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THE INTERNATIONAL SYSTEM OF UNITS (SI)*
Table 1 The SI Base Units
Quantity
Name
Symbol
length
meter
m
mass
kilogram
kg
time
second
s
electric current
ampere
A
thermodynamic temperature
kelvin
K
amount of substance
mole
mol
luminous intensity
candela
cd
Definition
“. . . the length of the path traveled by light in vacuum in
1/299,792,458 of a second.” (1983)
“. . . this prototype [a certain platinum – iridium cylinder] shall
henceforth be considered to be the unit of mass.” (1889)
“. . . 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.” (1967)
“. . . that constant current which, if maintained in two
straight parallel conductors of infinite length, of negligible
circular cross section, and placed 1 meter apart in vacuum,
would produce between these conductors a force equal to
2 107 newton per meter of length.” (1946)
“. . . the fraction 1/273.16 of the thermodynamic temperature
of the triple point of water.” (1967)
“. . . the amount of substance of a system which contains as
many elementary entities as there are atoms in 0.012 kilogram of carbon-12.” (1971)
“. . . the luminous intensity, in a given direction, of a source
that emits monochromatic radiation of frequency 540 10 12 hertz and that has a radiant intensity in that direction
of 1/683 watt per steradian.” (1979)
*Adapted from “The International System of Units (SI),” National Bureau of Standards Special Publication 330, 1972 edition. The definitions above were
adopted by the General Conference of Weights and Measures, an international body, on the dates shown. In this book we do not use the candela.
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APPE N DIX A TH E I NTE R NATIONAL SYSTE M OF U N ITS (SI)
Table 2 Some SI Derived Units
Quantity
Name of Unit
Symbol
area
volume
frequency
mass density (density)
speed, velocity
square meter
cubic meter
hertz
kilogram per cubic meter
meter per second
m2
m3
Hz
kg/m3
m/s
angular velocity
acceleration
angular acceleration
force
pressure
radian per second
meter per second per second
radian per second per second
newton
pascal
rad/s
m/s2
rad/s2
N
Pa
kg m/s2
N/m2
work, energy, quantity of heat
power
quantity of electric charge
potential difference, electromotive force
electric field strength
joule
watt
coulomb
volt
volt per meter (or newton per coulomb)
J
W
C
V
V/m
N m
J/s
A s
W/A
N/C
electric resistance
capacitance
magnetic flux
inductance
magnetic flux density
ohm
farad
weber
henry
tesla
F
Wb
H
T
V/A
A s/V
V s
V s/A
Wb/m2
magnetic field strength
entropy
specific heat
thermal conductivity
radiant intensity
ampere per meter
joule per kelvin
joule per kilogram kelvin
watt per meter kelvin
watt per steradian
A/m
J/K
J/(kg K)
W/(m K)
W/sr
Table 3 The SI Supplementary Units
Quantity
Name of Unit
Symbol
plane angle
solid angle
radian
steradian
rad
sr
s1
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SOME FUNDAMENTAL CONSTANTS OF PHYSICS*
Best (1998) Value
Constant
Symbol
Valuea
Computational Value
Uncertaintyb
Speed of light in a vacuum
Elementary charge
Gravitational constant
Universal gas constant
Avogadro constant
Boltzmann constant
Stefan – Boltzmann constant
Molar volume of ideal gas at STPd
Permittivity constant
Permeability constant
Planck constant
c
e
G
R
NA
k
s
Vm
e0
m0
h
3.00 10 8 m/s
1.60 10 19 C
6.67 10 11 m3/s2 kg
8.31 J/mol K
6.02 10 23 mol1
1.38 10 23 J/K
5.67 10 8 W/m2 K4
2.27 10 2 m3/mol
8.85 10 12 F/m
1.26 10 6 H/m
6.63 10 34 J s
2.997 924 58
1.602 176 487
6.674 28
8.314 472
6.022 141 79
1.380 650 4
5.670 400
2.271 098 1
8.854 187 817 62
1.256 637 061 43
6.626 068 96
exact
0.025
100
1.7
0.050
1.7
7.0
1.7
exact
exact
0.050
Electron massc
me
Proton massc
mp
Ratio of proton mass to electron mass
Electron charge-to-mass ratio
Neutron massc
mp/me
e/me
mn
Hydrogen atom massc
Deuterium atom massc
Helium atom massc
Muon mass
m1H
m2H
m4He
mm
9.11 10 31 kg
5.49 10 4 u
1.67 10 27 kg
1.0073 u
1840
1.76 10 11 C/kg
1.68 10 27 kg
1.0087 u
1.0078 u
2.0136 u
4.0026 u
1.88 10 28 kg
9.109 382 15
5.485 799 094 3
1.672 621 637
1.007 276 466 77
1836.152 672 47
1.758 820 150
1.674 927 211
1.008 664 915 97
1.007 825 031 6
2.013 553 212 724
4.002 603 2
1.883 531 30
0.050
4.2 10 4
0.050
1.0 10 4
4.3 10 4
0.025
0.050
4.3 10 4
0.0005
3.9 10 5
0.067
0.056
Electron magnetic moment
Proton magnetic moment
Bohr magneton
Nuclear magneton
Bohr radius
Rydberg constant
Electron Compton wavelength
me
mp
mB
mN
a
R
lC
9.28 1024 J/T
1.41 1026 J/T
9.27 1024 J/T
5.05 1027 J/T
5.29 1011 m
1.10 10 7 m1
2.43 1012 m
9.284 763 77
1.410 606 662
9.274 009 15
5.050 783 24
5.291 772 085 9
1.097 373 156 852 7
2.426 310 217 5
0.025
0.026
0.025
0.025
6.8 10 4
6.6 10 6
0.0014
a
Values given in this column should be given the same unit and power of 10 as the computational value.
b
Parts per million.
Masses given in u are in unified atomic mass units, where 1 u 1.660 538 782 10 27 kg.
c
d
STP means standard temperature and pressure: 0#C and 1.0 atm (0.1 MPa).
*The values in this table were selected from the 1998 CODATA recommended values (www.physics.nist.gov).
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SOME ASTRONOMICAL DATA
Some Distances from Earth
3.82 10 8 m
1.50 10 11 m
4.04 10 16 m
To the Moon*
To the Sun*
To the nearest star (Proxima Centauri)
2.2 10 20 m
2.1 10 22 m
10 26 m
To the center of our galaxy
To the Andromeda Galaxy
To the edge of the observable universe
*Mean distance.
The Sun, Earth, and the Moon
Property
Unit
Mass
Mean radius
Mean density
Free-fall acceleration at the surface
Escape velocity
Period of rotationa
Radiation powerc
kg
m
kg/m3
m/s2
km/s
—
W
Sun
1.99 10 30
6.96 10 8
1410
274
618
37 d at poles b 26 d at equator b
3.90 10 26
Earth
Moon
5.98 10 24
6.37 10 6
5520
9.81
11.2
23 h 56 min
7.36 10 22
1.74 10 6
3340
1.67
2.38
27.3 d
a
Measured with respect to the distant stars.
The Sun, a ball of gas, does not rotate as a rigid body.
c
Just outside Earth’s atmosphere solar energy is received, assuming normal incidence, at the rate of 1340 W/m2.
b
Some Properties of the Planets
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Plutod
Mean distance from Sun,
106 km
57.9
108
150
228
778
1430
2870
4500
5900
Period of revolution, y
0.241
0.615
1.00
1.88
11.9
29.5
84.0
165
248
Period of rotation,a d
58.7
243b
0.997
1.03
0.409
0.426
0.451b
0.658
6.39
Orbital speed, km/s
47.9
35.0
29.8
24.1
13.1
9.64
6.81
5.43
4.74
Inclination of axis to orbit
28#
3#
23.4#
25.0#
3.08#
26.7#
97.9#
29.6#
57.5#
Inclination of orbit to
Earth’s orbit
7.00#
3.39#
1.85#
1.30#
2.49#
0.77#
1.77#
17.2#
Eccentricity of orbit
0.206
0.0068
0.0167
0.0934
0.0485
0.0556
0.0472
0.0086
0.250
Equatorial diameter, km
4880
12 100
12 800
6790
143 000
120 000
51 800
49 500
2300
Mass (Earth 1)
0.0558
0.815
1.000
0.107
318
95.1
14.5
17.2
0.002
Density (water 1)
5.60
5.20
5.52
3.95
1.31
0.704
1.21
1.67
2.03
Surface value of g,c m/s2
3.78
8.60
9.78
3.72
22.9
9.05
7.77
11.0
0.5
Escape velocity,c km/s
4.3
10.3
11.2
5.0
59.5
35.6
21.2
23.6
1.3
Known satellites
0
0
1
2
67 ring
62 rings
27 rings
13 rings
4
a
Measured with respect to the distant stars.
Venus and Uranus rotate opposite their orbital motion.
c
Gravitational acceleration measured at the planet’s equator.
d
Pluto is now classified as a dwarf planet.
b
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CONVERSION FACTORS
Conversion factors may be read directly from these tables. For example, 1 degree 2.778 10 3 revolutions, so 16.7# 16.7 2.778 10 3 rev. The SI units are fully capitalized. Adapted
in part from G. Shortley and D. Williams, Elements of Physics, 1971, Prentice-Hall, Englewood
Cliffs, NJ.
Plane Angle
$
RADIAN
rev
60
1
1.667 10 2
3438
2.16 10 4
3600
60
1
2.063 10 5
1.296 10 6
1.745 10 2
2.909 10 4
4.848 10 6
1
6.283
2.778 10 3
4.630 10 5
7.716 10 7
0.1592
1
°
1 degree 1
1 minute 1.667 10 2
1 second 2.778 10 4
1 RADIAN 57.30
1 revolution 360
Solid Angle
1 sphere 4p steradians 12.57 steradians
Length
cm
1 centimeter 1
1 METER 100
1 kilometer 10 5
1 inch 2.540
1 foot 30.48
1 mile 1.609 10 5
1 angström 10 10 m
1 nautical mile 1852 m
1.151 miles 6076 ft
METER
km
in.
ft
mi
10 2
1
1000
2.540 10 2
0.3048
1609
10 5
10 3
1
2.540 10 5
3.048 10 4
1.609
0.3937
39.37
3.937 10 4
1
12
6.336 10 4
3.281 10 2
3.281
3281
8.333 10 2
1
5280
6.214 10 6
6.214 10 4
0.6214
1.578 10 5
1.894 10 4
1
1 fermi 10 15 m
1 light-year 9.461 10 12 km
1 fathom 6 ft
1 rod 16.5 ft
1 Bohr radius 5.292 10 11 m
1 mil 10 3 in.
1 parsec 3.084 10 km
1 yard 3 ft
1 nm 10 9 m
13
Area
METER2
1 SQUARE METER 1
1 square centimeter 10 4
1 square foot 9.290 10 2
1 square inch 6.452 10 4
cm2
ft2
in.2
10 4
1
929.0
6.452
10.76
1.076 10 3
1
6.944 10 3
1550
0.1550
144
1
1 square mile 2.788 10 7 ft2 640 acres
1 acre 43 560 ft 2
1 barn 10 28 m2
1 hectare 10 4 m2 2.471 acres
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APPE N DIX D CONVE RSION FACTORS
Volume
METER3
1 CUBIC METER 1
1 cubic centimeter 10 6
1 liter 1.000 10 3
1 cubic foot 2.832 10 2
1 cubic inch 1.639 10 5
cm3
L
ft3
in.3
10 6
1
1000
2.832 10 4
16.39
1000
1.000 10 3
1
28.32
1.639 10 2
35.31
3.531 10 5
3.531 10 2
1
5.787 10 4
6.102 10 4
6.102 10 2
61.02
1728
1
1 U.S. fluid gallon 4 U.S. fluid quarts 8 U.S. pints 128 U.S. fluid ounces 231 in.3
1 British imperial gallon 277.4 in.3 1.201 U.S. fluid gallons
Mass
Quantities in the colored areas are not mass units but are often used as such. For example, when we write 1 kg “”
2.205 lb, this means that a kilogram is a mass that weighs 2.205 pounds at a location where g has the standard value
of 9.80665 m/s2.
g
KILOGRAM
1 gram 1
1 KILOGRAM 1000
1 slug 1.459 10 4
1 atomic
mass unit 1.661 10 24
1 ounce 28.35
1 pound 453.6
1 ton 9.072 10 5
slug
u
oz
lb
ton
0.001
1
14.59
6.852 10 5
6.852 10 2
1
6.022 10 23
6.022 10 26
8.786 10 27
3.527 10 2
35.27
514.8
2.205 10 3
2.205
32.17
1.102 10 6
1.102 10 3
1.609 10 2
1.661 10 27
2.835 10 2
0.4536
907.2
1.138 10 28
1.943 10 3
3.108 10 2
62.16
1
1.718 10 25
2.732 10 26
5.463 10 29
5.857 10 26
1
16
3.2 10 4
3.662 10 27
6.250 10 2
1
2000
1.830 10 30
3.125 10 5
0.0005
1
1 metric ton 1000 kg
Density
Quantities in the colored areas are weight densities and, as such, are dimensionally different from mass densities.
See the note for the mass table.
KILOGRAM/
METER3
slug/ft3
1 slug per foot3 1
1 KILOGRAM
per METER3 1.940 10 3
1 gram per centimeter3 1.940
1 pound per foot3 3.108 10 2
1 pound per inch3 53.71
g/cm3
lb/ft3
515.4
0.5154
32.17
1.862 10 2
1
1000
16.02
2.768 10 4
0.001
1
16.02 10 2
27.68
6.243 10 2
62.43
1
1728
3.613 10 5
3.613 10 2
5.787 10 4
1
T
ime
Time
y
1 year 1
1 day 2.738 10 3
1 hour 1.141 10 4
1 minute 1.901 10 6
1 SECOND 3.169 10 8
lb/in.3
d
h
min
SECOND
365.25
1
4.167 10 2
6.944 10 4
1.157 10 5
8.766 10 3
24
1
1.667 10 2
2.778 10 4
5.259 10 5
1440
60
1
1.667 10 2
3.156 10 7
8.640 10 4
3600
60
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APPE N DIX D CONVE RSION FACTORS
Speed
ft/s
1 foot per second 1
1 kilometer per hour 0.9113
1 METER per SECOND 3.281
1 mile per hour 1.467
1 centimeter per second 3.281 10 2
1 knot 1 nautical mi/h 1.688 ft/s
km/h
METER/SECOND
mi/h
cm/s
1.097
1
3.6
1.609
3.6 10 2
0.3048
0.2778
1
0.4470
0.01
0.6818
0.6214
2.237
1
2.237 10 2
30.48
27.78
100
44.70
1
1 mi/min 88.00 ft/s 60.00 mi/h
Force
Force units in the colored areas are now little used. To clarify: 1 gram-force ( 1 gf) is the force of gravity that
would act on an object whose mass is 1 gram at a location where g has the standard value of 9.80665 m/s2.
dyne
1 dyne 1
1 NEWTON 10 5
1 pound 4.448 10 5
1 poundal 1.383 10 4
1 gram-force 980.7
1 kilogram-force 9.807 10 5
NEWTON
lb
pdl
gf
10 5
1
4.448
0.1383
9.807 10 3
9.807
2.248 10 6
0.2248
1
3.108 10 2
2.205 10 3
2.205
7.233 10 5
7.233
32.17
1
7.093 10 2
70.93
1.020 10 3
102.0
453.6
14.10
1
1000
kgf
1.020 10 6
0.1020
0.4536
1.410 10 2
0.001
1
1 ton 2000 lb
Pressure
atm
1 atmosphere 1
1 dyne per
centimeter2 9.869 10 7
1 inch of
watera at 4°C 2.458 10 3
1 centimeter
of mercurya
at 0°C 1.316 10 2
1 PASCAL 9.869 10 6
1 pound per inch2 6.805 10 2
1 pound per foot2 4.725 10 4
dyne/cm2
inch of water
cm Hg
PASCAL
lb/in.2
lb/ft2
1.013 10 6
406.8
76
1.013 10 5
14.70
2116
1
4.015 10 4
7.501 10 5
0.1
1.405 10 5
2.089 10 3
2491
1
0.1868
249.1
3.613 10 2
5.202
1.333 10 4
10
6.895 10 4
478.8
5.353
4.015 10 3
27.68
0.1922
1
7.501 10 4
5.171
3.591 10 2
1333
1
6.895 10 3
47.88
0.1934
1.450 10 4
1
6.944 10 3
27.85
2.089 10 2
144
1
a
Where the acceleration of gravity has the standard value of 9.80665 m/s2.
1 bar 10 6 dyne/cm2 0.1 MPa
1 millibar 10 3 dyne/cm2 10 2 Pa
1 torr 1 mm Hg
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APPE N DIX D CONVE RSION FACTORS
Energy, Work, Heat
Quantities in the colored areas are not energy units but are included for convenience. They arise from the relativistic
mass – energy equivalence formula E mc2 and represent the energy released if a kilogram or unified atomic mass
unit (u) is completely converted to energy (bottom two rows) or the mass that would be completely converted to
one unit of energy (rightmost two columns).
Btu
1 British
thermal unit 1
9.481
1 erg 10 11
1.285
1 foot-pound 10 3
1 horsepowerhour 2545
9.481
1 JOULE 10 4
3.968
1 calorie 10 3
1 kilowatthour 3413
1.519
1 electron-volt 10 22
1 million 1.519
electron-volts 10 16
8.521
1 kilogram 10 13
1 unified
atomic mass 1.415
unit 10 13
ft lb
erg
1.055
10 10
1
1.356
10 7
2.685
10 13
777.9
7.376
10 8
1
1.980
10 6
hp h
3.929
10 4
3.725
10 14
5.051
10 7
1
3.725
10 7
1.560
10 6
JOULE
1055
252.0
10 7
2.389
10 8
1.356
2.685
10 6
0.3238
6.413
10 5
1
0.2389
1
8.600
10 5
3.827
10 20
3.827
10 14
2.146
10 16
3.564
10 11
107
4.1868
10 7
3.600
10 13
1.602
10 12
1.602
10 6
8.987
10 23
3.088
2.655
10 6
1.182
10 19
1.182
10 13
6.629
10 16
1.341
5.967
10 26
5.967
10 20
3.348
10 10
4.1868
3.600
10 6
1.602
10 19
1.602
10 13
8.987
10 16
1.492
10 3
1.101
10 10
5.559
10 17
1.492
10 10
0.7376
kW h
eV
MeV
kg
2.930
10 4
2.778
10 14
3.766
10 7
6.585
10 21
6.242
10 11
8.464
10 18
1.676
10 25
6.242
10 18
2.613
10 19
2.247
10 25
6.585
10 15
6.242
10 5
8.464
10 12
1.676
10 19
6.242
10 12
2.613
10 13
2.247
10 19
1
10 6
106
5.610
10 35
1
5.610
10 29
1.174
10 14
1.113
10 24
1.509
10 17
2.988
10 11
1.113
10 17
4.660
10 17
4.007
10 11
1.783
10 36
1.783
10 30
1
9.320
10 8
932.0
cal
0.7457
2.778
10 7
1.163
10 6
1
4.450
10 26
4.450
10 20
2.497
10 10
4.146
10 17
1.661
10 27
u
7.070
10 12
670.2
9.037
10 9
1.799
10 16
6.702
10 9
2.806
10 10
2.413
10 16
1.074
10 9
1.074
10 3
6.022
10 26
1
Power
Btu/h
ft lb/s
hp
cal/s
kW
WATT
1 British thermal unit per hour 1
1 foot-pound per second 4.628
1 horsepower 2545
1 calorie per second 14.29
1 kilowatt 3413
1 WATT 3.413
0.2161
1
550
3.088
737.6
0.7376
3.929 10 4
1.818 10 3
1
5.615 10 3
1.341
1.341 10 3
6.998 10 2
0.3239
178.1
1
238.9
0.2389
2.930 10 4
1.356 10 3
0.7457
4.186 10 3
1
0.001
0.2930
1.356
745.7
4.186
1000
1
Magnetic Field
gauss
1 gauss 1
1 TESLA 10 4
1 milligauss 0.001
1 tesla 1 weber/meter2
Magnetic Flux
TESLA
milligauss
10 4
1
10 7
1000
10 7
1
maxwell
1 maxwell 1
1 WEBER 10 8
WEBER
10 8
1
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MATHEMATICAL FORMULAS
Geometry
⬆ is not equal to
Circle of radius r: circumference 2pr; area pr2.
Sphere of radius r: area 4pr 2; volume 43pr3.
Right circular cylinder of radius r and height h:
area 2pr 2 2prh; volume pr 2h.
Triangle of base a and altitude h: area 12ah.
is identical to, is defined as
is greater than (% is much greater than)
is less than ( is much less than)
is greater than or equal to (or, is no less than)
is less than or equal to (or, is no more than)
plus or minus
Quadratic Formula
If ax bx c 0, then x 2
& is proportional to
1b2 4ac
.
2a
b
the sum of
xavg the average value of x
Trigonometric Functions of Angle u
sin u y
r
cos u x
r
tan u y
x
cot u x
y
sec u r
x
csc u Trigonometric Identities
y axis
sin(90# u) cos u
cos(90# u) sin u
sin u/cos u tan u
r
r
y
θ
x axis
x
0
sin2 u cos2 u 1
y
csc2 u cot2 u 1
Pythagorean Theorem
sin 2u 2 sin u cos u
In this right triangle,
a2 b2 c2
c
cos 2u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u
a
b
Triangles
Angles are A, B, C
Opposite sides are a, b, c
Angles A B C 180#
sin B
sin C
sin A
a
b
c
2
2
2
c a b 2ab cos C
Exterior angle D A C
sec2 u tan2 u 1
C
b
B
equals approximately
is the order of magnitude of
cos a sin b
cos(a
b) cos a cos b ' sin a sin b
tan(a
b) tan a tan b
1 ' tan a tan b
sin b 2 sin 12(a
b) cos 12(a ' b)
cos a cos b 2 cos 12(a b) cos 12(a b)
D
cos a cos b 2 sin 12(a b) sin 12(a b)
c
Mathematical Signs and Symbols
equals
b) sin a cos b
sin a
a
A
sin(a
Binomial Theorem
(1 x)n 1 n(n 1)x2
nx
1!
2!
(x2 1)
Exponential Expansion
ex 1 x x2
x3
2!
3!
A-9
A-10
APPE N DIX E MATH E MATICAL FOR M U L AS
Logarithmic Expansion
Products of Vectors
ln(1 x) x 12 x2 13 x3 (|x| 1)
Let î, ĵ , and k̂ be unit vectors in the x, y, and z directions. Then
î î ĵ ĵ k̂ k̂ 1,
Trigonometric Expansions
(u in radians)
î î ĵ ĵ k̂ k̂ 0,
u5
u3
sin u u 3!
5!
u4
u2
2!
4!
cos u 1 u3
2u 5
tan u u 3
15
î ĵ k̂,
k̂ î ĵ
:
a ax î ay ĵ azk̂.
:
Let :
c be arbitrary vectors with magnitudes a,
a , b , and :
b, and c.Then
:
:
:
:
:
c ) (a
b ) (a
:
c)
a (b :
:
:
:
:
:
(sa
)b:
a (sb ) s(a
b)
Two simultaneous equations in unknowns x and y,
a1x b1 y c1
and
a2x b2 y c2,
have the solutions
a1c2 a2c1
.
a1b2 a2b1
c1
c2
b1
b2
a1
a2
b1
b2
a1
a2
c1
c2
a1
a2
b1
b2
c1b2 c2b1
a1b2 a2b1
(s a scalar).
Let u be the smaller of the two angles between :
a and
b . Then
:
: :
:
ab b:
a axbx ayby azbz ab cos î
a b b a ax
bx
:
:
:
î
and
y
ĵ k̂ î,
Any vector :
a with components ax, ay, and az along the
x, y, and z axes can be written as
Cramer’s Rule
x
î ĵ ĵ k̂ k̂ î 0,
ay
by
:
ĵ
ay
by
k̂
az
bz
az
a az
a
ĵ x
k̂ x
bz
bx bz
bx
(aybz by az)î (azbx bzax)ĵ
(axby bxay)k̂
:
:
b | ab sin u
|a
:
:
:
:
:
a (b :
c ) b ( :
c :
a) :
c (a
b)
:
:
:
:
: :
:
a (b :
c ) (a
c )b (a
b):
c
ay
by
APPE N DIX E MATH E MATICAL FOR M U L AS
Derivatives and Integrals
In what follows, the letters u and v stand for any functions of
x, and a and m are constants. To each of the indefinite integrals should be added an arbitrary constant of integration.
The Handbook of Chemistry and Physics (CRC Press Inc.)
gives a more extensive tabulation.
1.
2.
3.
1.
dx
1
dx
4.
2.
du
d
(au) a
dx
dx
5.
d
du
dv
(u v) dx
dx
dx
6.
3.
4.
d m
x mxm1
dx
d
1
5.
ln x dx
x
7.
8.
6.
dv
du
d
(uv) u
v
dx
dx
dx
7.
d x
e ex
dx
8.
d
sin x cos x
dx
10.
9.
d
cos x sin x
dx
11.
10.
d
tan x sec2 x
dx
12.
11.
d
cot x csc2 x
dx
13.
12.
d
sec x tan x sec x
dx
14.
13.
d
csc x cot x csc x
dx
15.
du
d u
14.
e eu
dx
dx
du
d
15.
sin u cos u
dx
dx
d
du
16.
cos u sin u
dx
dx
9.
dx x
au dx a
u dx
xm dx dx
ln |x|
x
u
dv
dx uv dx
18.
19.
20.
0
21.
v dx
v
du
dx
dx
ex dx ex
sin x dx cos x
cos x dx sin x
tan x dx ln |sec x|
sin2 x dx 12 x 14 sin 2x
eax dx 1 ax
e
a
1
(ax 1) eax
a2
xeax dx x2eax dx xneax dx x2neax dx 2
0
17.
u dx xm1
( m ( 1)
m1
0
16.
(u v) dx 1
(a2x2 2ax 2)eax
a3
n!
an1
p
13 5 (2n 1)
n1 n
2 a
A a
dx
ln(x 2x2 a2)
2x2 a2
1
x dx
2
(x2 a2)3/2
(x a2)1/2
dx
x
2 2
2 3/2
(x a )
a (x a2)1/2
2
x2n1 eax dx 2
n!
(a 0)
2an1
x dx
x d ln(x d)
xd
A-11
A
P
P
E
N
D
I
X
F
PROPERTIES OF THE ELEMENTS
All physical properties are for a pressure of 1 atm unless otherwise specified.
Element
Symbol
Atomic
Number
Z
Actinium
Aluminum
Americium
Antimony
Argon
Arsenic
Astatine
Barium
Berkelium
Beryllium
Bismuth
Bohrium
Boron
Bromine
Cadmium
Calcium
Californium
Carbon
Cerium
Cesium
Chlorine
Chromium
Cobalt
Copernicium
Copper
Curium
Darmstadtium
Dubnium
Dysprosium
Einsteinium
Erbium
Europium
Fermium
Flerovium*
Fluorine
Francium
Gadolinium
Gallium
Germanium
Gold
Ac
Al
Am
Sb
Ar
As
At
Ba
Bk
Be
Bi
Bh
B
Br
Cd
Ca
Cf
C
Ce
Cs
Cl
Cr
Co
Cn
Cu
Cm
Ds
Db
Dy
Es
Er
Eu
Fm
Fl
F
Fr
Gd
Ga
Ge
Au
89
13
95
51
18
33
85
56
97
4
83
107
5
35
48
20
98
6
58
55
17
24
27
112
29
96
110
105
66
99
68
63
100
114
9
87
64
31
32
79
A-12
Molar
Mass,
g/mol
(227)
26.9815
(243)
121.75
39.948
74.9216
(210)
137.34
(247)
9.0122
208.980
262.12
10.811
79.909
112.40
40.08
(251)
12.01115
140.12
132.905
35.453
51.996
58.9332
(285)
63.54
(247)
(271)
262.114
162.50
(254)
167.26
151.96
(237)
(289)
18.9984
(223)
157.25
69.72
72.59
196.967
Density,
g/cm3 at 20#C
10.06
2.699
13.67
6.691
1.6626 10 3
5.78
—
3.594
14.79
1.848
9.747
—
2.34
3.12 (liquid)
8.65
1.55
—
2.26
6.768
1.873
3.214 10 3 (0#C)
7.19
8.85
—
8.96
13.3
—
—
8.55
—
9.15
5.243
—
—
1.696 10 3 (0#C)
—
7.90
5.907
5.323
19.32
Melting
Point, #C
1323
660
1541
630.5
189.4
817 (28 atm)
(302)
729
—
1287
271.37
—
2030
7.2
321.03
838
—
3727
804
28.40
101
1857
1495
—
1083.40
—
—
—
1409
—
1522
817
—
—
219.6
(27)
1312
29.75
937.25
1064.43
Boiling
Point,
#C
Specific
Heat,
J/(g #C)
at 25#C
(3473)
2450
—
1380
185.8
613
—
1640
—
2770
1560
—
—
58
765
1440
—
4830
3470
690
34.7
2665
2900
—
2595
—
—
—
2330
—
2630
1490
—
—
188.2
—
2730
2237
2830
2970
0.092
0.900
—
0.205
0.523
0.331
—
0.205
—
1.83
0.122
—
1.11
0.293
0.226
0.624
—
0.691
0.188
0.243
0.486
0.448
0.423
—
0.385
—
—
—
0.172
—
0.167
0.163
—
—
0.753
—
0.234
0.377
0.322
0.131
APPE N DIX F PROPE RTI ES OF TH E E LE M E NTS
Element
Symbol
Atomic
Number
Z
Hafnium
Hassium
Helium
Holmium
Hydrogen
Indium
Iodine
Iridium
Iron
Krypton
Lanthanum
Lawrencium
Lead
Lithium
Livermorium*
Lutetium
Magnesium
Manganese
Meitnerium
Mendelevium
Mercury
Molybdenum
Neodymium
Neon
Neptunium
Nickel
Niobium
Nitrogen
Nobelium
Osmium
Oxygen
Palladium
Phosphorus
Platinum
Plutonium
Polonium
Potassium
Praseodymium
Promethium
Protactinium
Radium
Radon
Rhenium
Rhodium
Roentgenium
Rubidium
Ruthenium
Rutherfordium
Hf
Hs
He
Ho
H
In
I
Ir
Fe
Kr
La
Lr
Pb
Li
Lv
Lu
Mg
Mn
Mt
Md
Hg
Mo
Nd
Ne
Np
Ni
Nb
N
No
Os
O
Pd
P
Pt
Pu
Po
K
Pr
Pm
Pa
Ra
Rn
Re
Rh
Rg
Rb
Ru
Rf
72
108
2
67
1
49
53
77
26
36
57
103
82
3
116
71
12
25
109
101
80
42
60
10
93
28
41
7
102
76
8
46
15
78
94
84
19
59
61
91
88
86
75
45
111
37
44
104
Molar
Mass,
g/mol
Density,
g/cm3 at 20#C
178.49
(265)
4.0026
164.930
1.00797
114.82
126.9044
192.2
55.847
83.80
138.91
(257)
207.19
6.939
(293)
174.97
24.312
54.9380
(266)
(256)
200.59
95.94
144.24
20.183
(237)
58.71
92.906
14.0067
(255)
190.2
15.9994
106.4
30.9738
195.09
(244)
(210)
39.102
140.907
(145)
(231)
(226)
(222)
186.2
102.905
(280)
85.47
101.107
261.11
13.31
—
0.1664 10 3
8.79
0.08375 10 3
7.31
4.93
22.5
7.874
3.488 10 3
6.189
—
11.35
0.534
—
9.849
1.738
7.44
—
—
13.55
10.22
7.007
0.8387 10 3
20.25
8.902
8.57
1.1649 10 3
—
22.59
1.3318 10 3
12.02
1.83
21.45
19.8
9.32
0.862
6.773
7.22
15.37 (estimated)
5.0
9.96 10 3 (0#C)
21.02
12.41
—
1.532
12.37
—
Melting
Point, #C
2227
—
269.7
1470
259.19
156.634
113.7
2447
1536.5
157.37
920
—
327.45
180.55
—
1663
650
1244
—
—
38.87
2617
1016
248.597
637
1453
2468
210
—
3027
218.80
1552
44.25
1769
640
254
63.20
931
(1027)
(1230)
700
(71)
3180
1963
—
39.49
2250
—
Boiling
Point,
#C
Specific
Heat,
J/(g #C)
at 25#C
5400
—
268.9
2330
252.7
2000
183
(5300)
3000
152
3470
—
1725
1300
—
1930
1107
2150
—
—
357
5560
3180
246.0
—
2730
4927
195.8
—
5500
183.0
3980
280
4530
3235
—
760
3020
—
—
—
61.8
5900
4500
—
688
4900
—
0.144
—
5.23
0.165
14.4
0.233
0.218
0.130
0.447
0.247
0.195
—
0.129
3.58
—
0.155
1.03
0.481
—
—
0.138
0.251
0.188
1.03
1.26
0.444
0.264
1.03
—
0.130
0.913
0.243
0.741
0.134
0.130
—
0.758
0.197
—
—
—
0.092
0.134
0.243
—
0.364
0.239
—
A-13
A-14
APPE N DIX F PROPE RTI ES OF TH E E LE M E NTS
Element
Samarium
Scandium
Seaborgium
Selenium
Silicon
Silver
Sodium
Strontium
Sulfur
Tantalum
Technetium
Tellurium
Terbium
Thallium
Thorium
Thulium
Tin
Titanium
Tungsten
Unnamed
Unnamed
Unnamed
Unnamed
Uranium
Vanadium
Xenon
Ytterbium
Yttrium
Zinc
Zirconium
Symbol
Atomic
Number
Z
Molar
Mass,
g/mol
Sm
Sc
Sg
Se
Si
Ag
Na
Sr
S
Ta
Tc
Te
Tb
Tl
Th
Tm
Sn
Ti
W
Uut
Uup
Uus
Uuo
U
V
Xe
Yb
Y
Zn
Zr
62
21
106
34
14
47
11
38
16
73
43
52
65
81
90
69
50
22
74
113
115
117
118
92
23
54
70
39
30
40
150.35
44.956
263.118
78.96
28.086
107.870
22.9898
87.62
32.064
180.948
(99)
127.60
158.924
204.37
(232)
168.934
118.69
47.90
183.85
(284)
(288)
—
(294)
(238)
50.942
131.30
173.04
88.905
65.37
91.22
Density,
g/cm3 at 20#C
7.52
2.99
—
4.79
2.33
10.49
0.9712
2.54
2.07
16.6
11.46
6.24
8.229
11.85
11.72
9.32
7.2984
4.54
19.3
—
—
—
—
18.95
6.11
5.495 10 3
6.965
4.469
7.133
6.506
Melting
Point, #C
1072
1539
—
221
1412
960.8
97.85
768
119.0
3014
2200
449.5
1357
304
1755
1545
231.868
1670
3380
—
—
—
—
1132
1902
111.79
824
1526
419.58
1852
Boiling
Point,
#C
Specific
Heat,
J/(g #C)
at 25#C
1630
2730
—
685
2680
2210
892
1380
444.6
5425
—
990
2530
1457
(3850)
1720
2270
3260
5930
—
—
—
—
3818
3400
108
1530
3030
906
3580
0.197
0.569
—
0.318
0.712
0.234
1.23
0.737
0.707
0.138
0.209
0.201
0.180
0.130
0.117
0.159
0.226
0.523
0.134
—
—
—
—
0.117
0.490
0.159
0.155
0.297
0.389
0.276
The values in parentheses in the column of molar masses are the mass numbers of the longest-lived isotopes of those elements that are radioactive.
Melting points and boiling points in parentheses are uncertain.
The data for gases are valid only when these are in their usual molecular state, such as H2, He, O2, Ne, etc. The specific heats of the gases are the
values at constant pressure.
Source: Adapted from J. Emsley, The Elements, 3rd ed., 1998, Clarendon Press, Oxford. See also www.webelements.com for the latest values and
newest elements.
*The names and symbols for elements 114 (Flerovium, Fl) and 116 (Livermorium, Lv) have been suggested but are not official.