In the Name of God
(Applied Chemistry)
Dimensions & Units
Introduction
In science, a type of question often asked is how much? how big? In order to answer such questions it is important to have systems of measurement which are consistent and understood by all.
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1 Introduction
Inch, foot; based on human body
4000 B.C. Egypt; King’s Elbow=0.4633 m, 1.5 ft, 2 handspans, 6 hand-widths, 24 finger-thickness
AD 1101 King Henry I yard (0.9144 m) from his nose to the tip of his thumb
1528 French physician J. Fernel distance between Paris and Amiens
Introduction
1872, Meter (in Greek, metron to measure)- 1/10 of a millionth of the distance between the North Pole and the equator
Platinum (90%)-iridium (10%) X-shaped bar kept in controlled condition in Paris39.37 in
In 1960, 1,650,763.73 wave length in vacuum of the orange light given off by electrically excited krypton 86.
2 Dimensions & Units
Dimension - abstract quantity (e.g. length)
Dimensions are used to describe physical quantities
Dimensions are independent of units
Unit - a specific definition of a dimension based upon a physical reference (e.g. meter)
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Dimensions & Units
How long is the rod?
Rod of unknown length
Reference: Three rods of 1-m length The unknown rod is 3 m long. unit number The number is meaningless without the unit!
3 System of Units
A system of units is described by:
a set of fundamental dimensions from which all other dimensions may be derived, and a set of base units. A dimension is a property that can be measured such as distance, time, temperature, speed.
A unit is a basic division of a measured quantity and it enables to say how much of the quantity we have 10 miles, 2 hours etc.
In most systems, length (distance), weight, and time are fundamental quantities; or as has been now accepted as better in science and engineering, the substitution of mass for weight, as a better more
7basic parameter.
System of Units
Imperial System of Units
USCS (U.S. Customary System of Units)
SI (The International System of Units)
AES (American Engineering) systems of units
4 System of Units
Both imperial units and US customary units derive from earlier English units. Imperial units were mostly used in the British Commonwealth and the former British Empire but in most Commonwealth countries they have been largely supplanted by the metric system.
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System of Units
US customary units, however, are still the main system of measurement in the United States.
The U.S. customary units have common roots with the imperial units, which were used in the British Empire. Many U.S. units are virtually identical to their imperial counterparts, but the U.S. customary system developed from English units in use before the imperial system was standardized in 1824, and there are several numerical differences from the imperial system.
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5 System of Units
The International System of Units (abbreviated SI from French: Système international d'unités) is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten
France in 1840 legislated official adoption of the metric system and made its use be mandatory
In U.S., in 1866, the metric system was made legal, but its use was not compulsory
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Absolute and Gravitational Unit Systems
Absolute system
Dimensions used are not affected by gravity
Fundamental dimensions L,T,M
Gravitational System
Widely used used in engineering
Fundamental dimensions L,T,F
6 The International System of Units (SI)
Fundamental Dimension Base Unit
length [L] meter (m) mass [M] kilogram (kg) time [T] second (s) electric current [A] ampere (A) absolute temperature [q] kelvin (K) luminous intensity [l] candela (cd) amount of substance [n] mole (mol)
Supplementary SI Dimensions
Supplementary Dimension Base Unit
plane angle radian (rad)
solid angle steradian (sr)
7 Fundamental Units (SI)
Mass (kilogram): “a cylinder of platinum-iridium alloy maintained under vacuum conditions by the International Bureau of Weights and
Measures in Paris”
Fundamental Units (SI)
Time(second): “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom”
8 Fundamental Units (SI)
Length or Distance (meter): “the length of the path traveled by light in vacuum during a time interval of 1/299792458 seconds”
photon
Laser 1 m t = 0 s t = 1/299792458 s
Fundamental Units (SI)
Electric Current (ampere): “that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed one meter apart in a vacuum,would produce between these conductors a force equal to 2 ×
10-7 newtons per meter of length”
9 Fundamental Units (SI)
Temperature (kelvin): The kelvin unit is 1/273.16 of the temperature interval from absolutezero to the triple point of water.
Water Phase Diagram Pressure
Temperature 273.16 K
Fundamental Units (SI)
Amount of Substance (mole): “the amount of a substance that contains as many elementary entities as there are atoms in 0.012 kilograms of carbon 12”
10 Fundamental Units (SI)
Light or Luminous Intensity (candela): “the candela is the luminous intensity of a source that emits monochromatic radiation of frequency 540 × 1012 Hz and that has a radiant intensity of 1/683 watt per steradian.“
Supplementary Units (SI)
Plane Angle (Radian): “the plane angle between two radii of a circle which cut off on the circumference an arc equal in length to the radius:
11 Supplementary Units (SI)
Solid Angle (Steradian): “the solid angle which, having its vertex in the center of a sphere cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere”
The International System of Units (SI)
Prefix Decimal Multiplier Symbol
Atto 10-18 a Femto 10-15 f pico 10-12 p nano 10-9 n micro 10-6 m milli 10-3 m centi 10-2 c deci 10-1 d
12 The International System of Units (SI)
Prefix Decimal Multiplier Symbol
deka 10+1 da hecto 10+2 h kilo 10+3 k mega 10+6 M Giga 10+9 G Tera 10+12 T Peta 10+15 P exa 10+18 E
U.S. Customary System of Units (USCS)
Fundamental Dimension Base Unit
length [L] foot (ft) force [F] pound (lb) time [T] second (sec)
Derived Dimension Unit Definition
2 2 mass [FT /L] slug lbf sec /ft
13 American Engineering System of Units (AES)
Fundamenal Dimension Base Unit
length [L] foot (ft)
mass [m] pound (lbm)
force [F] pound (lbf) time [T] second (sec) electric charge [Q] coulomb (C) absolute temperature [q degree Rankine (oR) luminous intensity [l] candela (cd) amount of substance [n] mole (mol)
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SI System of Units
Force = (mass) (acceleration)
F=ma W=mg
14 SI System of Units: Force
Force = ma kg m s 2 = Newton
= N
SI System of Units: Work/Energy
Work/ Energy = Force X Distance
= N.m kg.m 2 s 2 = Joule
= J
15 SI System of Units: Power
Power = Work / Time N m Joule J s s s kg m 2 s 3 = Watt = W
SI System of Units: Stress/Pressure
Pressure = Force / Area N kg m / s 2 m 2 m 2 kg m s 2 = Pascal = Pa
16 Temperature Scale vs Temperature Interval
212oF
32oF
DT = 212oF - 32oF=180 oF
Scale Interval
Temperature Conversion
Temperature Scale oF 1.8o C 32 oR 1.8K 1 1 o C o F 32 K o R 1.8 1.8 Temperature Interval Conversion Factors
1.8 o F 1.8 oR 1 o F 1 o C F o C K o R K
17 Length
Name of unit Symbol Definition Relation to SI units angstrom Å ≡ 1×10 10 m ≡ 0.1 nm foot (International) ft ≡ 1/3 yd ≡ 0.3048 m ≡ 12 inches ≡ 0.3048 m inch (International) in ≡ 1/36 yd ≡ 1/12 ft ≡ 0.0254 m
1 meter (SI base unit) m ≡ Distance light travels in ⁄299792458 of ≡ 1 m a second in vacuum. micron µ ≡ 1×10 6 m
mile (international) mi ≡ 80 chains ≡ 5280 ft ≡ 1760 yd ≡ 1609.344 m
nanometer nm ≡ 1×10 9 m ≡ 1×10 9 m nautical mile NM; mi ≡ 1852 m ≡ 1852 m (international) quarter ≡ ¼ yd = 0.2286 m picometer pm ≡ 1×10 12 m
yard (International) yd ≡ 0.9144 m ≡ 3 ft ≡ 36 in ≡ 0.9144 m 35
Area
Name of unit Symbol Definition Relation to SI units
acre (international) ac ≡ 1 ch × 10 ch = ≡ 4 046.856 4224 m2 4840 sq yd square (roofing) ≡ 10 ft × 10 ft = 9.290 304 m2 square foot sq ft ≡ 1 ft × 1 ft ≡ 9.290 304×10 2 m2 square inch sq in ≡ 1 in × 1 in ≡ 6.4516×10 4 m2 square kilometre km2 ≡ 1 km × 1 km = 106 m2 square metre (SI unit) m2 ≡ 1 m × 1 m = 1 m2 square mile sq mi ≡ 1 mi × 1 mi = 2.589 988 110 336×106 m2 square yard sq yd ≡ 1 yd × 1 yd ≡ 0.836 127 36 m2 (International)
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18 Volume
Name of unit Symbol Definition Relation to SI units
barrel (Imperial) bl (Imp) ≡ 36 gal (Imp) = 0.163 659 24 m3
barrel (petroleum) bl; bbl ≡ 42 gal (US) = 0.158 987 294 928 m3
barrel (U.S. fluid) fl bl (US) ≡ 31½ gal (US) = 0.119 240 471 196 m3
cubic foot cu ft ≡ 1 ft × 1 ft × 1 ft ≡ 0.028 316 846 592 m3
cubic inch cu in ≡ 1 in × 1 in × 1 in ≡ 16.387 064×10 6 m3
cubic metre (SI unit) m3 ≡ 1 m × 1 m × 1 m ≡ 1 m3
cubic mile cu mi ≡ 1 mi × 1 mi × 1 mi ≡ 4 168 181 825.440 579 584 m3 cubic yard cu yd ≡ 27 cu ft ≡ 0.764 554 857 984 m3
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Volume
Name of unit Symbol Definition Relation to SI units
cup (metric) c ≡ 250.0×10 6 m3 = 250.0×10 6 m3 drop (metric) ≡ 1/20 mL = 50.0×10 9 m3 gallon (Imperial) gal (Imp) ≡ 4.546 09 L ≡ 4.546 09×10 3 m3 gallon (U.S. fluid; Wine) gal (US) ≡ 231 cu in ≡ 3.785 411 784×10 3 m3 litre L ≡ 1 dm3 ≡ 0.001 m3 pint (Imperial) pt (Imp) ≡ ⅛ gal (Imp) = 568.261 25×10 6 m3
pint (U.S. fluid) pt (US fl) ≡ ⅛ gal (US) = 473.176 473×10 6 m3
quart (Imperial) qt (Imp) ≡ ¼ gal (Imp) = 1.136 5225×10 3 m3 quart (U.S. fluid) qt (US) ≡ ¼ gal (US fl) = 946.352 946×10 6 m3
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19 Mass
Name of unit Symbol Definition Relation to SI units atomic mass unit, u; AMU ≈ 1.660 538 73×10 27 ± unified 1.3×10 36 kg carat (metric) ct ≡ 200 mg = 200 mg grain gr ≡ 1/7000 lb av ≡ 64.798 91 mg kilogram kg ≡ mass of the prototype ≡ 1 kg (SI base unit) near Paris (≈ mass of 1L of water) ounce (avoirdupois) oz av ≡ 1/16 lb = 28.349 523 125 g pound (avoirdupois) lb av ≡ 0.453 592 37 kg = 7000 ≡ 0.453 592 37 kg grains quarter (Imperial) ≡ 1/4 long cwt = 2 st = = 12.700 586 36 kg 28 lb av slug; geepound slug ≡ 1 gee × 1 lb av × 1 s2/ft ≈ 14.593 903 kg ton, long long tn or ton ≡ 2 240 lb = 1 016.046 9088 kg ton, short sh tn ≡ 2 000 lb = 907.184 74 kg tonne (metric) t ≡ 1 000 kg = 1 000 kg
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Force Name of unit Symbol Definition Relation to SI units dyne (cgs unit) dyn ≡ g·cm/s2 = 10 5 N kilogram- kgf; kp; Gf ≡ g × 1 kg = 9.806 65 N force; kip; kip-force kip; kipf; ≡ g × 1 000 lb = 4.448 221 615 klbf 2605×103 N newton (SI N A force capable of giving a mass = 1 N = 1 kg·m/s2 unit) of one kg an acceleration of one metre per second, per second. pound lb ≡ slug·ft/s2 = 4.448 230 531 N
pound-force lbf ≡ g × 1 lb = 4.448 221 615 2605 N
poundal pdl ≡ 1 lb·ft/s2 = 0.138 254 954 376 N
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20 Pressure or mechanical stress
Name of unit Symbol Definition Relation to SI units atmosphere atm ≡ 101 325 Pa (standard) bar bar ≡ 105 Pa barye (cgs unit) ≡ 1 dyn/cm2 = 0.1 Pa centimetre of cmHg ≡ 13 595.1 kg/m3 × ≈ 1.333 22×103 Pa mercury 1 cm × g 3 centimetre of cmH2O ≈ 999.972 kg/m × ≈ 98.063 8 Pa water (4 °C) 1 cm × g pascal (SI unit) Pa ≡ N/m2 = 1 Pa pound per square psi ≡ 1 lbf/in2 ≈ 6.894 757×103 Pa inch torr torr ≡ 101 325/760 Pa ≈ 133.322 4 Pa
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Energy, work, or amount of heat Name of unit Symbol Definition Relation to SI units
British thermal unit BTUIT = 1.055 055 852 (International Table) 62×103 J
calorie calIT ≡ 4.1868 J = 4.1868 J (International Table) electronvolt eV ≡ e × 1 V ≈ 1.602 177 33×10 19 ± 4.9×10 26 J erg (cgs unit) erg ≡ 1 g·cm2/s2 = 10 7 J foot-pound force ft lbf ≡ g × 1 lb × 1 ft = 1.355 817 948 331 4004 J horsepower-hour hp·h ≡ 1 hp × 1 h = 2.684 519 537 696 172 792×106 J inch-pound force in lbf ≡ g × 1 lb × 1 in = 0.112 984 829 027 6167 J joule (SI unit) J The work done when a force of one = 1 J = 1 m·N = newton moves the point of its 1 kg·m2/s2 = 1 C·V = application a distance of one metre 1 W·s in the direction of the force. 3 kilocalorie kcal; Cal ≡ 1 000 calIT = 4.1868×10 J kilowatt-hour; kW·h; ≡ 1 kW × 1 h = 3.6×106 J Board of Trade Unit B.O.T.U 42 .
21 Power or heat flow rate Name of unit Symbol Definition Relation to SI units
BTU (International BTUIT/h ≡ 1 BTUIT/h ≈ 0.293 071 W Table) per hour
BTU (International BTUIT/mi ≡ 1 BTUIT/min ≈ 17.584 264 W Table) per minute n
BTU (International BTUIT/s ≡ 1 BTUIT/s = 1.055 055 852 Table) per second 62×103 W
calorie (International calIT/s ≡ 1 calIT/s = 4.1868 W Table) per second foot-pound-force per ft lbf/h ≡ 1 ft lbf/h ≈ 3.766 161×10 4 hour W foot-pound-force per ft lbf/min ≡ 1 ft lbf/min = 2.259 696 580 minute 552 334×10 2 W foot-pound-force per ft lbf/s ≡ 1 ft lbf/s = 1.355 817 948 second 331 4004 W horsepower (metric) hp ≡ 75 m kgf/s = 735.498 75 W watt (SI unit) W The power which in one = 1 W = 1 J/s = second of time gives rise 1 N·m/s = to one joule of energy. 1 kg·m2/s3 43
Temperature
Name of unit Symbol Definition Conversion to kelvin degree Celsius °C °C ≡ K 273.15 [K] ≡ [°C] + 273.15 degree Fahrenheit °F °F ≡ °C × 9/5 + 32 [K] ≡ ([°F] + 459.67) × 5/9 degree Rankine °R; °R ≡ K × 9/5 [K] ≡ [°R] × 5/9 kelvin (SI base K ≡ 1/273.16 of the ≡ 1 K unit) thermodynamic temperature of the triple point of water.
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22 Plane angle
Name of unit Symbol Definition Relation to SI units degree (of arc) ° ≡ 1/360 of a revolution ≡ ≈ 17.453 π/180 rad 293×10 3 rad
grad; gradian; grad ≡ 1/400 of a revolution ≡ ≈ 15.707 gon 2π/400 rad ≡ 0.9° 963×10 3 rad radian (SI rad The angle subtended at the = 1 rad unit) center of a circle by an arc whose length is equal to the circle's radius. One full revolution encompasses 2π radians.
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How do dimensions behave in mathematical formulae?
Rule 1 - All terms that are added or subtracted must have same dimensions
D A B C All have identical dimensions
23 How do dimensions behave in mathematical formulae?
Rule 2 - Dimensions obey rules of multiplication and division
[M] [T2 ] 2 AB [T ] [L] D [L] C [M] 2 [L ]
How do dimensions behave in mathematical formulae?
Rule 3 - In scientific equations, the arguments of “transcendental functions” must be dimensionless.
A ln(x) C sin(x) x must be dimensionless B exp(x) D 3x
Exception - In engineering correlations, the argument may have dimensions Transcendental Function - Cannot be given by algebraic expressions consisting only of the argument and constants. Requires an infinite series x2 x3 ex 1 x ··· 2! 3!
24 Dimensionally Homogeneous Equations
An equation is said to be dimensionally homogeneous if the dimensions on both sides of the equal sign are the same.
Dimensionally Homogeneous Equations
Volume of the frustrum of a right pyramid with a square base b
h
B h V B2 Bb b2 3
3 L 2 2 2 3 L L L L L . 1
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