3 kg/m3] Oceans exchange heat to a depth of 100m, whereas rocks don't. This makes the ocean store 100 times more heat than the land. Energy radiation 1 Gray (Gy) = 1 J/kg 1 rad = 10 mGy = 1E-7 J absorbed per gram of material. Q = Quality factor: 1 for X-ray and gamma radiation; 10 for beta and 20 for alpha radiation RBE = Relative Biological Effectiveness= 1 rem = 1 rad x Q (above). Maximum permissible human dosage = 0.1 - 0.2 rem/yr = 1 - 2 mSv/yr (??) A sudden dose of 350 rem gives 50% mortality in 30 days. 1 Sievert = 1 Sv = 100 rem; 1 mSv = 0.1 rem. 1 Curie = 3.7E10 Becquerel (Bq) = 3.7E10 disintegrations/s = 1 gram of pure radium
Natural radiation hazard Average radiation exposure for a US resident: 3.6 mSv (higher in areas with radon emission) This is made up of: Radon 2 + food & water 0.4 + diagnostic X-rays 0.4 + cosmic radiation 0.26 (mSv). One flight from New York to HongKong = 0.1 mSv; Watching TV = 0.01 mSv/yr. Maximum occupational exposure per year: 50 mSv (equals 500 flights as above). Max for pregnant women: 5 mSv. Radiation hazard of 3.6 mSv equals a loss of life expectancy of 1.5 minutes or fewer than 2 cigarette puffs. (Source: Sci-Am Jan 2002 p 20)
Thermodynamics
First law, conservation of energy: Energy can neither be created nor destroyed. If energy in one form or one place disappers, it must show up in another form or another place. Energy can be converted from one form to another, but its total amount remains the same. Decrease in internal energy = heat produced + work done: dE = Q + W Second law, all forms of energy ultimately reduce to heat: All physical processes lead to a decrease in the availability of the energy involved. Processes involving energy transformations will not occur spontaneously unless there is a degradation of energy from a non-random to a random form. All spontaneous (natural) events act to increase the entropy within a system. Until
a system reaches its maximum entropy, it can do useful work. But as a system does work, its entropy increases until the system can no longer perform work. 1) In any transformation of energy, some of the energy is degraded. 2) No process is possible with 100% efficiency in transformation of energy. 3) No process is possible through the flow of heat from a colder body to a hotter one. 4) Once enrgergy has been used by a process, it cannot be used again by the same process. 5) High order energy (low entropy) converts into lower order energy (high entropy), and never the other way. 6) Order (low entropy) naturally degrades to disorder (high entropy). Third law, concerns absolute zero temperature: It states that it is impossible to reduce the temperature of any system to absolute zero. Ideal gases An 'ideal' gas is a conceptual one, behaving contrary to the second law, by not losing energy during expansion or compression. The volume of one mole of a substance, produces 22.4 litres of an ideal gas at OºC, 1 At, and 24.5 @ 25ºC
Thus 1 mole of oxygen, O2 = 2 x 16 = 32 g, produces 22.4 litres of gas with a density of 32/22,400 = 1.43E-3 g/cm3 @0ºC which is 700 times lighter than water. Air averages 28.96 g/mol = 1.29E-3 g/cm3 @0ºC = 773 times lighter than water.
Boyle Gay-Lussac: pressure x volume / temperature (K) = constant The volume of a gas increases proportionally to absolute temperature; The pressure of a gas increases proportionally as the volume decreases. ( P1 x V1 ) / T1 = ( P2 x V2 ) / T2 where situation (1) is before and situation (2) after a change.
Heating 22.4 litres of an ideal gas from 0º to 25ºC increases its volume to 22.4 x (273+25)/273 = 24.5 litres, see above.
To heat one mole of an ideal gas by one degree K, requires 8.3143 J. The abve laws are derived from the more universal equation P x V = R x T Where R = Gas constant = 8.317 J/mol/ºK, amount of heat to warm up one mole of a gas Henry's law: at a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid (for small concentrations, not saturated). e^p = e^{kc} or e p = e k c p = the partial pressure of the gaseous solute above the solution in atmospheres (approx kg/cm2) c = the concentration of the gaseous solute in the solution in mol/Litre or mol/kg for water k = the Henry's Law constant, which various with the kind of solvent and temperature. solubility decreases with increasing temperatures. Examples of the value for k at 26ºC: O2=769.2 ; CO2= 29.4; H2= 1282.1 L·atm/mol See http://www.henrys-law.org/
Stephan-Boltzman equation Defines the radiation emitting from an object of temperature T ºK: I = emissivity x 5.6704E-8 x T^4 W/m2 where emissivity = 1.00 for a black body; 0.997 for water; 0.969 for snow; desert sand 0.993.
Electricity
Coulomb's law constant = 1 / (4 x pi x e0) = 9.0E9 N m2 C-2 Permittivity constant ( in vacuum) e0 = 8.8541853E-12 C2 N-1 m-2 Permeability constant (in vacuum) m0 = 4 x pi x 1E-7 Wb A-1 m-1 Faraday's constant = 9.64867E4 C/mol Electronic charge = e = 1.6021917E-19 C 1 Ampere = 1 Coulomb/second.
Physical constants
Pi = 3.14159265 Natural logarithm e = 2.71828. ln(x) = 2.3026 log(x) Standard acceleration of gravity, g =0.98066 m/s/s Perfect gas constant, R = 8.3143 J/(g mol)/ºK. [Amount of energy needed to heat a gas] Avogadro's number, N = 6.022169E23 molecules/(g mol) [Number of molecules in a mole] Faraday constant, F = 9.648670E-4 C/mol Stefan-Boltzmann constant, sigma = 5.6697E-8 W/m2/ºK4 Also quoted as 5.6704E-8 Planck's constant, h = 6.6255E-34 Js [Amount of energy in an electro-magnetic wave; quantum theory] Boltzmann constant, k = 1.380622E-23 J/ºK. [Amount of radiated energy of a black body; entropy] Permeability of a vacuum, 4 x pi E-7 kg m /s/s/A/A
Frequencies, oscillation, electromagnetic spectrum
1 hertz (hz) = 1 cycle/sec (c/s) Speed of sound through air at earth's surface = 331.4 m/s Emission from neutral hydrogen = 1.42 Ghz
Biology
Humans Mean human mass worldwide = 50 kg, contains 1.9 kg nitrogen, which is 15% of body mass. Nitrogen content of proteins = 25% The mean human body contains 0.9 kg phosphorus, most of it in bones. Food energy per day = 2400 kcal = 10 MJ Maximum daily output of a manual worker = 0.5 kWh = 1.8 MJ Plants Dry weight (DM) of a crop is about 25% of wet weight. Carbon content of dried matter (DM) = 45% (containing carbohydrates CH2O chains) Nitrogen content of DM is 1.1 - 1.6%
Biomass calculations The ratios shown are not by weight but by the numbers of atoms. Average composition of terrestrial life: H:O:C:N:P:S = 2960:1480:1480:16:1.8:1 (E S Deevey Jr) Average composition of land plants: H:O:C:N:P:S = 1600:800:800:9:5:1 Average composition of marine plants: H:O:C:N:P:S = 212:106:106:16:2:1 Average composition of plant biomass: C:H:O= 1:(1.3-1.8):(0.5-0.6) (McDermitt&Loomis 1981) Redfield ratio required for phytoplankton: C:N:P= 106:16:1 Silicon requirement of diatoms: N:Si= 1:1 NPK fertiliser components of organisms: N:P:K= 100:18:22 with a yield of 20%. Carbohydrates = CH2O chains; glucose = C6H12O6: H:O:C = 12:6:6, or in mass 1:6:8 Energy conversion during photosynthesis/ respiration: 12 g carbon = 1 mole = 112 kcal = 469 kJ Photosynthesis: 6CO2 + 12 H2O + sunlight = C6H12O6 + 6O2 + 6H20 Respiration: C6H12O6 + 6O2 = 6CO2 + 6 H2O + heat (15.9kJ= 3.8kcal per gram carbohydrate) Methane production: 2CH2O = CH4 + CO2 Methane oxidation: 2CH4 + 3O2 = 2CO + 4H2O; 2CO + O2 = 2CO2. 1 kg dry organic matter (DM) = 0.45 kg C = 1.5 kg CO2 (with 15% variation) = 4200 kcal energy Rule of thumb: 4kcal per gram of dry organic matter 1 kg C corresponds to 2.2 kg dry organid matter. Wet weight (plants) = 4 x dry weight. Protein is 15% of body weight. N=25% of protein. gas exchange: 1 kg CO2 turnover = 0.73 kg O2 turnover
phytomass DM Gt NPP DM Gt/yr on land in oceans total 1840 107 3.9 55 1843.9 162
moles C/yr 4.0E15 2.1E15 6.1E15
watt/yr
kcal/yr
6.3E14 4.8E17 3.3E14 2.5E17 9.6E14 7.3E17
Animals 1 kg of beef requires 50,000 kg of water. A general equation to estimate the food requirement of marine mammal species (Innes et al 1987) Daily food cunsumption (kg biomass) Y = 0.123 x M ^ 0.80 or log(Y) = log(0.123) + 0.8 x log(M) Biochemical productivity Plant productivity increases 100% with 12ºC temperature increase Soil [note that facts relating to soil are traditionally expressed per hectare ha] Natural erosion before farming: 9E9 tonnes worldwide. In 1984: 25E9 tonnes Tennessee: losing 14 t/acre/yr = 35t/ha/yr. Nature puts back 5 t/acre/yr= 10t/ha/yr: Loss= 25t/ha/yr = 15m3/10,000m2/yr = 1.5mm /yr Conservation farming reduced soil loss from 14 to 10 t/acre/yr tropical rainforest = 180 t/acre in dry matter sediment yield from erosion by water 20-2600 t/sqkm/yr avg 30-92 m3/sqkm/yr =9 km3/yr particles + 9.4E9 ton dissolved = 6 cm/1000yr= 0.06 mm/yr (Last Frontier p367) Natural rate of weathering = 0.1mm/yr = 0.1 x 0.001 x 10,000 m3/ha/yr = 1m3/ha/yr = 100 m3/km2/yr = 1.5t/ha/yr Water Capillary rise of height h in a tube with diameter d : h = 0.3 / d (in cm) (approximately)
A hair-thin capillary of 0.1 mm = 0.01 cm causes water to rise by 30 cm.
NZ vital statistics
Total land area: 270,500 km2 = 27,050,000 ha Total seashore: 15,000 km Territorial sea out to 12 sea miles (22.2km): 160,000 km2 = 16,000,000ha Exclusive Economic Zone (EEZ): 4.3 Mkm2 (4.85 also quoted) = 4,300,000 km2 = 430,000,000 ha = 1.2 Mnm2 (square sea miles)
Continental shelf: 24,000,000 ha = 240,000 km2 only slightly smaller than NZ's land area Estuaries: 100,000 ha = 1000 km2 Sea area in cable protection zones: 1500 km2 Sea area in ammunition dumps, mining & oil drilling: 1500 km2 Sea area in marine reserves around mainland: 150 km2 = 15,000 ha Productivity of NZ coastal seas 2-8 t/km2 (D Pauly) NZ soil erosion: 270Mt/yr = 1000 t/km2/yr = 10 t/ha/yr = 6 m3/ha/yr is about six times the sustainable rate, which is severely underestimated. But most erosion by far comes from the 13Mha of agricultural land (about 50% of the land), which makes erosion 12 times the sustainable rate. But NZ has always had very low natural erosion and in recent years erosion has accelerated due to more torrential rains. so, NZ erosion may well top 400Mt/yr or over 20 times of what was natural. loss of Carbon by soil erosion, estimated at between 3 and 10 Mt C per year (AgResearch)
Mathematics refresher
Solve the linear equation: x2 + bx = c solution: x = SQR( b x b / 4 + c) - b / 2 Example: x2 - x = 1 solution: x = SQR( 1 / 4 + 1) - 1 / 2 = SQR( 5/4 ) - 0.5 = 0.618034 the divine number or golden section
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By Dr J Floor Anthoni (2001-?) www.seafriends.org.nz/books/units.htm Have you forgotten the physical constants of the physical world around us, and the formulas that define how things move? You are not alone. Have you never heard of them? Don't despair, because it is easy to understand how to apply them. The information on this page helps you making your own estimates and calculations (on the back of an envelope), while also furthering your understanding. Please note that this page is updated from time to time. . This page contains a collection of facts and rules, necessary for making ecological estimates and calculations. Print it out for your convenience when studying the topics on this web site.
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metric SI system: units, prefixes, standards dimension-less quantities: quantities that have no unit o ratios: %, ppt, ppm, billion, trillion, etc. o magic ratios and numbers: golden rule, natural logarithm, pi, aspects of paper, images, o angles: degrees, radians, minutes, seconds o density: water, air, Earth's crust, o averages: arithmetic and geometric averages o attenuation: Beer's Law, Lambert's Law, decay, absorption, attenuation o bel and decibel: measuring sound and expressing amplification and attenuation. o mathematical series: fibonacci, exponent, logarithm, scaling/ allometrics: how to scale effects up and down
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time/ frequency: second, minute, day, year, etc. o geological time: Geological Time Table o frequency: Hertz, Becquerel temperature: length, area, volume: oil measures: barrels, volume, mass biomass conversions: dry matter, carbon, energy o geometry: formulas for area and volume calculation o world dimensions: radius, area, volume, seas, etc. o cosmic dimensions: distances in space, etc. o speed, velocity, flow: knot, cfs, speed of sound, escape velocity, Poise mass, force, pressure: energy, work: energy statistics: energy use in 1975 energy conversion formulas: converting various mined fuels, coal, oil, gas. energy costs: the energetic cost of mining and manufacturing. o heat: specific heat (heat capacity), heat of fusion, heat of vaporisation
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radiation: energy per mass, Gray, Rad thermodynamics: the laws of thermodynamics electricity: voltage, current, charge, physical constants: frequencies: oscillation, electromagnetic spectrum biology: biological conversion factors and more NZ vital statistics: important environmental facts about New Zealand Mathematics refresher: how to solve those simple equations
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Jerrard H G (1963-1992): Dictionary of scientific units, including dimensionless numbers and scales. Chapman&Hall Rowlett, Russ (Internet): A dictionary of unitsof Measurements. www.unc.edu/~rowlett/units/ very extensive and accessible Harte, John (1939,1988): Consider a spherical cow, a course in environmental problem solving. Univ Science Books. Physics Fact Book: http://hypertextbook.com/facts/index-topics.shtml use of formula in sample problems, and interesting dimensions.
-- home -- Revised frequently: 20010412,20010830,20020110,20041110,20060901,
The metric system of units and measures
The international system of units (SI units) defines how physical properties are measured, previously known as the MKS (Metre, Kilogram mass and Seconds) system. Units consist of two parts, a prefix coding for magnitude (multiplication or division) and a unit of measure. The unit kg consists of the prefix k for 1000x and g for gram. As a rule, prefixes above 1000 are spelled with capital letters (M=mega=1,000,000x), and so are units derived from a person's name (W=Watt, J=Joule), but kilo k, hecto h and deka da are spelled in lower case. Units are always singular: An elephant has four feet. John is six foot tall. Mary travels to work about 20 kilometre. The bag contained 5 kilogram. Prefixes power of ten 24 prefix name Y 21 Z 18 15 E P 12 9 T G 6 M 3 k 2 h 1 da
yotta zetta exa peta tera giga mega kilo hecto deka
power of ten -24 prefix name y
-21 z
-18 -15 a f
-12 -9 p n
-6
µ
-3 m
-2 c
-1 d
yocto zepto atto femto pico nano micro milli centi deci
In computing, the K unit is used for 210=1024 instead of 1000. Here are the new conventions: power of two 10 20 approximate 3 power of ten prefix name 6 30 40 50 9 12 15 60 18 Ei
Ki Mi
Gi Ti Pi
kibi mebi gibi tebi pebi exbi
Basic SI units
Physical quantity length mass time temperature electric current luminous intensity amount of substance Derived SI units
Unit metre kilogram second Kelvin Ampere candela mole
Symbol m kg s ºK A cd mol
Physical quantity concentration density frequency energy, enthalpy force pressure power electric potential difference electric charge electric capacitance electric resistance electric conductance magnetic flux magnetic flux density (inductance) inductance luminous flux illumination activity
Unit mole per cubic metre kilogram per cubic metre Hertz Joule Newton Pascal Watt Volt Coulomb Faraday, farad Ohm Siemens Weber Tesla Henry lumen lux
Named after Heinrich Rudolf Hertz James Prescot Joule Isaac Newton Blaise Pascal James Watt Count Alessandro Volta Charles Augustin de Coulomb Michael Faraday Georg Simon Ohm Werner von Siemens Wilhelm Eduard Weber Nikola Tesla Joseph Henry Latin for giving light Latin for light Antoine-Henri
Symbol Definition mol /m3 kg /m3 Hz J N Pa W V C F (Ohm) S Wb T H lm lx Bq 1/s kg m2 /s2 kg m /s2 N /m2 J /s J /A/s (J/C) As C /V V /A A /V Vs WB /m2 V s /A cd s m lm /m2 1/s
absorbed dose dose equivalent
Becquerel Gray Sievert
Becquerel L H Gray Rolf Sievert
Gy Sv
J /kg J /kg
Russ Rowlett of the University of Carolina has a very extensive web site about this. The International Bureau of Standards in Paris also has an extensive web site. Bureau International des Poids et Mesure. A very precise unit converter converts one unit into another. Note! On the Seafriends web site, the powers of ten are often written as 1E6 rather than 10^6 or 106. 15000 = 15E3 = 1.5E4 Units are spelled the English way: metres rather than meters. Formulas are spelled out like e = m x c x c, rather than e=mc2. In scientific literature it is conventional to spell m/s as m s-1 , but this is not followed in this web site.
Dimension-less quantities Ratios
% = percent or procent = per hundred = 0.01 o /oo = promille = per thousand = 0.001 ppt = parts per thousand = .001 = 1E-3 ppm = parts per million = .000001 = 1E-6 ppb = parts per billion (Am) = 1E-9 1 billion (American) = 1,000,000,000 = 1 thousand millions = 1E9 1 trillion (American) = 1 million millions = 1E12 1 quadrillion (American) = 1 thousand millions = 1E15 1 billion (European) = 1 million millions = 1E12 1 trillion (European) = 1 million million millions = 1E18 1 order of magnitude = 10 x to decimate originally means, to take 10% (= x 0.9), but is often used as meaning to leave 10% (= x 0.1). Magic numbers and ratios One (1): One is perhaps the most beautiful number of all; the number of symmetry. Figures drawn with an aspect ratio of one, are all perfectly symmetric: the circle, square, polygons, stars and so on. It is the only number for which n / 1 = 1 / n and n = n0 = n1 = nn = nm. Square root of two: The international paper size A0, A1, A2, A3, A4, A5, ... series was designed such that an A0 sized sheet cut in half, would produce two A1 sheets of equal size and equal ratio between height and width (=aspect). There is only one mathematical number that has
this property: the square root of 2. SQR(2) = 1.4142. In this series of paper sizes, the height/width ratio is SQR(2). When cut in half, becomes SQR(2) / 2 = SQR(2) / (SQR(2) * SQR (2)) = 1 / SQR(2). Thus the ratio remains identical, although the cut sheets need to be turned ninety degrees. This is the only number/ratio which inverses when divided by two: 1 / n = n / 2 thus n x n = 2 or n = SQR( 2 ) Golden section: the division of a number so that the whole is to the greater part as that part is to the smaller part. It is also called the divine proportion, first defined by Euclid, and is used by architects in designing buildings (first used for the Parthenon). 1.61803 to 1 or 1 to 0.61803. A rectangle with these proportions is called a golden rectangle and is thought to be more pleasing to the eye. It is calculated by solving the equation 1 / n = n + 1 or n2 + n - 1 = 0 The golden section can directly be calculated as: (1 + SQR(5)) / 2 (see mathematics refresher below) Graphically it is drawn as follows: First draw a square ABCD. Then, find the midpoint M of side AB. Next, use a compass to extend AB to a point E so that ME = MC. Rectangle AEFD is a golden rectangle. To divide AB according to the golden section, use a compass to find a point G on AE so that EF = EG. Fibonacci series The Italian mathematician Leonardo Fibonacci defined the Fibonacci series 0,1,1,2,3,5,8,13,21,34,55,89,144, ..... where each following number is the sum of the two previous (2=1+1, 3=2+1, 144=89+55, ..). It is a series that has a constant progression as found in things that grow in nature. In fact, each following number can be calculated from the previous by multiplying with 1.618033988 (dividing by 0.618033988) and rounding the result. Note the remarkable mathematical quality 1 / n = 1 + n or the Golden Section above. Two (2): The lowest number that makes counting and arithmetic possible. The only number for which n x n = n + n = nn . It is the number that stands for duplication, replication and redundancy. A rectangle with aspect of two, becomes a smaller rectangle with the same ratio when folded or halved twice. Natural number or natural logarithm: e = 2.71828 . e is calculated by the series : e = 1 + 1/1 + 1/2 + 1/(2x3) + 1/(2x3x4) + .... + 1/(2x3x4x..n) ... to infinity. The natural logarithm is an important number for mathematics, especially when relating to physical quantities. Many formulas have been derived with it. e is also related to the most compact notation for information. Where we use the digital (0-9) number system and computers the binary (0-1) number system, the value system based on e expresses information in the most compact way, although it is continuous and not discrete (consisting of steps). Throughout the universe, e is an important quantity. Pi: the ratio of circumference to the diameter of a circle 3.14159... The number obtained by dividing 22 by 7 22/7=3.1428571.. is almost the same, but slightly larger. Pi can be calculated exactly as follows: pi / 2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x ..... x (n+1)/n x n/(n+1) . (John Wallis series), or by a
shorter series, which converges more slowly: pi / 4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 .... (Leibniz series (1673), also discovered by James Gregory (1671)) Aspect ratios of various images: 1.3333 or 4/3 the television frame, Super-8 film and 16mm film formats. Thought to be the ideal proportion matching our eyesight. Modern digital widescreen format is 4x4/3x3= 16/9. 1.5 or 3/2 the 35mm format for both movies (across) and still pictures (along, 36/24 mm). Other film formats: 6/4 cm, 9/6 cm, ? Cinemascope Perfect numbers A number is called perfect, if the sum of its factors adds up to the number itself, like: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. There are only four such perfect numbers: 6; 28; 496; 8128. (Nichomachus Gerasa, 100 AD). The 6th and 7th are: 8,859,869,056 and 137,438,691,328 (Pietro Cataldi, 1603) What is the fifth perfect number?
Angles A circle is 360 degrees = 2 x pi radians One degree = 60 minutes = 60 arcminutes One minute = 1 arcminute = 60 secondes = 60 arcseconds 1 minute over a meridian = 1 nautical mile = 1.852 km Density Density or specific weight is a ratio between weight and volume. Traditionally gram / cubic centimetre = kg / litre or ton/m3. Multiply by 27.7 to get lb/in3, and by 1.33 to get ton/yd3. The density of clean water is defined as 1.000. A rock with density 2.80 weighs 2800 kg / cubic metre = 2.8 t / m3 Density of air at the surface is 0.00013 Densest substance on Earth is iridium 22.5, followed by osmium 22.4, platinum 21.4, rhenium 21.0, neptunium 20.4, plutonium 19.8, gold 19.3, tungsten 19.3, uranium 18.9, ... mercury 13.53, ... lead 11.4, ... iron 7.86, .... Averages The arithmetic average of a, b, c = (a + b + c ) / 3 The geometric average of a,b,c = (a x b x c ) / 3 An arithmetic series is: a + (a+d) + (a+2d) + (a+3d) + .... A geometric series is: a + (a x r) + (a x r x r) + ... An exponential series is: ex = 1 + x + ( x2 ) / ( 1 x 2 ) + ( x3 )/ (1 x 2 x 3 ) + ... ( xn ) / n! A logarithmic series is: ln(1+x) = x - (x2) / 2 + (x3) / 4 - (x4) / 4 + ... Attenuation
Beer's Law and Lambert's Law When a quantity changes continually with a fixed rate of change, it follows either exponential growth or logarithmic decay, depending on whether the factor is larger or smaller than one. Example: 1,2,4,8,16,32,64,128,256,... is exponential growth with a fixed factor of 2. Example: 256,128,64,32,16,... is logarithmic decay with a fixed factor of 0.5. Examples from the natural world are: absorption of light/sound by a medium, radio active decay. The difference between Beer and Lambert is that the first was concerned with the density of the medium, whereas the second was concerned with the distance travelled through the medium. Both concerns are captured in the same formula: Id = I0 x e x x d or ln( Id / I0 ) = x x d Where I0 = starting intensity, Id = intensity at distance d and ex = rate of change for a distance equal to d = 1, and x = the decay constant. If x is larger than 0, there is exponential growth, but if it is less than zero, there is logarithmic decay. For absorption by a medium, x can be written as a constant times concentration : k x c To convert from the natural logarithm ln( to the decimal logarithm log(, correct by: e= 2.71828 Id = I0 x 10 ( x x d ) / e or log( Id / I0) = ( x x d ) / log( e ) = 2.3029 x x x d If the rate of change is known for a unit distance, then the exponent x can be calculated as: x = ln( unit rate of change) For the 1,2,4,8 series above, x can be calculated from the rate of change = 2: 2 = 1 x e x x1 or x = ln(2) = 0.693. Likewise, x = -0.693 for the 256,128,64,.. series. Bel and decibel Originally used for measuring sound amplitudes, the bel and decibel are also used for measuring amplification and abatement. The Bel unit is named after the inventor of the telephone, Alexander Graham Bell (1847-1922). One Bell louder means ten times louder, measured by amplitude. 1 bel = 1 B = log ( I / I0 ) = 10 decibels ; where I = current intensity and I0 is previous intensity, or a sound level of 2E-10 bar. 1 decibel = 1 dB = 10 x log( I / I0 ) The range of human hearing extends from 0 - 120 dB, at which level sound becomes painful, while also causing permanent deafness. The critical level for the start of ear damage is 85 dB. Conversation: 30-60 dB; jet plane 160 dB; Noise from trucks can exceed 80 dB measured at the roadside. People begin to complain when sound levels in their neighbourhood reach 35-40 dB. Mathematical series This section contains the various mathematical series important to nature.
Fibonacci: Leonardo Fibonacci (1175-1250), was an Italian mathematician who helped introduce the Hindu-Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) into western Europe, particularly the zero. He is also known as the originator of a special series of numbers, now called the Fibonacci sequence or the Fibonacci numbers. 1, (1+1=2), (1+2=3), (2+3=5), (3+5=8), (5+8=13), ..... Natural logarithm, e: e = 1 + 1/1 + 1/(1x2) + 1/(2x3x4) + 1/(2x3x4x5) + ....
Scaling / allometrics
Allometrics (Gk: allo=other; metron= measure) is the ability to scale size up and down according to the relationship between size and some other measure. Many of these relationships appear throughout all chapters of this page. For instance, the volume of a sphere is related to its radius as 4/3 x pi x r x r x r. With this formula, the volume of any sphere can be calculated. It is an exact and mathematically provable formula. However, a number of relationships have been established empirically (by experiment), and these are listed here. Note that empirical formulas are subject to some uncertainty. Organisms; weight m in kg. Basal Metabolic Rate of an organism, related to weight: P = 73.3 x m ^ 0.75 (kcal/day), where m= kg. Mammal brain weight, excluding apes & humans = 0.01 x m ^ 0.70 (kg) Ape brain weight = (0.03-0.04) x m ^ 0.66 (kg) Life span (not humans nor birds) = 11.8 x m ^ 0.20 (years) Life span for birds = 28.3 x m ^ 0.19 (years) Humans; weight m in kg. Heart weight = 0.005.8 * m ^ 0.98 (kg) Lung weight = 0.0113 x m ^ 0.98 (kg) about twice the weigh of the heart. Brain weight = 0.085 x m ^ 0.66 (kg) Tidal volume = 7.69 x m ^ 1.04 (ml) Vital capacity = 56.7 x m ^ 1.03 (ml) Blood volume = 65.6 x m ^ 1.02 (ml) Muscle mass = 0.40 x m ^ 1.00 (kg) Skeletal mass = 0.0608 x m ^ 1.08 (kg) Heart rate = 241 x m ^ -0.25 (beats/min) Note: heart rate decreases with weight Blood circulation time = 17.4 x m ^ 0.25 (seconds) Respiratory rate = 53.5 x m ^ -0.26 (per min) Respiratory rate decreases with weight. Oxygen consumption = 1.94 x m ^ 0.79 (ml/s) Glucose turnover = 5.59 x m ^ 0.75 Impact damage Nuclear weapon (power in Mt) area of destruction = 400 x (Mt) ^ 0.67 (km2). (Steel, 1995) Thus a bomb of 10 Mt TNT destroys 1870 km2, equal to an asteroid of 25m radius, density 3
and speed 20,000m/s. Asteroid impact energy for radius r (m), speed v (m/s), density d = (5.0E-13) x r ^3 x v ^2 x d (Mt TNT)
Time, frequency
1 day = 1440 minutes = 86400 sec. The day lengthens by 1 second every 62500 years. exact duration of a year = 365.25 days 1 yr = 8760 hr = 31,560,000 sec = 3.156E7 s 1 eon = 1E9 years. Life started 3.6 eons ago (3,600,000,000 years). The Big Bang 15 eons ago. Earth = 4.6 eons Geological time See Geologic Time Table. Frequency 1 hertz = 1 Hz = 1/s (per second) = 1 c/s (cycles/second) = 2 x pi rad/s = 6.28318 rad/s 1 revolution per minute (rpm) = 0.105 rad/s unit of (radioactive) activity = Becquerel = 1 Bq = 1/s (per second) 1 Curie = 1 Ci = 3.7E10 Bq
Temperature
Temperature is internal energy in a substance, the movement of its molecules. degrees Celsius = degrees Kelvin - 273.16 = (5 / 9) x (degrees Fahrenheit - 32) degrees Fahrenheit = (9 / 5) x degrees C + 32 0 ºK = -273.15 ºC Gas constant = 8.317 J/mol/ºK, amount of heat to warm up a gas Temperature of sun's surface = 5780ºK (as an ideal blackbody radiator); of its interior = 15,000,000 ºK If the sun was burning coal, it would have burnt up in 1500 years.
Length, area, volume
Length 1 m = 39.37 inches (in) = 3.281 feet (ft) = 1.094 yards (yd) 1 inch = 2.5400 cm
1 ft = 0.3048 m 1 km = 1000 m = 0.6214 statute mile (mi) = 0.5400 nautical mile 1 statute mile = 1.6093 km =1609.3 m 1 fathom = 6 ft = 1.829 m 1 furlong (fur) = 201 m 1 league = 1 nautical mile (n mile) = 1.852 km 1 chain = 66 ft = 20.1168m 1 Aengstrom (AE) = 1E-8 m = 0.1 nm = 100 pm (pico metre) Book printing: 1 inch = 6 pica = 72 point Area 1 m2 = 10.76 ft2 = 1.196 yd2 1 km2 = 1000000 m2 = 100 hectares (ha) = 247.1 acres = 0.3861 mi2 1 square nautical mile = 3.4299 km2 1 square mile = 2.59 km2 1 are = 100 m2 1 hectare (ha) = 10,000 m2 = 0.01 km2 1 square inch (in2) = 6.45 cm2 1 square foot (ft2) = 929 cm2 = 0.0929 m2 1 perch = 25.3 m2 1 acre = 0.405 ha = 4050 m2 1 rood (rd) = 0.101 ha Volume 1 litre (l) = 1000 cm3 = 33.81 fluid ounces = 1.057 US quarts 1 fluid ounce = 29.57353 ml (USA) = 28.4 ml (Imperial) 1 cu ft = 0.0283 m3; 1 in3 = 16.4 cm3; 1 yd3= 0.765 m3 1 Imperial pint = 0.5682 l 1 Imperial gallon = 1.2009 US gallons = 4.5461 litre 1 US gallon = 0.8327 Imperial gallon = 3.7854 litre 1 m3 = 1000 l = 35.31 ft3 = 264.2 US gallons = 6.290 barrels 1 acre foot = 1230 m3 =43,560 cu ft 1 bushel (US) = 1 bu = 8 (Imperial?) gallon = 35.23907 litre 1 wheat bushel (US) = 60 pound = 27.216 kg. Barley bushel = 48 pound = 21.772 kg. Oats bu = 32 pound = 14.515 kg 1 rye bu = 56 pound = 25.401 kg. Volume of one mole of a chemical substance at surface temp & pressure = 22.414 litre/mol = 6.0249E23 molecules
Oil quantities and measurements 1 m3 = 6.29 barrels (6.7 barrels/tonne for crude oil; 8.5 for motor spirits; avg 7.3) 1 tonne crude oil = 7.3 barrels average = 256 Imp gall = 301 US gall 1 barrel per day = 50 tonnes oil / yr (average)
1 barrel = 34.97 Imperial gallons = 42.00 US gallons = 159.0 l 1 km3 = 1E9 m3 = 0.2399 mi3 = 810600 acre-feet 1 mile-per-gallon US (mpg) = 0.4251 km/l kilometres-per-litre. 50mpg = 21.26 km/l 1 mile-per-gallon IMPERIAL (mpg) = 0.3541 km/l kilometres-per-litre. 50mpg = 17.71 km/l Sulfur content of gasoline is approximately 350 ppm. Gas quantities and measurements 1 Tcf= 1 Trillion-cubic-foot (USA)= 1E12 cuft = 2.83E10 m3 1 mole of an ideal gas = 22.4136 litre @ 0º C = 24.4651 l @ 25ºC
Geometry pi = 3.14159 Circumference of a circle = 2 x pi x r, where pi = 3.14159 and r = radius Area of a circle = pi x r x r. Area of an ellipse with axes a and b = pi x a x b Surface area of a sphere = 4 x pi x r x r Volume of a sphere = 4/3 x pi x r x r x r Volume of a cone with height h and base radius r = pi x h x r x r / 3 Volume of a pyramid with base area b and height h: = h x b / 3 Surface area of a triangle with base b and height h = b x h / 2 Area of a regular pentagon with side a, = 1.721 x a x a. Hexagon = 2.598 x a x a. Octagon = 4.828 x a x a Area of a trapezium with parallel sides a and b and height h = 0.5 x ( a + b ) x h The distance between two points (x1,y1) and (x2,y2) = SQR( (x2 - x1)2 + (y2 - y1)2 ) Equation or formula for a parabola: y2 = 4 x a x x, where a = focus Equation for an ellipse: x2 / a2 + y2 / b2 = 1 Equation for a hyperbola: x2 / a2- y2 / b2 = 1 World dimensions Radius: equatorial = 6378.163 km. polar = 6356.177 km Circumference: equatorial = 40074 km. polar = 39942 km Mean elevation of continents = 800 m (840m). Mean depth of oceans = 3800m (3729m*, 3730m, 3795m). Contains 97% of all water (not ice). Total surface = 510.1E6 km2. Surface of oceans = 361.3E6 km2 (70.8%) Mixed surface layer avg depth= 75m. Mixed layer volume= Surface area of ice-free land = 133E6 km2 Productive land 88E6 km2 = 8.8 Gha.(66%). Sea ice= 33E6km2. Continents & Islands: 148.8E6 km2. Eurasia= 0.536E14 m2. Africa= 0.298E14 m2. N-America + Central= 0.238E14 m2. S America= 0.179E14 m2. Antarctica= 0.149E14 m2. Oceania= 0.089E14m2. Volume of the earth = 1.083E12 km3 *) see formula above. Average density = 5.518. Internal temperature estimated at 5000ºK Mass of the earth = 5.976E24 kg. Mass of the oceans = 1.35E21 kg = 1.350E9 km3 *
Mass of ice caps and glaciers = 2.9E19 kg. Mass of lakes and streams = 1.3E17 kg. Mass of water in atmosphere= 1.3E16 kg. Mass of the atmosphere = 5.2E18 kg (5.14E18). Mass of N2 in atmosphere 3.95E18 kg. Mass of all organisms (wet weight) = 5E15 kg= 1.3E15 kg dry weight.
* Turekian, Karl K (1968): Oceans. Prentice-Hall.
Cosmic dimensions Mean distance to sun = 149.6E6 km = 497 s = 8.28 minutes Mean distance to moon = 384,000 km = 3.84E8 m = 1.3 light-seconds Geostationary orbit = 35,900 km from surface = 42,200 km from earth's centre (?) Mass of moon = 7.4E22 kg Mass of Earth = 600E22 kg. Diameter of the sun = 109 x Earth's =1,380,000 km = 864,000 km Mass of sun = 330,000 x mass of Earth = 200E26 kg ? Each second, the sun turns 700 million tonnes of hydrogen into 695 million tons of helium. The difference is radiated out. The sun rotates around its axis in 27 days. Sunspot cycles: 11 and 22 years. The more sunspots, the less energy reaches Earth (0.3% max variation; UV 30%). Year 2002 = year of maximum sunspots. In 220 million years, the sun completes an orbit around the centre of the milky way The moon does 12 cycles around the Earth in 354 days Velocity of light in vacuum (Einstein's constant), c = 2.9979E8 m/s (approx 300,000 km/s) 1 light year = 9.454E12 km Speed, velocity, flow 1 knot = 0.5144 m/s = 1.852 km/hr =1.151 mi/hr 1 mile per hour (mph) = 1.61 km/h = 0.447 m/s 1 cfs (cubic foot per second) = 448.9 gal/min = 8.931E5 m^3/yr = 724.0 acre-ft/yr Speed of sound in dry air = 331 m/s Speed of sound in the ocean is between 1450 and 1570 m/s. Speed of sound increases 1.3 m/s for each 0.1% increase in salinity and increases also 0.17 m/s for each bar or 10 m depth Escape velocity from Earth = 11.2 km/s. For orbital velocities see oceans/currents/atmosphere dynamic viscosity : 1 Poise = 100 cP = 100 m Pa s kinematic viscosity: 1 Stokes = 100 cSt = 1 cm2 /s = 1E-4 m2 /s Ocean currents: 1 Sv = 1E6 m3/s
Stokes' Law of settling velocities, as defined in the gram-cm metric system
[from Karl K Turekian (1968) Oceans. Prentice-Hall] How fast does a small particle sink? velocity v = [ D x D x ( dp - df ) x g ] / [ 18 x p ] cm/s where D= diameter cm, dp= density of particle g/cm3, df= density of fluid g/cm3, g= acceleration of gravity = 980cm/sec2, p=viscosity in poise g/sec cm. For a sphere of quartz (dp=2.6) falling through ocean water (df=1.0) at a temperature of 10ºC (p=0.0140 poise) and diameter D=0.2 mm=0.02 cm, the terminal velocity is 2.48 cm/s. Over this diameter, a correction (Oseen's equation) is necessary due to turbulence. This equation predicts a lower speed.
radius microns 1 10
velocity time to fall to cm/s ocean bottom 4000m 0.00025 51 years 0.025 185 days 1.8 days 2.7 days
100 (Stokes) 2.5 100 (Oseen's) 1.7
See also Heezen& Hollister's diagram of current velocities for erosion, transportation and sedimentation, oceano/soil51.gif For particles larger than 1mm their speed increases ten fold for a hundred fold increase in diameter thus following a square root function. A 1mm particle sinks at about 20cm/s. A one metre boulder would sink at 6.3 m/s or 22.7 km/h, at fast bicycling speed.
Mass, force, pressure
1 kg (weight/force) = g Newtons, where g = acceleration of gravity = 9.8066 m/s/s 1 gram (g) = 15.43 grains = 0.03527 ounces avoirdupois = 0.03215 troy ounces 1 ounce avoirdupois (oz) = 28.34952 g 1 kilogram (kg) = 2.205 pounds (lb) = 35.27 ounces avoirdupois (oz) 1 lb = 0.4536 kg; 1 lbf = 4.45 N 1 stone = 14 lb = 6.35 kg 1 metric ton (t, tonne) = 1000 kg = 1.102 short tons = 0.9842 long tons = 2204.6 lb 1 long tonne = 1016.047 kg, previously also used for deadweight tonne. 1 deadweight ton (dwt) = 1 t (the difference between an empty and fully laden ship, including fuel & passengers) 1 newton (N) = 1E5 dynes = 0.1020 kg force =0.2248 lb force 1 bar = 100 kPa = 1E5 Pa = 1E5 N/m2 = 10.20 t/m2 = 0.9869 atmosphere = 750 torr 1 atmosphere = 1.013 bar = 14.7 lb/in2 = 76.0 cm of mercury = 1.013E6 dyne/cm/cm = 1013 kPa 1 inch of mercury (inHg) = 33.9 mb = 3.390 Pa 1 torr = 1mm mercury = 1.333 mbar 1 ton/in2 = 15.4 MPa 1 pound/acre = 1.12085 kg/ha 1 pound/in2 (psi) = 1 lbf/in2 = 6895 Pa =6.805E-2 atm = 51.71 mm Hg = 6.859E-2 bar
1 atomic unit of mass = 1 u = 1.66043E-27 kg Mass of proton = 1.672614E-27 kg Mass of neutron = 1.674920E-27 kg Mass of electron = 9.109558E-31 kg Standard acceleration of free fall on Earth = 9.80665 m/s/s Universal gravitational constant G = 6.672E-11 N m2 /kg2 (in calculating the attraction between two masses) Atmosphere at sea level: mass density = 1.225 kg/m3. Pressure = 1.01325 bar Molecules/litre = 2.547E22. Average temperature = 15.0 ºC Molecular weight of air = 28.96 g/mole. Oxygen = 2 x 16 = 32. Nitrogen = 2 x 14 = 28. Centripetal acceleration c = - (m x v x v ) / r , where v = velocity, m = mass and r = radius. (m/s/s) Gravity attraction of Earth (free fall acceleration) c = ( G x mE ) / r x r , where G= gravitational constant, mE= mass of Earth c= (6.67E-11 x 6E24 ) / r x r = 400E12 / r x r (attraction of 1 kg by Earth's gravity) for each orbit, the two above must balance : 400E12 / (r x r) = v x v / r , thus v = SQR(400E12/ r)
Energy, work, power
1 Joule (J) = 1 N m = 2 kg m/s = 1E7 ergs = 1 watt-second (Ws) = 0.2390 calories (cal) One joule is the amount of energy used when a force of 1 newton moves an object 1 metre in the direction of the force. 1 erg = 1 g cm cm = 1E-7 J 1 food calorie = 1 kcal = 4184 J 1 kilojoule (kJ) = 1000 J = 0.9484 Btu (British thermal units) = 737.6 foot pounds (ft lb) = 2390 cal = 0.2778E-3 kWhr 1 Btu is the quantity of heat needed to raise the temperature of 1 pound of water by 1 °F. 1 British thermal unit (Btu) = 1055.056 J 1 therm = 100,000 Btu = 106 MJ 1 horsepower (hp) = 0.746 kW; 1 kW = 1.34 hp 1 kilowatt-hour = 3600 kJ = 3.6 MJ = 3413 Btu = 859.845 kilocalories (kcal) = 1.98E6 footpounds = 0.034 therms 1 kilowatt-year = 31.56E9 J 1 kilowatt (kW) - 1 kJ/s = 1.341 horsepower (Hp) 1 quad = 1 quadrillion BTU = 1E15 Btu = 1.055E18 J = 1.055 EJ = 2.930E11 kWh = 0.001 Q 1 langley = 1 cal/cm2 ; 1 Langley/minute = 0.6974 kW/m2 1 electronvolt = 1.602E-19 J (1.502???)
1 litre gasoline = 35.2E6 J = 8413 kcal. 1 kiloton of TNT explosives = 4.2E12 J; 1 ton TNT = 4.2E9J; 1kg TNT = 4.2E6 J (Ehrlich, 1977) 1 tonne of coal = 8000 kWh = 28.8 GJ (Average coal burns at 28.8 MJ/kg) Combustion of dry biomass = 15-30 kJ/g
A small car example: a small motor car's engine is about 40 kW energy output = 40 kJ/s = 40 x 1.341 = 53.6 Hp In one hour (=3600 s), this car can produce 3600 x 40 kJ = 144000 kJ = 144E6 J = 144/4.2E6, equal to 34.2 kg of TNT explosives. An internal combustion motor of this capacity uses about 10 litre fuel per hour (352E6 J), travelling at 120 km/hr (at 32 KW): An internal combustion motor runs at near 30% efficiency, requiring an amount of fuel equal to 3x its energy output. The energy content of gasoline is about 46 MJ/kg or about 37 MJ/litre. The car above would have consumed 144 x 3 MJ = 11.7 litres. Energy statistics relating to the airplane attack on the World Trade Centre in New York, 11 September 2001: Towers: 410m tall; total weight 1.25 Mt; Collapse energy = 0.5 x 410 x 1.25E9 x 9.8= 2.5E12 J = 597 t TNT (Sci-Am: 500t ). Fuel: Boeing 767 fuel capaciry 23,980 Gall (USA) = 90,800 litres = 2.7E12 J = 640 t TNT; Together 1280t TNT (Sci-Am: 1170 t) Impact: direct explosion 3000 gall = 180 t TNT; fire afterwards 990 t TNT. Takeoff weight of airplane 412,000 lb = 187 t. Typical cruising speed: 530 mph = 853 km/h = 237 m/s. Kinetic energy of each plane : 0.5 x 187000 x 237 x 237 = 5.2E9 J = 1.24t TNT; Together 2.5 ton TNT (Sci-Am: 2 t). By comparison: Tomahawk cruise missile = 0.5 t TNT; Hiroshima bomb = 20,000 t TNT; a typical tornado = 5,100 t TNT. (Source: Scientific American Nov 2001 p 10) Note that Sci-Am arrives at slightly different values for energy equivalents.
Solar energy striking top of atmosphere = 1.72E17 W. Solar energy reaching surface = 8.6E16 W Solar energy per m2 on outside of atmosphere: average 1.360 kW/m2 ; 1.406 in June solstice and 1.314 in December solstice. Ultraviolet 0-380nm = 10.0%; visible 380 - 760 nm = 44.8%; infrared > 760nm = 45.2%. Evaporation of water = 4.13E15 W. Gross photosynthesis = 2E14 W. Global gross primary production (V Smil) 1E14 W. A modern power station = 1 million kW = 1 GW = 1E9 J/s = 1 GJ/s = 1000 MJ/s This power station consumes 375 tonnes of coal per hour or 3 million kW. Total generating efficiency is about 30% Total electricity generating power of the USA = 8.3E11 W = 8.3E5 MW = 8.3E5 MJ/s. Per year = United Kingdom consumes an amount of energy equal to 330 million tonnes of coal/ year = 2.64E12 kWh = 9.5E18 J
Theoretical amount of energy from uranium U-235 burner-type fission = 79E12 J/kg (2,700,000 x coal), but these reactors run at 1-2% fuel utilisation (50,000 x coal). A breeder-type of fission can run at 40-70% fuel utilisation (1,500,000 x coal). Because Uranium ore contains about 0.2% U3O8 ore, 1 kg of uranium-containing ore produces as much energy as 100 kg of coal in a standard fission reactor.
Daily intake of a well-nourished Westerner = 3 kWh = 10.8 MJ Maximum daily output of a manual worker = 0.5 kWh = 1.8 MJ Work output of a horse in an 8-hour day = 12 kWh = 43.2 MJ Energy statistics for 1975 Energy to feed 1 person for 1 day (2400 food calories) Nonfood energy use per person per day, world average Nonfood energy use per person per day, USA average 1 tank of gasoline (15 USgal = 56.8 litre) 1 barrel of oil (42 USgal = 158.9 litre) 1 metric ton of coal (= 1000 kg), average Boeing 707 flight, San Francisco - New York 1 kg Uranium-235, completely fissioned (used-up) Summer thunderstorm Fuel input to a 1000 MW power plant for 1 day Hydrogen bomb (1 Megaton TNT equivalent) Total human non-food energy use per day Sunlight striking top of atmosphere, 1 day (From Ehrlich, 1977). Note! enormous discrepancies exist, JFA Energy densities of various fuels and food Substance MJ/kg Substance Hydrogen 114.0 Pure plant oils Gasolines 46-47 Butter Crude oils 42-44 Pure protein Natural gases 33-37 Pure carbohydrates Ethanol 29.6 Cereal grains Best bituminous coals 27-29 Lean meats Common steam coals 22-24 Fish Good lignites 18-20 Potatoes Air-dried wood 14-15 Fruits Cereal straw 12-15 Vegetatbles Human feces 1.8-3.0 Urine 0.1-0.2 Source: V. Smil: Energies Typical energy costs of common materials (MJ/Kg) Material MJ/kg Made from Material MJ/kg Aluminium 227Bauxite Polyethylene 87-115 Energy MJ 10 200 1,000 2,000 5,900 29,000 1,400,000 79,000,000 160,000,000 260,000,000 4,000,000,000 800,000,000,000 15,000,000,000,000
MJ/kg 37-38 29-30 23.0 17.0 15.2-15.4 5-10 2.9-9.3 3.2-4.8 1.5-4.0 0.6-1.8
Made from Crude oil
342 Clay Polystyrene 62-108 Crude oil 2-5 Clay and Polyvinylchloride 85-107 Crude oil 5-9 limestone Sand 0.08-0.1 Riverbed 60-125 Sulfide ore Silicon 230-235 Silica 18-35 Sand, etc. Steel 20-50 Iron 20-25 Iron ore Sulfuric acid 2-3 Sulfur 0.07- Sedimentary Titanium 900-940 Ore concentrate 0.1 rock Water 0.001Streams, 230-70 Ore concentrate Wood 0.01 reservoirs 87-115 Standing timber 3-7 Standing timber Source: V Smil: Energies. 1 tonne of coal = 0.585 tonnes of petroleum = 7400 kWh = 708 m3 natural gas = 1430 m3 town gas 1 tonne of petroleum = 1.7 tonne coal = 12500 kWh = 1189 m3 natural gas = 2450 m3 town gas 1 tonne coal = 1.0 t brown coal NZ = 2.0-3.0 t brown coal elsewhere = 2.0 t peat = 0.77 t crude oil = 0.66 t petroleum products = 750 t natural gas = 8000 kWh (United Nations statistics) 1 tonne oil = 1.5 t coal = 4.9 t lignite = 3.3 t peat = 1167 m3 natural gas = 12000 kWh (British Petroleum statistics) 1 tonne tropical topsoil @ 1.45% Cn = 14.5 kg carbon. 1 tonne loam soil @ 3-7% C = 30-70 kg C. About 735 joules of energy are required to lift 15 kg of oil 5 meters out of the ground just to overcome gravity.
Bricks Cement Copper Glass Iron Limestone Nickel Paper
Energy conversion Potential energy E = m x h x g (J) where m = mass (kg) and h = height (m) and g= 9.81 = acceleration of gravity Kinetic energy E = 0.5 x m x v x v where m= mass and v = velocity Converting mass to energy: E = m x c x c where m = mass and c = speed of light 1 g of matter = 9E13 J = 25 GWh 1 atomic unit of mass = 1 u = 1.49E-10 J= 931 MeV 1 electron = 0.000549 u = 0.510 MeV Electrical noise power P = k x T x B watt, where k = Boltzman's constant, T = temperature ºK and B = bandwidth (hz) To photosynthesise 1 kg of carbon, 4.77E8 J in potential energy is obtained, = 1.14E8 cal. per gram of carbohydrate: 15.9kJ= 3.8 kcal. To electrolyse 1 mole of a substance, 96,500 Coulombs of electricity are required for each of its valence. e.g. 96,500 Coulombs to electrolyse 1 mole of copper (Cu++) = 63.54 deposits 31.77 gram. Heat Heat is the energy passed from one substance to another. Heat capacity = specific heat = heat storage (per kg per degree)
Heat of fusion of water (melting energy) at 0 ºC = 79.71 cal/g = 333.52 J/g Heat of vaporization of water (evaporation energy) at 100 ºC = 539.55 cal/g = 2256 j/g Latent heat of evaporation at 10ºC ? = 2.5E6 J/kg Specific heat of water = 1.000 = calories to warm 1 g by 1 ºC = 1.000 cal = 4.184 J Specific heat of sea water at 3.5% salinity = 0.932 cal = 3899 J/kg/ºC (roughly 4000) [density = 1000kg/m3] Specific heat of rock = 800 J/kg/ºC (about 5 times less than water). [density is 3000kg/m3] Specific heat of air = 1030 J/kg/ºC [density is 1.3 kg/m3] Oceans exchange heat to a depth of 100m, whereas rocks don't. This makes the ocean store 100 times more heat than the land. Energy radiation 1 Gray (Gy) = 1 J/kg 1 rad = 10 mGy = 1E-7 J absorbed per gram of material. Q = Quality factor: 1 for X-ray and gamma radiation; 10 for beta and 20 for alpha radiation RBE = Relative Biological Effectiveness= 1 rem = 1 rad x Q (above). Maximum permissible human dosage = 0.1 - 0.2 rem/yr = 1 - 2 mSv/yr (??) A sudden dose of 350 rem gives 50% mortality in 30 days. 1 Sievert = 1 Sv = 100 rem; 1 mSv = 0.1 rem. 1 Curie = 3.7E10 Becquerel (Bq) = 3.7E10 disintegrations/s = 1 gram of pure radium
Natural radiation hazard Average radiation exposure for a US resident: 3.6 mSv (higher in areas with radon emission) This is made up of: Radon 2 + food & water 0.4 + diagnostic X-rays 0.4 + cosmic radiation 0.26 (mSv). One flight from New York to HongKong = 0.1 mSv; Watching TV = 0.01 mSv/yr. Maximum occupational exposure per year: 50 mSv (equals 500 flights as above). Max for pregnant women: 5 mSv. Radiation hazard of 3.6 mSv equals a loss of life expectancy of 1.5 minutes or fewer than 2 cigarette puffs. (Source: Sci-Am Jan 2002 p 20)
Thermodynamics
First law, conservation of energy: Energy can neither be created nor destroyed. If energy in one form or one place disappers, it must show up in another form or another place. Energy can be converted from one form to another, but its total amount remains the same. Decrease in internal energy = heat produced + work done: dE = Q + W Second law, all forms of energy ultimately reduce to heat: All physical processes lead to a decrease in the availability of the energy involved. Processes involving energy transformations will not occur spontaneously unless there is a degradation of energy from a non-random to a random form. All spontaneous (natural) events act to increase the entropy within a system. Until
a system reaches its maximum entropy, it can do useful work. But as a system does work, its entropy increases until the system can no longer perform work. 1) In any transformation of energy, some of the energy is degraded. 2) No process is possible with 100% efficiency in transformation of energy. 3) No process is possible through the flow of heat from a colder body to a hotter one. 4) Once enrgergy has been used by a process, it cannot be used again by the same process. 5) High order energy (low entropy) converts into lower order energy (high entropy), and never the other way. 6) Order (low entropy) naturally degrades to disorder (high entropy). Third law, concerns absolute zero temperature: It states that it is impossible to reduce the temperature of any system to absolute zero. Ideal gases An 'ideal' gas is a conceptual one, behaving contrary to the second law, by not losing energy during expansion or compression. The volume of one mole of a substance, produces 22.4 litres of an ideal gas at OºC, 1 At, and 24.5 @ 25ºC
Thus 1 mole of oxygen, O2 = 2 x 16 = 32 g, produces 22.4 litres of gas with a density of 32/22,400 = 1.43E-3 g/cm3 @0ºC which is 700 times lighter than water. Air averages 28.96 g/mol = 1.29E-3 g/cm3 @0ºC = 773 times lighter than water.
Boyle Gay-Lussac: pressure x volume / temperature (K) = constant The volume of a gas increases proportionally to absolute temperature; The pressure of a gas increases proportionally as the volume decreases. ( P1 x V1 ) / T1 = ( P2 x V2 ) / T2 where situation (1) is before and situation (2) after a change.
Heating 22.4 litres of an ideal gas from 0º to 25ºC increases its volume to 22.4 x (273+25)/273 = 24.5 litres, see above.
To heat one mole of an ideal gas by one degree K, requires 8.3143 J. The abve laws are derived from the more universal equation P x V = R x T Where R = Gas constant = 8.317 J/mol/ºK, amount of heat to warm up one mole of a gas Henry's law: at a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid (for small concentrations, not saturated). e^p = e^{kc} or e p = e k c p = the partial pressure of the gaseous solute above the solution in atmospheres (approx kg/cm2) c = the concentration of the gaseous solute in the solution in mol/Litre or mol/kg for water k = the Henry's Law constant, which various with the kind of solvent and temperature. solubility decreases with increasing temperatures. Examples of the value for k at 26ºC: O2=769.2 ; CO2= 29.4; H2= 1282.1 L·atm/mol See http://www.henrys-law.org/
Stephan-Boltzman equation Defines the radiation emitting from an object of temperature T ºK: I = emissivity x 5.6704E-8 x T^4 W/m2 where emissivity = 1.00 for a black body; 0.997 for water; 0.969 for snow; desert sand 0.993.
Electricity
Coulomb's law constant = 1 / (4 x pi x e0) = 9.0E9 N m2 C-2 Permittivity constant ( in vacuum) e0 = 8.8541853E-12 C2 N-1 m-2 Permeability constant (in vacuum) m0 = 4 x pi x 1E-7 Wb A-1 m-1 Faraday's constant = 9.64867E4 C/mol Electronic charge = e = 1.6021917E-19 C 1 Ampere = 1 Coulomb/second.
Physical constants
Pi = 3.14159265 Natural logarithm e = 2.71828. ln(x) = 2.3026 log(x) Standard acceleration of gravity, g =0.98066 m/s/s Perfect gas constant, R = 8.3143 J/(g mol)/ºK. [Amount of energy needed to heat a gas] Avogadro's number, N = 6.022169E23 molecules/(g mol) [Number of molecules in a mole] Faraday constant, F = 9.648670E-4 C/mol Stefan-Boltzmann constant, sigma = 5.6697E-8 W/m2/ºK4 Also quoted as 5.6704E-8 Planck's constant, h = 6.6255E-34 Js [Amount of energy in an electro-magnetic wave; quantum theory] Boltzmann constant, k = 1.380622E-23 J/ºK. [Amount of radiated energy of a black body; entropy] Permeability of a vacuum, 4 x pi E-7 kg m /s/s/A/A
Frequencies, oscillation, electromagnetic spectrum
1 hertz (hz) = 1 cycle/sec (c/s) Speed of sound through air at earth's surface = 331.4 m/s Emission from neutral hydrogen = 1.42 Ghz
Biology
Humans Mean human mass worldwide = 50 kg, contains 1.9 kg nitrogen, which is 15% of body mass. Nitrogen content of proteins = 25% The mean human body contains 0.9 kg phosphorus, most of it in bones. Food energy per day = 2400 kcal = 10 MJ Maximum daily output of a manual worker = 0.5 kWh = 1.8 MJ Plants Dry weight (DM) of a crop is about 25% of wet weight. Carbon content of dried matter (DM) = 45% (containing carbohydrates CH2O chains) Nitrogen content of DM is 1.1 - 1.6%
Biomass calculations The ratios shown are not by weight but by the numbers of atoms. Average composition of terrestrial life: H:O:C:N:P:S = 2960:1480:1480:16:1.8:1 (E S Deevey Jr) Average composition of land plants: H:O:C:N:P:S = 1600:800:800:9:5:1 Average composition of marine plants: H:O:C:N:P:S = 212:106:106:16:2:1 Average composition of plant biomass: C:H:O= 1:(1.3-1.8):(0.5-0.6) (McDermitt&Loomis 1981) Redfield ratio required for phytoplankton: C:N:P= 106:16:1 Silicon requirement of diatoms: N:Si= 1:1 NPK fertiliser components of organisms: N:P:K= 100:18:22 with a yield of 20%. Carbohydrates = CH2O chains; glucose = C6H12O6: H:O:C = 12:6:6, or in mass 1:6:8 Energy conversion during photosynthesis/ respiration: 12 g carbon = 1 mole = 112 kcal = 469 kJ Photosynthesis: 6CO2 + 12 H2O + sunlight = C6H12O6 + 6O2 + 6H20 Respiration: C6H12O6 + 6O2 = 6CO2 + 6 H2O + heat (15.9kJ= 3.8kcal per gram carbohydrate) Methane production: 2CH2O = CH4 + CO2 Methane oxidation: 2CH4 + 3O2 = 2CO + 4H2O; 2CO + O2 = 2CO2. 1 kg dry organic matter (DM) = 0.45 kg C = 1.5 kg CO2 (with 15% variation) = 4200 kcal energy Rule of thumb: 4kcal per gram of dry organic matter 1 kg C corresponds to 2.2 kg dry organid matter. Wet weight (plants) = 4 x dry weight. Protein is 15% of body weight. N=25% of protein. gas exchange: 1 kg CO2 turnover = 0.73 kg O2 turnover
phytomass DM Gt NPP DM Gt/yr on land in oceans total 1840 107 3.9 55 1843.9 162
moles C/yr 4.0E15 2.1E15 6.1E15
watt/yr
kcal/yr
6.3E14 4.8E17 3.3E14 2.5E17 9.6E14 7.3E17
Animals 1 kg of beef requires 50,000 kg of water. A general equation to estimate the food requirement of marine mammal species (Innes et al 1987) Daily food cunsumption (kg biomass) Y = 0.123 x M ^ 0.80 or log(Y) = log(0.123) + 0.8 x log(M) Biochemical productivity Plant productivity increases 100% with 12ºC temperature increase Soil [note that facts relating to soil are traditionally expressed per hectare ha] Natural erosion before farming: 9E9 tonnes worldwide. In 1984: 25E9 tonnes Tennessee: losing 14 t/acre/yr = 35t/ha/yr. Nature puts back 5 t/acre/yr= 10t/ha/yr: Loss= 25t/ha/yr = 15m3/10,000m2/yr = 1.5mm /yr Conservation farming reduced soil loss from 14 to 10 t/acre/yr tropical rainforest = 180 t/acre in dry matter sediment yield from erosion by water 20-2600 t/sqkm/yr avg 30-92 m3/sqkm/yr =9 km3/yr particles + 9.4E9 ton dissolved = 6 cm/1000yr= 0.06 mm/yr (Last Frontier p367) Natural rate of weathering = 0.1mm/yr = 0.1 x 0.001 x 10,000 m3/ha/yr = 1m3/ha/yr = 100 m3/km2/yr = 1.5t/ha/yr Water Capillary rise of height h in a tube with diameter d : h = 0.3 / d (in cm) (approximately)
A hair-thin capillary of 0.1 mm = 0.01 cm causes water to rise by 30 cm.
NZ vital statistics
Total land area: 270,500 km2 = 27,050,000 ha Total seashore: 15,000 km Territorial sea out to 12 sea miles (22.2km): 160,000 km2 = 16,000,000ha Exclusive Economic Zone (EEZ): 4.3 Mkm2 (4.85 also quoted) = 4,300,000 km2 = 430,000,000 ha = 1.2 Mnm2 (square sea miles)
Continental shelf: 24,000,000 ha = 240,000 km2 only slightly smaller than NZ's land area Estuaries: 100,000 ha = 1000 km2 Sea area in cable protection zones: 1500 km2 Sea area in ammunition dumps, mining & oil drilling: 1500 km2 Sea area in marine reserves around mainland: 150 km2 = 15,000 ha Productivity of NZ coastal seas 2-8 t/km2 (D Pauly) NZ soil erosion: 270Mt/yr = 1000 t/km2/yr = 10 t/ha/yr = 6 m3/ha/yr is about six times the sustainable rate, which is severely underestimated. But most erosion by far comes from the 13Mha of agricultural land (about 50% of the land), which makes erosion 12 times the sustainable rate. But NZ has always had very low natural erosion and in recent years erosion has accelerated due to more torrential rains. so, NZ erosion may well top 400Mt/yr or over 20 times of what was natural. loss of Carbon by soil erosion, estimated at between 3 and 10 Mt C per year (AgResearch)
Mathematics refresher
Solve the linear equation: x2 + bx = c solution: x = SQR( b x b / 4 + c) - b / 2 Example: x2 - x = 1 solution: x = SQR( 1 / 4 + 1) - 1 / 2 = SQR( 5/4 ) - 0.5 = 0.618034 the divine number or golden section
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