From CHAPTER 8 of Lab Math

Reference Tables and Equations

GREEK SYMBOLS

Greek		Computer key	(United States pronunciation) And some standard uses of uppercase and lowercase symbols
A α	alpha	A a	Cutoff for statistical significance
B β	beta	B b	(1 – β) is the power of a statistical test.
Γ γ	gamma	G g	Micrograms
Δ δ	delta	D d	CHANGE IN; change in
E ɛ	epsilon	E e
Z ζ	zeta	Z z	(“zate-ah”)
H η	eta	H h	(“ate-ah”) Symbol for viscosity
Θ θ	theta	Q q	(“thate-ah”)
I ι	iota	I i
K κ	kappa	K k
Λ λ	lambda	L l	Microliters Symbol for wavelength
M μ	mu	M m	(“myoo”) Micro-Symbol for population mean
N ν	nu	N n	Symbol for frequency
Ξ ξ	xi	X x	(“zi” or “k-sigh”)
O ο	omicron	O o
Π π	pi	P p	3.1416; the circumference of a circle divided by its diameter
P ρ	rho	R r	Symbol for density
Σ σ	sigma	S s	SUM; symbol for population standard deviation
T τ	tau	T t	Time constant
Y υ	upsilon	U u
Φ φ	phi	F f	(“fee”)
X χ	chi	C c	(“ki”)
Ψ ψ	psi	Y y	(“sigh”)
Ω ω	omega	W w	SYMBOL FOR OHMS; symbol for angular frequency

PREFIXES FOR UNITS

Prefix	Abbreviation	Multiplier
yotta-	Y	10²⁴
zetta-	Z	10²¹
exa-	E	10¹⁸
peta-	P	10¹⁵
tera-	T	10¹²
giga-	G	10⁹
mega-	M	10⁶
kilo-	k	10³
hecto-	h	10²
deca-	da	10¹
deci-	d	10⁻¹
centi-	c	10⁻²
milli-	m	10⁻³
micro-	μ	10⁻⁶
nano-	n	10⁻⁹
pico-	p	10⁻¹²
femto-	f	10⁻¹⁵
atto-	a	10⁻¹⁸
zepto-	z	10⁻²¹
yocto-	y	10⁻²⁴

PREFIXES FOR NOMENCLATURE

Prefix	Multiplier
hemi-	1/2
mono-	1
di-, bi-, bis-	2
tri-, tris-	3
tetra-	4
penta-	5
hexa-	6
hepta-	7
octa-	8
nona-	9
deca-	10

SYSTÈME INTERNATIONALE UNITS: SI UNITS

Property	Symbol for property	Dimension	SI unit	Symbol for unit	Symbol in base units
Length	l, d	L	meter	m	m
Mass	m	M	kilogram	kg	kg
Time	t, τ	T	second	s	s
Electric current	I	A	ampere	A	A
Temperature	T	Θ	kelvin	K	K
Amount of substance	n	N	mole	mol	mol
Luminous intensity	I_V	J	candela	cd	cd
Plane angle	α	–	radian	rad	m m⁻¹
Solid angle	Ω	–	steradian	sr	m² m⁻²
Area	A	L²	meters squared	m²	m²
Volume	V	L³	meters cubed	m³	m³
Volume		L³	liter	l or L	10⁻³ m³
Frequency	f	T ⁻¹	hertz	Hz	s⁻¹
Radioactivity		T ⁻¹	becquerel	Bq	s⁻¹
Rate, speed, velocity	U, ν	L T ⁻¹	meters/second	m s⁻¹	m s⁻¹
Angular velocity	ω	T ⁻¹	radians/second	rad s⁻¹	m m⁻¹ s⁻¹
Acceleration	a	L T ⁻²	meters/second squared	m s⁻²	m s⁻²
Molarity	M	N L⁻³	mol/liter	M	mol 10³ m⁻³
Density	ρ	M L⁻³	kilograms/meter cubed	kg m⁻³	kg m⁻³
Concentration	c	M L⁻³	kilograms/liter	kg L⁻¹	kg 10³ m⁻³
Force, weight	F, w	M L T⁻²	newton	N	kg m s⁻²
Pressure	p	M L⁻¹ T ⁻²	pascal	Pa; N m⁻²	kg m⁻¹ s⁻²
Energy, work	E, W	M L² T ⁻²	joule	J; N m	kg m² s⁻²
Power	P	M L² T ⁻³	watt	W; J s⁻¹	kg m² s⁻³
Electrical charge	Q	T A	coulomb	C	s A
Electrical potential	V	M L² T ⁻³ A⁻¹	volt	V; W A⁻¹	kg m² s⁻³ A⁻¹
Electrical resistance	R, Ω	M L² T ⁻³ A⁻²	ohm	Ω; V A⁻¹	kg m² s⁻³ A⁻²
Electrical capacitance	C	M⁻¹ L⁻² T ⁴ A²	farad	F; C V⁻¹	kg⁻¹ m⁻² s⁴ A²
Electrical field strength		M L T ⁻³ A⁻¹	volts/meter	V m⁻¹	kg m s⁻³ A⁻¹
Inductance		M L² T ⁻² A⁻²	henry	H; Wb A⁻¹	kg m² s⁻² A⁻²
Conductance		M⁻¹ L⁻² T³ A²	siemen	S; A V⁻¹	kg⁻¹ m⁻² s³ A²
Flux density		MT ⁻² A⁻¹	tesla	T; Wb m⁻²	kg s⁻² A⁻¹
Magnetic flux		M L² T ⁻² A⁻¹	weber	Wb, V s	kg s⁻² A⁻¹ m²
Luminous flux		J	lumen	lm; cd sr	cd
Illuminance		L⁻² J	lux	lx; lm m⁻²	m⁻² cd
Luminance		L⁻² J	candelas/meter squared	cd m⁻²	m⁻² cd
Heat capacity/entropy	S	M L² T ⁻² Θ⁻¹	joules/kelvin	J K⁻¹	kg m² s⁻² K⁻¹
Specific entropy		L² T ⁻² Θ⁻¹	joules/kilogram kelvin	J kg⁻¹ K⁻¹	m² s⁻² K⁻¹
Thermal conductivity		M L T ⁻³ Θ⁻¹	watts/meter kelvin	W kg⁻¹ K⁻¹	kg m s⁻³ K⁻¹

The seven fundamental units (SI base units) are assumed to be mutually independent; that is, none of the base units can be constructed by arranging any of the others. The fundamental units are shown in boldface letters.

OTHER UNITS

Property	Symbol for property	Dimensions	Other unit	Symbol for unit	To convert to SI
Plane angle	θ, α		degree	°	rad = degrees ÷ 57.3
Mass	m	M	dalton	D	kg = D ÷ (6.022142 × 10²⁶)
Temperature	T	Θ	degree Celsius	°C	K = °C + 273.15
Temperature	T	Θ	degree Fahrenheit	°F	K = (5/9)°F + 255.37
Area	A	L²	hectare	ha	m² = 0.0001 ha
Volume	V	L³	cubic cm	cc	mL = cc
Force	F	M L T ⁻²	dyne	dyn	N = 105 dyne
Pressure	p	M L⁻¹ T ⁻²	atmosphere	atm	Pa = 9.86926 × 10⁻⁶ atm
Energy, work	E, W	M L² T ⁻²	calorie	cal	J = 0.239 cal
Energy, work	E, W	M L² T ⁻²	erg	erg	J = 107 erg

Constants

Constant	Symbol	Value
Angstrom	Å	10⁻¹⁰ m
Atomic mass unit	u	1.660538921(73) × 10⁻²⁷ kg
Avogadro constant	N_A, L	6.02214129(27) × 10²³ mol⁻¹
Avogadro’s number	A	6.02214129(27) × 10²³
Base of natural log (ln)	e	2.718281828459
Boltzmann constant	k	1.3806488(13) × 10⁻²³ J K^–1
Complex numbers	i	i = ; i² = −1
Electron volt	eV	1.602176565(35) × 10⁻¹⁹ J
Elementary charge	e	1.602176565(35) × 10⁻¹⁹ C
Faraday constant	F	9.64853365(21) × 10⁴ C mol⁻¹
Gravitational acceleration (standard)	g_n	9.80665 m s⁻²
Gravitational constant	G	6.67384(80) × 10⁻¹¹ N m² kg⁻²
Mass of a neutron	m_n	1.674927351(74) × 10⁻²⁷ kg
Mass of a proton	m_p	1.67262777(74) × 10⁻²⁷ kg
Mass of an electron	m_e	9.10938191(40) × 10⁻³¹ kg
Molar volume of ideal gas at STP	V_m	22.413968(20) × 10⁻³ m³ mol⁻¹
Mole	mol	6.02214129(27) × 10²³
Permeability of a vacuum (magnetic constant)	μ_o	4π × 10⁻⁷ N A⁻²
Permitivity constant (electric constant)	ɛ_o	8.854187817 × 10⁻¹² F m⁻¹
Pi	π	3.14159265358979323846264338327950
Planck constant	h	6.62606957(29) × 10⁻³⁴ J s
Planck constant over 2π (h bar)		1.054571726(47) × 10⁻³⁴ J s
Rydberg constant	R	10,973,731.568539(55) m⁻¹
Speed of light in a vacuum	c	2.99792458 × 10⁸ m s⁻¹
Speed of sound (dry air, 0°K)	C	331.45 m s⁻¹
(water, 20°K)		1470 m s⁻¹
Standard temperature and pressure	STP	273.15° K and 101.325 kPa
Stefan–Boltzmann constant	σ	5.670373(21) × 10⁻⁸ W m⁻² K⁻⁴
Universal or molar gas constant	R	8.3144621(75) J mol⁻¹ K⁻¹

See physics.nist.gov/cuu/Constants/index.html.

The number in parentheses after the mantissa is the standard certainty in the last digits of the mantissa. For example, Avogadro’s constant is written 6.02214129(27) × 10²³ mol⁻¹, which means 6.02214129 × 10²³ ± 0.00000027 × 10²³ mol⁻¹. If no uncertainty is indicated, the number is exact.

USEFUL EQUATIONS FROM GEOMETRY, ALGEBRA, AND TRIGONOMETRY

Line

ALGEBRA

Arithmetic with fractions

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To find the distance d between two points (x₁, y₁), and (x₂, y₂), use the Pythagorean theorem.

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To find the roots of a polynomial of the form y = ax² + bx + c, use the Quadratic equation y = 0, where

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The term b² − 4ac is called the discriminant; it can tell you how many solutions there will be.

If b² − 4ac > 0, there are two real solutions.

If b² − 4ac = 0, there is one real solution.

If b² − 4ac < 0, there are two complex (imaginary) solutions.

Logarithms

log_kx = m such that k^m = x. Usually, k = 10.

Laws

log ab = log a + log b,

log a/b = log a – log b,

log a/b = –log b/a,

log aⁿ = n log a,

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ln x (the natural log of x) = m such that e^m = x,

ln e = 1.

GEOMETRY

Triangle

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Perimeter of a triangle = a + b + c.

Sum of the internal angles = 180° = π radians.

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Sides of a right triangle = c² = a² + b² (the Pythagorean theorem).

Parallelogram

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Perimeter of a parallelogram = 2a + 2b.

Area of a parallelogram = b × h = ab sin α.

Volume of a box = b × h × d.

Regular n-gon

This is an n-sided polygon with equal sides of length b and equal angles of 360°/n or 2π/n radians. See the following example:

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Perimeter = nb.

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Circle

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Circumferene of a circle = 2πr.

Area of a circle = πr².

Surface area of a sphere = 4πr².

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Ellipse

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r₁ + r₂ = a constant.

Area of an ellipse = πab.

An ellipsoid is an ellipse rotated around the short axis (b).

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Right circular cylinder

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Surface area of a cylinder (not including the flat ends) = 2πrh.

Volume of a cylinder = πr2h.

Half cone

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Surface area of a cone (not including the flat end) = πrs.

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Torus

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Surface area = π²(b² – a²).

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TRIGONOMETRY

For a right triangle

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Function		Abbreviation	Definition in terms of sin and cos
Sine α	sin
Cosine α	cos
Tangent α	tan
Cotangent α	ctn
Secant α	sec
Cosecant α	csc

A mnemonic device for remembering the first three lines of this table is SOH CAH TOA: Sine is Opposite over Hypotenuse; Cosine is Adjacent over Hypotenuse; Tangent is Opposite over Adjacent. Some trigonometric equations follow.

One angle, α

sin² α + cos² α = 1,

sin(–α) = –(sin α),

cos(–α) = cos α,

sin(180° – α) = sin α,

cos(180° – α) = –cos α,

sin(90° – α) = cos α,

cos(90° – α) = sin α.

Two angles, α and β

sin(α + β) = sin α cos β + cos α sin β,

cos(α + β) = cos α cos β = sin α sin β.

THE ELECTROMAGNETIC SPECTRUM

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HALF-LIVES OF RADIONUCLIDES

Radionuclide	Half-life	Uncertainty	Units
³H	4500	±8	days
¹⁴C	5730		years
¹⁸F	1.82951	±0.00034	hours
²²Na	950.97	±0.15	days
²⁴Na	14.9512	±0.0032	hours
³²P	14.262		days
³³P	25.34		days
³⁵S	87.51		days
⁴⁴Ti	22,154	±456	days
⁴⁵Ca	162.61		days
⁴⁶Sc	83.831	±0.066	days
⁵¹Cr	27.7010	±0.0012	days
⁵⁴Mn	312.028	±0.034	days
⁵⁷Co	272.11	±0.26	days
⁵⁸Co	70.77	±0.11	days
⁵⁹Fe	44.5074	±0.0072	days
⁶⁰Co	1925.12	±0.46	days
⁶²Cu	9.67	±0.03	min
⁶⁵Zn	244.164	±0.099	days
⁶⁷Ga	3.26154	±0.00054	days
⁷⁵Se	119.809	±0.066	days
⁸⁵Kr	3934.4	±1.4	days
⁸⁵Sr	64.8530	±0.0081	days
⁸⁸Y	106.626	±0.044	days
⁹⁹Mo	65.9239	±0.0058	hours
^99mTc	6.00718	±0.00087	hours
¹⁰⁹Cd	463.26	±0.63	days
^110mAg	249.950	±0.024	days
¹¹¹In	2.80477	±0.00053	days
¹¹³Sn	115.079	±0.080	days
^117mSn	14.00	±0.05	days
¹²³I	13.2235	±0.0019	hours
¹²⁵I	59.41	±0.13	days
¹²⁵Sb	1007.56	±0.10	days
¹²⁷Xe	36.3446	±0.0028	days
¹³¹I	8.0197	±0.0022	days
^131mXe	11.934	±0.021	days
¹³³Ba	3853.6	±3.6	days
¹³³Xe	5.24747	±0.00045	days
¹³⁴Cs	753.88	±0.15	days
¹³⁷Cs	11015.	±20.	days
¹³⁹Ce	137.734	±0.091	days
¹⁴⁰Ba	12.7527	±0.0023	days
¹⁴⁰La	40.293	±0.012	hours
¹⁴¹Ce	32.510	±0.024	days
¹⁴⁴Ce	284.558	±0.038	days
¹⁵²Eu	4945.5	±2.3	days
¹⁵³Gd	239.472	±0.069	days
¹⁵³Sm	46.2853	±0.0014	hours
¹⁵⁴Eu	3138.2	±6.1	days
¹⁵⁵Eu	1738.97	±0.49	days
¹⁶⁶Ho	26.7663	±0.0044	hours
¹⁶⁹Yb	32.0147	±0.0093	days
¹⁷⁷Lu	6.64	±0.01	days
¹⁸¹W	121.095	±0.064	days
¹⁸⁶Re	89.248	±0.069	hours
¹⁸⁸Re	17.021	±0.025	hours
¹⁹²Ir	73.810	±0.019	days
¹⁹⁵Au	186.098	±0.047	days
¹⁹⁸Au	2.69517	±0.00021	days
²⁰¹Tl	3.0456	±0.0015	days
²⁰²Tl	12.466	±0.081	days
²⁰³Hg	46.619	±0.027	days
²⁰³Pb	51.923	±0.037	hours
²⁰⁷Bi	11,523.	±19.	days
²²⁸Th	698.60	±0.36	days

For radiation safety information, see http://www.epa.gov/rpdweb00; http://www.radiationanswers.org.

Source: http://www.nist.gov/pml/data/halflife-html.cfm.

FUNCTIONAL CHEMICAL GROUPS

Name	Formula	Comment
Acetyl	CH₃O
Aldehyde	RCHO	Functional group is carbonyl; R is an H, alkyl, or aryl group
Alkane	C_nH_2n+2	Also known as aliphatic hydrocarbons
Alkene	C_nH_2n	Compounds with C = C functional groups
Alkyl	C_nH_2n+1	A group derived from an alkane minus one H
Alkyne	C_nH_2n−2	Compounds with C–C functional groups
Amino	–NH₂
Aryl		Any group containing one or more fused benzene rings, less one H
Benzene	C₆H₆	Cyclic, with delocalized electrons
Bromo	–Br	Halogen
Carbonyl	–C = O	Functional group of aldehydes and ketones
Carboxyl	–COOH	Acids containing carboxyls are called carboxylic acids.
Chloro	–Cl	Halogen
Ester	–COOR	R is an H, alkyl, or aryl group
Ethanol	CH₃–CH₂OH	Produced by fermentation
Ethyl	–CH₂–CH₃	Alkyl
Fluoro	–F	Halogen
Hydroxyl	–OH	When part of a C-containing molecule, this defines an alcohol.
Iodo	–I	Halogen
Ketone	RR′CO	Functional group is carbonyl; R and R′ are alkyl and/or aryl groups
Methanol	CH₃OH
Methyl	–CH₃	Alkyl
n-Butyl	–CH₂–CH₂–CH₂–CH₃	Alkyl
n-Propyl	–CH₂–CH₂–CH₃	Alkyl
Nitro	–NO₂

GENETIC CODE

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AMINO ACIDS

Amino acid			Molecular formula	MW
Alanine	Ala	A	CH₃–CH(NH₂)–COOH	89.09
Arginine	Arg	R	HN = C(NH₂)–NH–(CH₂)3–CH(NH₂)–COOH	174.20
Asparagine	Asn	N	H₂N–CO–CH₂–CH(NH₂)–COOH	132.12
Aspartic acid	Asp	D	HOOC–CH₂–CH(NH₂)–COOH	133.10
Cysteine	Cys	C	HS–CH₂–CH(NH₂)–COOH	121.15
Glutamine	Gln	Q	H₂N–CO–(CH₂)₂–CH(NH₂)–COOH	146.15
Glutamic acid	Glu	E	HOOC–(CH₂)₂–CH(NH₂)–COOH	147.13
Glycine	Gly	G	NH₂–CH₂–COOH	75.07
Histidine	His	H	NH–CH = N–CH = C–CH₂–CH(NH₂)–COOH	155.16
Isoleucine	Ile	I	CH₃–CH₂–CH(CH₃)–CH(NH₂)–COOH	131.17
Leucine	Leu	L	(CH₃)₂–CH–CH₂–CH(NH₂)–COOH	131.17
Lysine	Lys	K	H₂N–(CH₂)₄–CH(NH₂)–COOH	146.19
Methionine	Met	M	CH₃–S–(CH₂)₂–CH(NH₂)–COOH	149.21
Phenylalanine	Phe	F	Ph–CH₂–CH(NH₂)–COOH	165.19
Proline	Pro	P	NH–(CH₂)₃–CH–COOH	115.13
Serine	Ser	S	HO–CH₂–CH(NH₂)–COOH	105.09
Threonine	Thr	T	CH₃–CH(OH)–CH(NH₂)–COOH	119.12
Tryptophan	Trp	W	Ph–NH–CH = C–CH₂–CH(NH₂)–COOH	204.23
Tyrosine	Tyr	Y	HO–p-Ph–CH₂–CH(NH₂)–COOH	181.19
Valine	Val	V	(CH₃)₂–CH–CH(NH₂)–COOH	117.15

See www.chemie.fu-berlin.de/chemistry/bio/amino-acids_en.html.

NEUTRAL RESIDUES OF AMINO ACIDS (AMINO ACIDS IN PROTEINS)

Amino acid		Molecular weight of neutral residue (g/mol)	Properties of side chain	Genetic code
Alanine	A	71.1	Nonpolar	GCU, GCC, GCA, GCG
Cysteine	C	103.1	Nonpolar	UGU, UGC
Aspartic acid	D	115.1	Acidic	GAU, GAC
Glutamic acid	E	129.1	Acidic	GAA, GAG
Phenylalanine	F	147.2	Nonpolar	UUU, UUC
Glycine	G	57.0	Nonpolar	GGU, GGC, GGA, GGG
Histidine	H	137.1	Basic	CAU, CAC
Isoleucine	I	113.2	Nonpolar	AUU, AUC, AUA
Lysine	K	128.2	Basic	AAA, AAG
Leucine	L	113.2	Nonpolar	UUA, UUG, CUU, CUC, CUA, CUG
Methionine	M	131.2	Nonpolar	AUG
Asparagine	N	114.1	Polar, uncharged	AAU, AAC
Proline	P	97.1	Nonpolar	CCU, CCC, CCA, CCG
Glutamine	Q	128.1	Polar, uncharged	CAA, CAG
Arginine	R	156.2	Basic	CGU, CGC, CGA, CGG, AGA, AGG
Serine	S	87.1	Polar, uncharged	UCU, UCC, UCA, UCG, AGU, AGC
Threonine	T	101.1	Polar, uncharged	ACU, ACC, ACA, ACG
Valine	V	99.1	Nonpolar	GUU, GUC, GUA, GUG
Tryptophan	W	186.2	Nonpolar	UGG
Tyrosine	Y	163.2	Polar, uncharged	UAU, UAC

THE MITOCHONDRIAL GENETIC CODE

		Mitochondria
Codon	Nuclear	Yeast	Drosophila	Plant	Mammal
AGA	Arg	Arg	Ser	Arg	Arg
AGG	Arg	Arg	Ser	Arg	Arg
AUA	Ile	Met	Met	Ile	Met
CUA	Leu	Thr	Leu	Leu	Leu
UGA	STOP	Trp	Trp	STOP	Trp

The genetic code of mitochondria varies with species and is different from the nuclear genetic code.

Buffers

Buffer	pK_a at 25°C	pH range at 25°C
Oxalic acid (pK₁)	1.27
H₃PO4 (pK₁)	2.15
Citric acid (pK₁)	3.13
Oxalate– (pK₂)	4.27
Acetic acid	4.76
MES	6.1	5.5–6.7
NaHCO₃	6.35
Bis-Tris	6.5	5.8–7.2
ADA	6.6	6.0–7.2
PIPES	6.76	6.1–7.3
ACES	6.8	6.1–7.5
MOPSO	6.9	6.2–7.6
Imidazole	7.0	6.2–7.8
BES	7.1
MOPS	7.15	6.5–7.9
TES	7.4	6.8–8.2
HEPES	7.47	6.8–8.2
DIPSO	7.6	7.0–8.2
HEPPSO	7.8	7.1–8.5
POPSO	7.8	7.2–8.5
HEPPS (EPPS)	8.0	7.3–8.7
Tricine	8.1	7.4–8.8
Tris	8.1	7.0–9.0
Trizma	7.2–9.0	6.91–8.70
Bicine	8.26	7.6–9.0
Glycylglycine	8.4	7.5–8.9
TAPS	8.4	7.7–9.1
AMPSO	9.0	8.3–9.7
	9.25
CHES	9.3	8.6–10.0
CAPSO	9.6	8.9–10.3
AMP	9.7	9.0–10.5
Glycine	9.78
	10.33
CAPS	10.4	9.7–11.1
Piperidine	11.12
	12.38

Birth of the Blog

This blog, like the book Lab Math from which it springs (incompletely formed), will be about numbers. I will endeavor to:

1. showcase the basic and the practical, not the challenging or even the advanced;

2.. provide straightforward guidance for the unenthusiastic (“just do it exactly this way”);

3.. provide refreshers for those needing refreshment (whether they know it or not.)

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